Welcome back to the next lecture on Fundamentals of Statistical Thermodynamics. So far we have developed equations which connect internal energy with molecular partition function and entropy with molecular partition function. Both these thermodynamic quantities internal energy and entropy are very important in thermodynamics. Internal energy is the total energy of the system added up in all the forms and entropy is a measure of degree of disorder or ordered state of a system. Why both these thermodynamic quantities are important? Because work or heat which both are equivalent form of energy when these are done on the system or these are done by the system these affect the internal energy. Work is done at the cost of internal energy. Similarly entropy if there is a change in entropy that can be connected with the creation of more order or more disorder. Going back to the second law of thermodynamics where we discussed that the change in entropy of an isolated system is positive for a spontaneous process. When I said I meant in change in entropy of the system plus change in entropy of the surroundings has to be positive if the process is to be spontaneous. Therefore connecting internal energy with molecular partition function and entropy with molecular partition function is very important and it tells us how to determine these values by using spectroscopy. Today we will use the derived equations to further discuss their applications. Let us take up the first example in one of the previous lectures we talked about heat capacity. So again revisiting heat capacity is a very important thermodynamic quantity because it connects one thermodynamic signature or one thermodynamic quantity at one temperature to the same thermodynamic quantity at another temperature. So therefore it is a connector for the thermodynamic quantities at different temperature. Not only that a literal definition of heat capacity is the amount of heat required to change the temperature of a system by one degree one Kelvin. That means it also directly talks about the strength of the system. If we impose constant volume constraints then the definition that we need to use is C v is equal to del u by del t at constant volume. And in the previous one of the previous lectures we have shown that u is equal to u 0 plus 3 n by 2 beta. We also then derived this expression in terms of Boltzmann constant temperature etc. And then we equated it equal to 3 by 2 nRT. This is for the translational motion of a particle of a molecule or of a molecule. I am talking about a monatomic type of gas in three dimensions. Substituting the numbers we got C v equal to 12.5 joules per Kelvin per mole. Let us highlight some of the comments. This value agrees almost exactly with the experimental data on monatomic gases at normal pressures. This is theoretical prediction. Second is experimental data. And we have earlier talked about how to determine heat capacity experimentally. We need calorimeters. Under constant volume conditions bomb calorimeter. Under constant pressure conditions the other type of calorimeters which can work under constant pressure conditions. Second point that we need to be careful in this case about is that the value that we got 12.5 joules per Kelvin per mole. It applies only to monatomic gas, monatomic perfect gas. And for more complex molecules means for diatomic molecule for triatomic molecule. There are other modes of motion which we need to consider. Monatomic only translational degree of freedom is there. Electronic can also be there but that can be discussed later. But when you go to diatomic and we go to triatomic and more complex molecules then not only translational degree of freedom is there. There is rotational degree of freedom. There is vibrational degree of freedom in addition to electronic contributions. So, this is a very simple example. Now let us go to some other type of questions. The question is to show that heat capacity of n two level systems that is C v is given by the following expression. C v is equal to n k into beta e square exponential beta e divided by 1 plus exponential beta e square. And we know beta is equal to 1 by k t. So, that means the expression that we are going to derive can allow us to calculate the value of C v as a function of temperature and from the knowledge of energy separation. The given system is a two level system. And let us say separation between the ground state and the upper or the excited state is e. So, that is why C v basically by definition C v is equal to del u del t at constant volume. Now, either we can differentiate with respect to temperature and result can also be expressed in terms of beta, beta and temperature anyway are connected to each other. And sometimes instead of differentiating with respect to temperature it becomes easier to differentiate with respect to beta. So, you see d by d t I can write d by d t as d beta by d t into d by d beta mathematically that is allowed. And since beta is equal to 1 over k t therefore d beta over d t is going to be minus 1 over k t square. Therefore, in place of d beta by d t I can write minus 1 over k t square and then I can write differentiation with respect to beta. Why this kind of transformations? Why this kind of derivatives? Because sometimes it becomes easier if we work in terms of d by d beta. You can work in terms of d by d t and then convert into beta choice is yours. But as I said that sometimes if we choose an appropriate derivative it is easier to get results. Now, let us come back to the problem that we need to solve. We have this two level ground state and upper state. The first step will be to write partition function which will be 1 plus exponential minus beta. For ground state 1 for upper state the contribution is exponential minus beta. C v is a derivative of internal energy. So, therefore first of all let us talk about internal energy. u minus u 0 is equal to minus n by q del q del beta at constant volume q is 1 plus exponential minus beta i. So, therefore let us now work on this minus n by q minus n by q q is 1 plus exponential minus beta i and into derivative of q with respect to beta which is exponential minus beta e and into it is going to be minus e. So, what I have now is u minus u 0 is equal to minus minus is plus. So, it is n e exponential minus beta e divided by 1 plus exponential minus beta e. I can further simplify it I can multiply numerator and denominator and numerator with exponential plus beta e exponential plus beta e why because if I multiply by exponential plus beta e this upper term becomes 1. So, I have n e into 1 that I am not writing divided by exponential beta e plus 1 just to make the things little more simpler. We have now the expression for internal energy and if we take it is derivative with respect to temperature or with respect to beta then we will get the heat capacity. What we have is now u minus u 0 is equal to n e minus u 0. So, this is n e over exponential beta e plus 1 that is what we have c v is equal to del u del t at constant volume and we just showed that it is equal to minus 1 over k t square into I can write del u del beta at constant volume. We just showed that now let us act upon this. So, what we have c v is equal to minus 1 over k t square into let us take the derivative of internal energy with respect to beta n e is anyway there constant into minus 1 over exponential beta e plus 1 square into exponential beta e into e to the power of e. Let us further simplify it. So, c v is equal to what we have minus minus becomes plus. So, I have n then I have e square I have exponential beta e over k t square into exponential beta e over k t square into exponential beta e plus 1 whole square. I have combined the terms and I have n into e square into exponential beta e divided by k t square exponential beta e plus 1. We can now further simplify it by noting that beta is equal to 1 over k t. Now, I can write this as beta over t is equal to 1 over k t square. So, I want to simplify my result I have c v is equal to n instead of 1 by k t square let me write beta by t into e square exponential beta e divided by exponential beta e plus 1 square. Now, remember this is the derivative that I can write another term from here 1 over k t 1 over k t is equal to beta which means 1 over t is equal to k times beta. I can use it over here 1 over t is equal to k times beta that means what I have now is once I substitute over here c v is equal to 1 over t is k times beta. So, I have n k and another beta term will comes beta e square let me combine all this beta e square into exponential beta e by exponential beta e by exponential beta e plus 1 square. So, I have this result c v is equal to n k into beta e whole square exponential beta e divided by exponential beta e plus 1 this is what we were asked to show remember that beta is equal to 1 over k t. Now, we can briefly talk about the effect of temperature on this result we will use this expression that we derived and let us see what happens when temperature approach is 0 when temperature approach is 0 then beta is equal to 1 over k t that means beta will approach a value of infinity and you see you have infinity in the denominator also. So, c v will approach a value of 0 you have infinity in the denominator and when t approaches infinity then beta is 1 over k t that means beta approaches a value of 0 again you see here you have 0 then also that means c v will approach a very small value and if you work out actually take a maxima also it will show a maxima. So, in general what I want to say here is that depending upon the system given to you for example, here we were talking about n 2 level systems and we discussed that each level is a non degenerate and therefore, we draw an expression for the molecular partition function for that and then we developed further equations alright. This is showing that the heat capacity of n 2 level systems is given by some kind of expression which permits us to calculates c v as a function of temperature from the knowledge of energy levels or energy separation I hope this derivation is clear. Now, let us go to another type of question we have a similar system now n 2 level systems evaluate the molar entropy of n 2 level systems and then plot the resulting expression what is the entropy when 2 states are equally thermally accessible in the previous question we talked about internal energy and then we talked about connection of internal energy with partition function and eventually an expression for the heat capacity. Now we will be talking about entropy and entropy is connected to molecular partition function through this equation s is equal to u minus u naught by t plus n k log q and remember that u minus u naught is also connected to molecular partition function u minus u naught for the same system which is given to us n 2 level systems we have just discussed that u minus u naught is equal to we derived the expression which is n times e over n minus exponential beta e plus 1 this is the expression that we derived in the previous lecture let I can take you back to the previous lecture and this is where look at the this expression that u minus u naught we have derived this expression is equal to n times e divided by exponential beta e plus 1 let us make use of this now. So, that means u minus u 0 by t will be equal to what which will be n e over t times exponential beta e plus 1 and since we decided that instead of t let us talk in terms of beta that means beta is equal to 1 over k t or we use 1 over t is equal to k times beta we will use that that means u minus u 0 by t that means u minus u 0 by t is equal to 1 over t is k beta. So, n k k beta I am writing instead of 1 over t e over exponential beta e plus 1 this is u minus u 0 by t now we can write the expression for entropy s is equal to u minus u 0 by t is n k beta e divided by exponential beta e plus 1 plus n k log q plus n k log q q is what it is a tool I am going to write the expression for entropy s is equal to n k into beta e divided by exponential beta e plus 1 that is u minus u 0 by t plus n k log q n k log q n k n k is common. So, therefore, I can take it on the left hand side and then express in terms of s by n k once we do that what we have is the following expression s by n k is equal to beta e over 1 plus exponential beta e plus log 1 plus exponential minus beta e that is what I was saying that this n k and n k is common and you can take it on the other side and eventually the expression that you get is s upon n k is equal to beta e over 1 plus exponential beta e plus log into 1 plus exponential minus beta e we have now entropy as a function of temperature it is not a linear or very simple function you see beta 1 over t is appearing here 1 over t is appearing here 1 over t is appearing here. So, therefore, we can easily talk about in the extreme limits that is when temperature approach is 0 or when temperature approach is infinity, but remember that the expression that we have got here is for n two level systems a ground state and an excited state which is separated by energy e. We have now connected the entropy of such a system with temperature energy separation by this expression we need to further discuss the variation of entropy with temperature in this case in the limits of t approaching 0 and t approaching infinity, but that we will do in the next lecture. Thank you very much.


Welcome back. We have been considering the effect of chemical reaction on the mass transfer rate within the framework of the surface renewal theories and in the last lecture, we set up the equations in the framework of these theories that describe the instantaneous reaction regime. So, you recall that these were the equations. So, the instantaneous reaction regime is one in which A diffuses from the interface and B diffuses from the bulk and they meet and consume each other at a reaction plane which is located somewhere within the surface element. So, this reaction plane itself starts at the gas liquid interface at the instant the liquid element arrives at the gas liquid interface and as the element ages at the interface it is as it spends more and more time at the interface the reaction plane progressively moves inward into the liquid element. So, this becomes a moving boundary problem and if you look at the equations that were set up. So, there is a region between 0 and zeta 1 of theta where zeta 1 is the location in non-dimensional terms of the reaction plane. So, the region between 0 and the reaction plane the interface and the reaction plane is occupied by A and since there is no B there cannot be a reaction there. So, it is a case of pure diffusion. The region between the reaction plane and the rest of the element is occupied by B and once again since there is no A there there cannot be a reaction and this is the equation that governs the pure diffusion of B in that region. At the reaction plane itself because the reaction is assumed to be instantaneous the concentration of A and B both go to 0 and the rate of supply of B from the in the negative x direction if you like is in stoichiometric requirement as compared to the rate of supply of A in the positive x direction. So, that is why there is a sign difference here. So, now this is the formulation of the problem and in order to solve this we require a little more information and that is because this is a moving boundary problem we need to track the boundary movement. Now, this can be done in various ways and birds toward light foot in their in the first edition of their book transport phenomena suggest a very elegant way of doing this and we will simply state that before proceeding further. So, this approach consists in recognizing that at the reaction plane the following identity holds we have the concentration of A which is a function of zeta and theta and since we are considering the reaction plane it is zeta 1 which is itself a function of theta is identically equal to 0. So, A at the reaction plane is a function of the single variable theta and that is a constant value and it is 0.  So, this can be rearranged to get the rated which zeta 1 moves with theta and that equation can be used along with the previously shown equations in order to arrive at a solution to these equations. So, that solution is analytically possible and it is the details of the solution are available in the books that I mentioned in the previous lecture the book by Dan Quartz that is Gas Liquid Reactions and the book on Transport Phenomena by Burt Stewart and Lightfoot. We will not go into the details of the solution except to mention that the solutions you know looking at the nature of the equations the solutions can be expected to be of the error function type and. So, we formulate the error function solutions in terms of the variable x divided by or zeta divided by 2 root theta in non-dimensional terms and the constants are evaluated using the conditions that I have laid out earlier. So, now it turns out when you solve these equations that the rate of the instantaneous rate of absorption or the instantaneous flux which is given by negative of the diffusivity multiplied by the concentration gradient at the interface. So, this flux has the same kind of dependence on time as the flux in the case of physical mass transfer. In other words if you look at the absorption rate at every point during the life of the surface element the absorption flux is greater than the physical absorption flux by a constant factor. Since this factor which enhances the instantaneous mass transfer rate is a constant ultimately when you average these instantaneous rates weighted with the I of t dt function whether you assume the Higbee function or the Danckwerts function for the I of t dt that is the plug flow or the mixed flow assumption at the interface. It does not really matter as far as the effect of reaction on the mass transfer rate is concerned or in other words in so far as the calculation of enhancement factor is concerned. So, if you can define an instantaneous enhancement factor which is the rate by which is the factor by which the instantaneous flux is higher than the physical mass transfer flux at every time during the life of the surface element. This instantaneous enhancement factor is independent of time and therefore, the average enhancement factor over the entire life of the element whether you average the enhancement factors using the Higbee function or the Danckwerts function it gives you the same result. So, that result is what I will give you next. So, that is given by an expression of this kind error function of beta divided by square root of dA where this function beta where this parameter beta comes from the transcendental function exponential of beta squared divided by dB error function complement beta divided by square root of dA equals dB rather q square root of dA upon dB e to the power beta squared by dA error function of beta divided by square root of dA. So, this is the equation that is implicit in beta which has to be solved by a process of trial and error knowing the other quantities like q and dA and dB and once you get beta you substitute this in the enhancement factor expression and this is the same expression in both the surface renewal theories that we have considered that is Higbee and Danckwerts. So, the message that we are trying to convey here is that within the framework of these theories the surface renewal theories e infinity turns out to be or the maximum enhancement factor available to your system turns out to be a function of dA by dB and the parameter q. Recall that it was a function only of q in the film theory and here it there is an additional parameter which is the ratio of diffusivities. We have said that often this ratio is not very far from unity the rate of I mean the diffusivity of A ratioed with respect to the diffusivity of B is a number something like 1 in many cases and in that in those situations the e infinity becomes a function of q alone in the ambit of surface renewal theories as it does in the case of the film theory. Moreover there is also while these expressions look very different from the expression that we obtained for the film theory it turns out that if the value of the enhancement factor is large then there is a very simple approximation that obtains to this complicated expression here and that is e infinity equals square root of dA upon dB multiplied by 1 plus q and if you recall that without this diffusivity function what remains was the enhancement factor predicted in the instantaneous reaction regime by the film theory. So, the factor that distinguishes the surface renewal theory expression from the film theory expression is just this factor square root of the diffusivity ratio. Now, the this is as I said it is an approximation and the error that you make in calculating enhancement factor by this expression is of the order of 1 over 2 e infinity. So, the larger the value of the enhancement factor is the in other words the larger the value of q is the smaller the error that you make in using this expression. So, now, we are in a we have gone through the entire gamut of regimes in the surface renewal theory framework and we are now in a position to put the entire story down in terms of a plot of enhancement factor versus square root of m or also called as Hata number. So, we have Hata number of 1 somewhere there let us say Hata number of 3 somewhere here and so, the enhancement factor remains 1. So, this is the story according to surface renewal theories and so, this is a slow reaction regime where we said that there is absolutely no difference between the surface renewal theories and the film theory and about 1 the enhancement factors start to lift off the ground. So, this is 1 and we have a region here which I will show by dotted lines which is the transition slow to fast where there is a bit of a difference between the film theory and the surface renewal theory. So, this is the film theory asymptote would be here slightly below, but the difference is not much and as the Hata number becomes larger and larger you have the fast reaction asymptote obtaining E is equal to root m and depending on the value of the instantaneous enhancement factor you have got branches coming out of this asymptote and these are increasing values of E infinity and the E infinity itself is given by normally by this expression here with beta being calculated by that expression, but for larger values of E infinity we can use this approximation. So, we have the slow reaction regime where film theory is equal to the surface renewal theory is identical with the surface renewal theory. We have the transition regime where to a good approximation you can calculate the enhancement factor by either the expression given by the film theory or the expression given by the surface renewal theory. The two differ by a few percent at most and then in the fast reaction regime once again you have got a situation where the surface renewal theory and the film theory agree completely for reasons that we have elaborated on earlier and then in the instantaneous reaction regime there is a difference if the diffusivities of a and b are very different from each other. So, there is this region what we have normally called as the transition fast to instantaneous. So, this regime so before this regime we have the fast reaction regime E is equal to root m and beyond this we have got the instantaneous reaction regime E is equal to E infinity. So, how do we calculate the enhancement factor in this region? One can of course, graphically do an interpolation between this asymptote and the appropriate asymptote here, but it has been shown that a good approximation is to use the same expression that we derived with the film theory using making that is making a pseudo first order rate assumption except that the pseudo first order rate constant as is calculated as k C b i using the value of the concentration of b at the interface rather than as k C b b which uses the concentration of b in the bulk. So, the expression that we had derived then if you recall is square root of m E infinity minus E divided by E infinity minus 1 divided by tan h of the same quantity m E infinity minus E divided by E infinity minus 1. So, this equation obtains for the transition regime fast to instantaneous. So, the only difference between the way you calculate enhancement factor in the surface renewal theory and the way you do it in film theory is that while you are doing using the same expression the value of E infinity that you use comes from the penetration theory or the surface renewal theory expression which is different as we have seen when the diffusivities are different as compared to the film theory situation. So, the value of I mean the definition of E infinity is what changes in this expression depending on what theory you use. So, we have completed the discussion of the various reaction regimes in both the film theory and the surface renewal theory and an important point that it makes the entire discussion makes is that while the film theory is supposed to be less accurate as compared to the surface renewal theory in so far as the prediction of the mass transfer coefficient on the diffusivity is more correct in the surface renewal theories when it comes to predicting the effect of reaction on the mass transfer rate in the slow reaction regime in the transition slow to fast to a good approximation and certainly in the fast reaction regime that is in the entire pseudo first order situation. The film theory is nearly as good as the surface renewal theories for the purposes of predicting the enhancement factor. So, this is convenient because in more complex cases it is always simpler to use the film theory because there are ordinary differential equations and finite fields and so on as compared to the surface renewal theories. The only places where there is a difference is where the concentration of B starts to make a difference to the overall absorption rate that is in the second order regimes. So, having come so far now let us see whether we can relax some of the assumptions that we made at the very beginning. There are two important assumptions that we made number 1 was that there was no gas phase resistance and number 2 that the reaction is second order that is first order with respect to A and first order with respect to B. So, taking the first taking the second assumption first that is let us look at the effect of different reaction orders. So, we will see the effect of relaxing the assumptions. So, the first assumption is that the assumption of reaction order. So, we have assumed that A plus nu B going to C C is the reaction and we have assumed that the rate the intrinsic rate of this expression of this reaction is given by K C A C B that is the case for which we have developed the theory. So, far supposing the reaction is mth order in A and nth order in B and the rate constant is appropriately designated as K m n what difference does this make and how good are the theories that we have developed so far for the second order reaction in cases such as this. So, it turns out that all that needs to be done is to redefine the Herta number in the following manner. So, it is the physical mass transfer coefficient in the denominator and 2 divided by m plus 1 diffusivity of A the rate constant C A star raise to the power m minus 1 C B bulk raise to the power n. So, provided you calculate your root m in that manner this root m can be plugged into all expressions of the enhancement factor that we have derived and you get reasonable results. So, this is a very good way of extending the theory to reactions of other than second order. The second assumption that we want to examine is the assumption of negligible gas phase resistance and this turns out to be quite simple if the gas phase resistance is not negligible all that it means is that the value of C A star that we have been using in the theoretical expressions has now to be calculated as being in equilibrium with the interfacial partial pressure of the gas which in this case would be different from the bulk partial pressure of the gas. So, we do that by we calculate the interfacial partial pressure by equating the flux of the gaseous solute from the gas side to the flux on the liquid side. In other words we use the usual balance for two phase mass transfer if P stands for the partial pressure of A and P A B is the partial pressure of A in the bulk of the gas that is in the interior of the bubble if you like and P A I is the value of the partial pressure at the gas liquid interface then this would be equal to the rate at which the gas is taken away and that is K L into C A I minus C A B where C A is the concentration C A I is the concentration of A in the liquid at the gas liquid interface minus C A bulk is the concentration of A in the bulk. And usually as we have seen if the enhancement factor needs to be considered in if E is greater than 1 it usually implies that C A B is close to 0. So, either you have a situation where the rate is given by K L into C A I minus C A B or you have a situation where it is given by K L into A E into C A I C A B being equal to 0. And the C A I and P A I are in equilibrium with each other being the concentrations the partial pressure and the concentration on the two sides of the interface or we can write C A I is the C A star corresponding to P A I and this could be given by Henry's law or whatever a suitable thermodynamic expression. Therefore, the essential theory remains the same except that wherever we have had C A star being calculated as being in equilibrium with the bulk partial pressure of the gas we replace that by the quantity that is calculated as being in equilibrium with the partial pressure at the interface which itself is given by this expression here. So, that is all that needs to be done in order to take gas phase resistance into account. In order to see how these things work let us take an example and so we look at this example here for the instantaneous reaction regime and the transition to the instantaneous reaction. So, the example is about calculation of the maximum enhancement factor and the actual enhancement factor. So, let me read the problem carbon dioxide is being absorbed from a gas into a solution of sodium hydroxide at 20 degree centigrade in a packed tower. At a certain point in the tower the partial pressure of carbon dioxide is 1 bar and the concentration of sodium hydroxide is 0.5 kilo moles per meter cubed. Other data are as follows the physical mass transfer coefficient is 10 to the power minus 4 meters per second interfacial area per unit volume of packed space is 100 meter inverse that is 100 meter square per meter cubed of packed space. The concentration of A at the interface is given as 0.04 kilo moles per meter cube we assume that the gas phase resistance is negligible. The second order rate constant of the reaction is 10 to the power 4 meter cube per kilo mole second and the diffusivity of A and B are this should be diffusivity of B. A and B are given find the maximum enhancement possible and the actual enhancement and find also the actual absorption rate in units of kilo moles per second per unit volume of packed space. So, the reaction stoichiometry is given here. So, 1 mole of carbon dioxide reacts with 2 moles of sodium hydroxide. So, how do we attempt this example? The maximum enhancement factor we note first of all that the diffusivities of A and B are quite different from each other and in fact the ratio of the diffusivity of B to the diffusivity of A in this case turns out to be 1.7 if you calculate this ratio from the given values of the diffusivities. Therefore, because this is significantly different from 1 we have to use the surface renewal theories in order to calculate the maximum enhancement factor and rather than use that complex expression we will first see whether E infinity can be calculated by the approximate expression that we had which is square root of D A upon D B 1 plus Q. Q itself is given by D B C B B divided by nu D A C A star and putting in the values D B upon D A is 1.7 and C B B is 0.5 kilo moles per meter cube, nu the stoichiometric factor is 2 moles of NaOH per mole of carbon dioxide multiplied by C A star that is 0.04. This value turns out to be 10.625 not a particularly large number in this case we have said that Q is often of the order of 100 or more, but carbon dioxide has a relatively larger solubility as compared to gases such as oxygen, hydrogen etcetera. So, this value is a moderate value of about 10 or 11. If we substitute this value of Q in the expression for E infinity above we calculate the value of E infinity as 8.91. So, remember that this is an approximate value since we have used this expression and if we want to estimate the error we can do this as 1 over 2 E infinity and so approximately the error is going to be of the order of 1 over 18 or thereabouts. So, that is an acceptable value of the error because often the errors in the mass transfer coefficient itself are of a larger magnitude than this. So, let us proceed further and calculate the actual enhancement factor as the problem requires us to do. So, in order to calculate the actual enhancement factor we need to first estimate the regime. So, that we know which expression to use. So, this question can be answered by calculating the value of the Hata number or root m which for a second order reaction can be calculated in the following manner. This is the standard definition for a second order reaction and if you plug in the values turns out that 1 over m is 1 over k l which is 10 to the power minus 4 meters per second all units are in S i therefore, we do not have to do any conversions 1.8 multiplied by 10 raise to minus 9 is the value of diffusivity in square meters per second k is 10 to the power 4 second order rate constant multiplied by 0.5 is the value of the sodium hydroxide concentration. So, this gives you a value of 30. Now, so comparing this with the value of q we find that this is greater than 10.625. So, we have the situation root m greater than q. Now, in order to assume instantaneous reaction we would require that root m be far greater than q while 30 is more than 10.625 is the difference large enough that we can assume instantaneous reaction. So, we are not sure of that therefore, let us not assume that e is equal to e infinity and proceed to calculate the value of e using the transition regime expression which is this here square root of m into e infinity minus e divided by e infinity minus 1 this also under the square root divided by tan h of the same quantity right. In order to simplify our calculations and in order to put in place an iterative scheme for calculating the value of enhancement factor we will calculate a first approximation as e is equal to the numerator. In other words we assume that the value of this modified Hata number if you like m times e infinity minus e divided by e infinity minus 1 is large enough that is it is larger than 3. So, that we can assume the the hyperbolic tangent of this quantity to be nearly equal to 1. So, if we do that then we can calculate the first approximation as root m divided by e infinity minus 1 into square root of e infinity minus e where I have just separated out the values that we know from the value that we do not know. So, this is square root of well square root of m we know to be 30 divided by square root of e infinity is 8.91. So, we have 7.91 here multiplied by e infinity minus e under the square root and this gives you a value of this number here turns out to be 10.67 and therefore, we have the equation e equals 10.67 times square root of e infinity minus e and e infinity we know to be 8.91. So, we can square this and obtain the value of e as from this quadratic expression 10.67 square into e infinity minus e. This is a standard quadratic equation which we can solve and this will give you a value of e as 8.30 which is not too far from 8.91 which is the value that we had estimated for the instantaneous enhancement that is the maximum value of the enhancement factor. So, what we are saying is that even for root m a factor of 3 higher than q the reaction is almost totally in the diffusion control regime that is in the instantaneous reaction regime. So, now, since we made the approximation of the denominator that is the hyperbolic tangent term in the denominator being equal to 1 we can test out that assumption by calculating the second approximation where we calculate this quantity square root of m which is 30 square root of m is 30 e infinity minus 1 e infinity minus e divided by e infinity minus 1. So, this turns out to be a value that is sufficiently large that this is greater than 3. Therefore, the tan h of this quantity here is approximately equal to 1. So, the second approximation also turns out to be 8.30. So, we have converged on a value of the enhancement factor as 8.30 we accept that value and now we are ready to calculate the absorption rate. So, we do that in the following manner absorption rate is the absorption flux that is K L C star which is the physical absorption flux multiplied by the actual enhancement factor. This is the chemical absorption flux the actual absorption flux, but we want the rate in units of moles per centimeter cubed of packed space per second. Therefore, we can multiply this by the interfacial area ASP which is the square meters of area available per unit volume of packed space. So, if we can substitute this and calculate the value I will leave that to you. One point that we should note is that in our consideration of the various regimes and calculation of rates and so on. We have come across several definitions of this quantity which we have called the interfacial area. So, in our theoretical development we used a hat which we called as the interfacial area per unit volume of liquid. In one of the earlier examples we had the case of a sparse reactor in which the term interfacial area per unit volume of dispersion was introduced. Now, remember that the volume of dispersion includes the volume of liquid and the volume of the held up gas. So, this quantity of the this definition of the interfacial area is a little different from the interfacial area a hat that we have used in the theoretical development. And in this example we have encountered the interfacial area per unit volume of packed space. So, basically it does not make a difference to your calculation of the flux at all because that is given by this expression here. Depending on what units you want the rate to be in you have to multiply this by the appropriate type of interfacial area per unit volume. So, here the rate is required in moles per unit volume of packed space per second. So, use the interfacial area as square meters per meter cubed of packed space. So, that is a matter that is fairly trivial and does not need to engage our attention any further. So, now this is the rate expression that you would use in the design of the packed bed reactor itself. In other words in any design exercise you would do a mass balance on the flowing phase for a continuous equipment such as this in which you say that if the equipment is operating at steady state you would say that in any slice of the liquid in any slice of the packed bed there is a certain amount of solute that is entering and there is a certain amount of solute that is leaving and the balance is being absorbed within that slice of the packed bed. So, the absorption rate would be the rate of absorption per unit volume of the packed space multiplied by the volume of the slice. So, this rate of absorption per unit volume of the packed space is what we have calculated here. So, this is the local rate expression that would go into any of your macroscopic balances or reactor level balances. So, that completes our discussion of the gas liquid reactions. So, let us summarize what we have seen in the past 6 or 7 lectures. So, we have seen we started out by looking at how does mass transfer occur from a gas to an agitated liquid and we considered this in terms of two possible mechanisms. One is a steady state mechanism what we called as the film theory which assumes that the entire resistance to gas liquid gas liquid mass transfer is located in a thin film of thickness delta located at the gas liquid interface. Outside of delta the liquid is in a state of continuous mixing and because of the hydrodynamic forces there and because of that the concentration is uniform in that region. So, there is a concentration variation that goes from C A star to the bulk concentration that is prevalent in the rest of the liquid and this concentration drop occurs entirely in the diffusion film. So, that is the assumption on which the film theory is built and further we said that because this film is expected to be very very small and we by later on with reference to the available values of the mass transfer coefficient we estimated the volume of liquid in this film to be about 0.1 percent or so of the total volume of the liquid. So, because of the volume contained in this film is so small we are justified in treating this film to be always in a state of steady state. In other words any changes either on the gas side in terms of the changing partial pressure or on the liquid side in terms of the changing values of C A B the film is able to immediately adjust itself on an instantaneous basis to these changes therefore, the diffusing solute always proceeds as though the conditions were studied. So, we have a very simple equation to solve the steady state diffusion equation which is a second order ordinary differential equation with the constant boundary conditions. So, this theory predicts that the mass transfer coefficient is proportional to the linear power of the diffusivity and then we considered an alternative mechanism for the mass transfer of A from the gas side to the liquid side and this mechanism assumed that the action of turbulence is not to restrict the distance over which the concentration drop occurs as assumed in the case of the film theory, but it is to actually periodically throw elements of liquid from the bulk to the gas liquid interface and depending on the nature of the hydrodynamic field there is a certain time period during which individual elements of liquid stay at the gas liquid interface and then leave. Because these time periods are expected to be short the in general the processes assumed to be of an unsteady state nature. So, here that is number 1 and the second thing that happens is that because the time of exposure is small the depth of penetration is also small and therefore, the surface element can be assumed to be infinitely thick from the point of view of the diffusing solute. So, we get to solve a partial differential equation of the second order first order in time and second order in distance in order to calculate the absorption flux in a single surface element as a function of the time it has spent at the interface. In order to calculate the average absorption rate at that location we have to consider the unit gas liquid interface at that location which itself is a mosaic of several surface elements of various surface edges. So, the absorption rate in these different elements has to be averaged in order to get the overall absorption rate and we can do this averaging by assuming two types of surface age functions or age distribution functions if you like. And these give rise to two different theoretical pictures one which was originally proposed by Higbee where he said that every element of liquid spends exactly the same amount of time at the gas liquid interface as every other element. The second one due to Danckwerts who said that the picture at the gas liquid interface is more like in a well mixed vessel where elements of liquid are arriving randomly and departing randomly from the gas liquid interface. So, irrespective of which distribution function you use it turns out that the mass transfer coefficient is predicted to be predicted to have a square root dependence on the diffusivity. And if you compare these predictions of the film theory and the surface renewal theory with the experimental data it turns out that the surface renewal theories are closer to the actual picture than the film theory. But irrespective of that we should realize that neither of these theories is able to predict the physical mass transfer rate in any real sense because each of these theories has a parameter that usually in most realistic contacting situations cannot be calculated from first principles. Therefore, the theories are in some sense useless in their ability to predict mass transfer rate is concerned. But the usefulness of the theories is in their ability to predict the effect of chemical reaction on the mass transfer rate what is called as the enhancement factor. So, this is the business that we address next and we saw that whether you are a proponent of the film theory or whether you are a proponent of the surface renewal theories the effect of chemical reaction on mass transfer depends on a value of depends on the value of a parameter called as a Hata number which is the relative rate of reaction to the rate of diffusion. So, this there is a definition of the Hata number that arises on the non-dimensionalization of the relevant diffusion reaction equations. And as the Hata number increases the reactions are the reactions being considered are of ever increasing severity with respect to the mass transfer rate or the diffusion rate. So, we have the slow reaction regime and we have the fast reaction regime and in between we have the transition from slow to fast reaction. And all through this these three regimes the slow reaction the transition and the fast reaction the assumption of pseudo first order rate holds because the value of the Hata number or the value of m that is the square of the Hata number is much less than the relative abundance factor which governs whether concentration of B is going to be uniform right up to the interface or not. So, in all of these regimes it does not really matter as to whether you use the film theory to calculate the enhancement factor or the surface renewal theories to calculate the enhancement factor. Fair enough there is a bit of a difference in the transition regime, but that difference is of the order of a few percent and if you consider the uncertainties in the values of the physical mass transfer coefficient itself. So, this error is usually subsumed in the errors with which you can calculate the rate of absorption overall. So, the point is that as long as you have got a pseudo first order situation or as long as you have a situation in which the concentration of B does not play a role in the absorption rate expressions. The actual mechanism of mass transfer turns out to be unimportant the film theory and the surface renewal theory predict the much the same kind of values for the enhancement factor. So, then for larger values of root m that is for reactions which are of even higher severity than the ones that we have considered so far. The reaction is fast enough to deplete the concentration of B close to the interface and this is where the differences between the two sets of theories starts to surface and here by and large we should go with the more realistic theory that is the surface renewal theory. Fortunately, it turns out that for most practical situations the expressions from the surface renewal theories are not very different or not very difficult rather to evaluate the enhancement factors from and therefore, we can use the surface renewal theories with without too much difficulty. So, these are the various ways in which we can calculate the enhancement factor and once you have calculated the enhancement factor the local rate of absorption at any point within the equipment is given by the local mass transfer coefficient and multiplied by the driving force multiplied by this enhancement factor. So, you had the expression K L into C A star minus C A B for the physical mass transfer rate and all you got to do now is to put in this value of the enhancement factor as a multiplicator to this expression much as we use the effectiveness factor in the case of gas solid reactions to modify the intrinsic rate expression. So, there the basic case was the case of reaction and the effectiveness factor which had a value of less than 1 multiplied this intrinsic reaction rate to give you the actual reaction rate. Here, the base case is the base of mass transfer and this has to be multiplied by an enhancement factor which has a value greater than unity in order to give the actual rate of mass transfer. So, once you know the actual rate of mass transfer this is the rate expression that goes into various equipment level balances which you use either for analysis of process equipment or for design of process equipment. So, with that we have now completed setting up the requisite apparatus for doing an analysis or design of gas liquid reaction equipment where the enhancement factor calculations have to precede the writing the expression for the local rate. So, we shall close this set of lectures here and thank you for your attention.


Today, I have the pleasure of speaking with Alan Stevens from Coral Mining. How are you today, Alan? Very well. Thank you, Tracy. Okay, Alan. Let's get right to it. You just put out your maiden resource estimate for the Merrimaca project. Did I pronounce that properly? You did. And of course, this is in Chile. Tell us more about this. Merrimaca is a new discovery that Coral made earlier this year. We've drilled only about 50 holes, and we've started with our resource estimates right away. And we expect to get it into production as quickly as possible. So I understand the Merrimaca project is, of course, close to your processing plant. Can you tell us more about this? The processing plant is called Ivan. The processing plant will produce cathode copper, which means that the final product we will produce is copper metal, as opposed to concentrate, which is the other way of producing copper. So you have a very aggressive timeline then towards production. Is that correct? It is correct. We've drilled our first holes in this deposit in March of last year, and we hope to be producing copper by sometime in 2018, 2019, which is an extremely fast. So Alan, for those of us that may be new to Coral mining, can you give us a bit of an overview about the company? Certainly. Coral was set up in 2005. The objective was to become a mining company and to be producing copper as quickly as possible. It's taken us a bit longer than we expected, but we are now a producing copper company and we're looking to grow that production over the next two or three years. Well in addition to your aggressive timeline, you actually had a very interesting comment in your about paragraph. You said you're looking for corotype deposits. Can you tell us what that means? Corotype deposits are really any deposit that we can identify at whatever stage of development in Chile that we can see a timeline to production as quickly as possible. Okay, so in addition to the most recent news on Miramaca, maybe you can give us a little bit more of an overview on this project. Miramaca is a new discovery, a very well located part of Chile. We're fast tracking it to production. It'll be a very nice project for a company of our size and it'll produce good copper at a very low price. And of course in your operations update, which I absolutely loved, it's very thorough. I like how Coral handles news release updates for shareholders. You also mentioned a company or a project called Berda. Can you tell us a little bit more about that? Berda is an existing operating mind that we put into production last year. It's a project we discovered back in 2011 and we're currently expanding the output to approximately 5,000 tons of copper per year and we expect that to reach that level of production in the middle of 2017. So Alan, copper, we have an aggressive timeline. We have a processing facility. Can you tell us what else we as shareholders might be looking forward to in 2017? I think Coral is a copper growth story. We like to characterize ourselves as being a growth company that happens to be producing copper. Our business is discovering copper deposits and taking them all the way through to production ourselves. Most companies are either exploration companies or they're mining companies. We're going to be a company that finds our own deposits and puts them into production. So Alan, thank you so much for updating us on Coral Mining. Thank you Tracy for the opportunity.


From VOA Learning English, this is the education report in special English. Tests of school children in South Africa show that they are not doing as well as they should. The government tested about 7 million children in 24,000 schools. This year, for the first time, testing included students in the ninth grade. There was a sharp drop in average scores from first grade to ninth grade in both mathematics and language. The news was not all bad, however. The country's basic education department said young students appeared to make gains compared to last year. This is especially true of third graders, but the gains seemed to have disappeared in the ninth grade. At that level, only about 2 percent scored more than 50 percent in math. A spokesman for the basic education department says although the results have been criticized, they show hopeful developments. Students from poor families scored well compared to wealthier students. And he said the study confirms that school attendance is high. Tabo Kupa serves on the Executive Committee of the Youth League of the ruling African National Congress. He said the results for the ninth grade are disappointing. He added that the test results call for officials to act. This year, South Africa's basic education department was accused of failing to provide textbooks to students in the poor northern Limpopo province. An employee was accused of dropping about 700 of the badly needed books into a river in Limpopo. To educate an estimated 12 million students, the nation spends 207 billion rand or about 23 billion dollars. Education is South Africa's biggest budget item and education spending is set to increase. For VOA Learning English, I'm Alex Villareal.


Everybody forced here ripping dual case 2022 and 2021 Spectre football hobby Random team style results are on your screen right now. There's a link in chat You can always find him a late sports cards calm links of the results quick link is the bottom of the website So I'm trying to say Let's get started we'll be eating rattlesnake eggs out here. I'm gonna open both cases right now Get all the boxes on the table That's what I want to know Few rattlesnake eggs you are from Florida Arizona, which is basically just for the west They're from the Everglades you eat nothing normal. Yeah, you're like, oh our there are Thanksgiving possum and our Here's fine It's overrated people. It's like oh so good. It's like you're just frying it dipping in honey mustard You could dip most that you can fry most things dip it in honey mustard. It's gonna taste pretty good Deep fry one of those Frickin empty cans out there and dip it in honey mustard. It'd be good. I guess we're start with 2022, right? Should we just alternate? Has anyone ever done that before? Does anyone ever done that? Are you allowed to do that? I'm gonna do it I'm gonna do it back and for 2022 and 2021. Oh, I feel like a rebel right now y'all Breaking the law breaking the law Breaking the law breaking the law Will I though want to see a Garrett Wilson rookie to 60 nice start for the Jets It's gonna be a long recap Well, I'll top of the eight in Hutchinson. I'll see as much as I can Drake London 2022 99 triple Jersey rookie for the Falcons Everyone's about to be put up. Everyone's about to be shooketh as as as somebody younger than me would say Anoradian RPA Chris Olave green ink NFC South and NFC South from Saints 33 35 So tired, dude That's a sick car. 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To 75 on the Bears aspiring RPA with a Believe that'd be a number patch Okay, let's do a box of 2020. We're just gonna go back and forth. Oh man. Oh man What's our nation? So it is like a dueling case for a school back and forth Omar Jackson silver for the Ravens the 149 Cadarius Tony number to 25 rookie Giants High voltage to eight Nia our orange patch Brandon cooks Texans and Pat's auto is JC Jackson 22 25 I Swear every time every night like 10 p.m. My face starts getting more itchy Calls card dude, we need to get I don't know what we need to get we got to get something in here Scrub it dub dub Jalen hurts to 20 for the Eagles Tom Brady 1099 for the bucks Jamar Chase rookie building blocks triple relic bucks It's a 99. Yeah, it's card dust They're worth five of five Kyle Trask psychedelic radiant rookie patch auto you go bucks Kyle Trask Last week Kyle quit the bands Andre silver the Pat's rookie there a Ron Dale Moore rise above Cardinals rookie number 11 to 25 Devontae Smith to 99 triple relic rookie for the Eagles and Steve Largent number to 10 on the Seahawks illustrious legends auto Steve Largent Eric Stokes rookie for the Packers to 75 danger us Seahawks to 99 To 99 she over O Jersey card bangles and Oda pay away rookie auto for the Ravens number 13 of 75 Back to 2022 we go Personals are full tonight. They've been full for a little while now Kenzie's live at 10 a.m. Eastern with more group breaks and Sam's live at 5 p.m. Eastern tomorrow with more group breaks as well Got F1 baseball basketball and football tomorrow Can't we'll be live with on-the-fly group breaks and personals tomorrow. He's still live right now personals also Jonathan Taylor to 15 on the orange die-cut Colts Snoop Conor rookie Jags Jerry Rice one of eight on the a lot on the Ball legend, what does that say? Something legends elite legends Think materials elite legends Like spider web like a stained glass or something neon spider Jerry Rice patch guard Niners to eight and a Sam Hal to a 25 Pink RPA commanders You got asked a question by Carl movie 75 DJ more Panthers It's to it's either Shawshank redemption or save in private Ryan Our rookie Spectress Village Jones jr. To 25 for the Bears Yeah, so they'll sign with like Like a color match Marker pretty much is probably like some neon sort of highlighters marker. Yeah, the pinks. They'll do that in spectra Traylon Berks to 50 triple relic rookie Titans Cowboys Jake Ferguson five at 10 rookie auto Orange there Big movie guy man, but those are two or Shawshank is green definitely now in the 60 chargers Carl What's your all-time favorite movie? quarter-real Patterson Skye Morton 99 chiefs triple rookie Jersey. What's everybody's favorite movie? Tom Rathman on the Super Bowl auto Niners Shawshank Mac Jones to 99 there we go Pat's Which can in silver we've got a Michael Vic to 75 max impact Jersey card Falcons blazing saddles is a good one That's classic Mel Brooks right there and Simeon rice Super Bowl auto for the Bucks Don't doc Saints Jaws and Jurassic Park are great answers to man. Those movies are so frickin Infinitely rewatchable Oh, I bet that was like the Spielberg masterclass right there Both of those movies Mine probably today's the dean the pick of destiny. I'm just kidding Sean Alexander 8 of 60 for the Seahawks even though that's a That's a good one Anthony Schwartz to 60 rookie patch card for the Browns The Goonies the Goonies is great Mac Jones to 99 on the building blocks rookie triple relic there you go Pat's Mack rookie and Mills mafia to 125 RPA Texans Hardball hardball is good hardball is a Man, I'm gonna be down move we get anybody it'll get you to 149 Vikings down and cook Rise above neon splatter rookie to two at low rams Seven of eight DJ shark 24 25 spectacle patch card for color patch. Yeah, saving pride Ryan sets the tone real quick Immediately Bob greasy numbered 18 of 30 on the illustrious legends auto for the Dolphins Careful Sean this card greasy He didn't his whole career is under the feet You're like yeah, it is the losses didn't count The 50 Joe burrow Order bangles neon die-cut Joe burrow bangles 28 of 99 Shorty Nelson to 7 or 25 Jersey car for the Packers Redemption Anthony Schwartz rookie patch auto for the Browns these exact redemption should expire in about four or five months here Yes, that's correct Harrison. I've never seen that don't plan on it She's on say Williams at 35 rookie for the Broncos Quinn Williams jets to 25 To 60 Tori Holt Jersey card rams So do you offer Sean a baby Ruth? I think you and he's It's a 35 green RPA Elijah more jets. That's really sick I love it Josh Berlin I'm Josh Berlin One of mine. It gotta be my top five is no country for old men That movie is just like one of the most tense movies I've ever seen 599 Kirk Cousins like literally not a dull moment of life without tension in that movie Vikings Kirk Cousins Brian Robinson jr. Dude twins with Danny DeVito and Otto Schwarzenegger Is I saw that for the first time ever like this past year. Oh, man. It's so good It's a 75 Josh Allen Jersey card for the bills There will be blood. Oh, that's a really good one that movie the movie itself isn't quite as good But the acting and like the writing is outrageous It's 125 rookie Jersey auto jets breeze haul Yeah, a lot of Daniel Day Lewis like you really can't do much wrong. He's just incredible Tarantino yellow Tarantino movies are great. Yeah.  That's a permanent timeout Christian McCaffrey to 50 for the Panthers Rookie Spectres Trevon Walker to 99 Jags River runs through it is good Whoa, this is a big one right here Nike swoosh gold vinyl monumental memorabilia George Pickens Steelers rookie boom Who's got the Berg the Pittsburgh the iceberg lettuce Pittsburgh where you at? I can't read right now Bruce There you go, Bruce Crazy right there Trent McDuffie to 50 rookie auto chiefs Maybe I'll get Netflix again. I haven't had her a few months, but I do want to see all quiet on the Western front That was one of my favorite books honestly in like high school in like college I read it a couple times he's a con what a 40 for the Dolphins Like the only book I've read multiple times Or more than twice I should say Eli Mitchell 1500 points and it's everybody here and Alec Pierce to 25 rookie dual patch Colts Favorite 90s movie there's a lot of the 99 Drake London rookie Heat heat is a good movie Same private Ryan Jurassic Park to Money two or two Harrison Smith a laundry tag nebula Holy crap what a box we had the two and two one casino is good. Yeah Lowers big Harrison Smith get that magged It's a 99 irradiant RPA pier strong junior Pats It's pretty solid That's definitely a super 90s movie though killer clowns from outer space is a great movie Never seen that Bloodsport. Yeah, one of my uncles like well bloodsport alive. 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It probably could be it very well could be 15 Rob Gronk Pats Fields rookie for the Bears to 60 nice card Conner. Oh my god, dude Point break is another one too. We're talking for going along those lines. Oh, dude. You're seeing point You're seeing one right? You got to see Conner and point break I think I might own them both actually in college my roommates and I were on this kick up Just like watching the cheesiest corniest movies that you could possibly think of and I'm buying Conner and point break as you I was like you can't find any words You buy the 75 Jersey card Tyler Boyd Bengals Yeah, right Friday 13 part 7 Jason returns once again after also dying four other times again RPA to to outlaw Yeah Thomas Saints to 149 crazy TJ Hawkinson numbered to 60 on the Jersey card Lions a Three of five Javante Williams Michael Carter rookie psychedelic dual-patch auto Broncos and jets random at the end We're I mean dumb and dumber is one of them. We're talking about that movie like all night We talked about all week dumb and dumber is one of the best airplane is great That's a classic though That like Set the standard for a lot of calm Saturday the 14th Not Friday the 13th, but Saturday the Friday the 13 Saturday the 14 George Kittle to 99 Niners Make it a large farva Zach Wilson to 75 Jersey card jets you want a large farva dude national treasure is to is So fun Charles Woodson auto to 10 Raiders icons auto ETS ET is fantastic ET is great Yeah, you can pretty much take any movie with a John Williams soundtrack, and it's probably great Tony Romo Chad Johnson Bengals to 40 Cortland Sutton to 15 on the patch card Broncos Cheating John up and smoke up and smoke Broncos and Super Bowl auto Harry Carson for the Giants Independence day. Yeah, that movie's okay, but it's just a very like It's an icon as an iconic blockbuster whoa dude retro Spectacle Tony Romo.  It's like he's on a Russian submarine or something Randy Moss to 99 Jersey card Pats Player one patch Pats Ryan Robbins, Jr. 75 RPA commanders and Micahsicki Dolphins to 15 Monty Python the holy grails of classic ever seen Thursday. Oh, yeah. No, I've seen Friday. I haven't seen Thursday though John Lynch bucks to 99 bucks legends rookie aura James Cook 37 to 50 bills Sunday Spectacle Jersey Jamar Chase Bengals 11 to 15 As far as musicals goes grease is good. My wife hates that movie though for some reason I have no idea why Alec Pearson sky more 12 out of 30 dual patch auto Colch chiefs random at the end get a lot of these You're like musicals and Victor I don't know man. Some of those old school ones. I kind of dig Like singing in the rain I'm watching that when I was a kid with my grandma. It's just old like Like super old timey style like even like singing in the rain was later on It's like stage on screen kind of Nico Collins Texans rookie there 35 Zack Wilson jets number to 75 Jets Elijah Ward to 99 rookie dual Jersey card. No one's said Princess Brighette. Have they That's a great one redemption bears Justin Fields Rookie patch auto neon blue Big card bears. Hopefully he signs man. I'd be sick once you get that one Miles Garrett to 149 Browns Brett far of a retrospect number to 99 Packers Teenage Mutant Ninja Turtles in the 90s, dude Like I love that movie. Yeah, but it's a objectively that movie kind of stinks man He's really bad, but it's so good. You know what I mean to 75 rookie dual rally cremander Stevenson Pats I'm not sure Felix. I don't believe so. Hopefully he does time the Wallace to 15 on the RPA Ravens Right at 13. Well, it's almost over. It's it's Saturday the 14th now for the East Coast Des Fitzpatrick rookie Titans to 75 to to out well Rams rookie to 99 Broncos Bradley chubb to 99 Jersey card max impact And Raiders Dave Casper auto to 49.  Let's do that. That's more fun That'll spark that'll spark debate What movie do people dislike that a lot of people other people really like or is like really beloved? Greg Newsom the second rookie Browns to 15 The yummy brown Washington number to 60 Jalen Darden to 99 a rookie Jersey card bucks Titanic why is that Carl? I Think that's an overrated of course because it's you know got one so much money one of these award things overrated Yeah, but what's gonna happen the ship gonna sink rookie auto does do some for the Bears number to 35 sup Riley Super prediction earlier. I said Bangles Eagles. I'm gonna stick with that for now. I It's so tough to pick All right, we're down to the last half of the break here for beach Never saw goonies and don't care to I Mean, I'm not gonna talk to the force on watch movie, but it's definitely a good, you know like young adult like adventure movie It's it's a lot of fun But if you don't want to watch it, I want you to travel to 20 for the Falcons Hangovers overrated for sure. I agree with that Not that I dislike it overrated Mike Evans Next era rookie Jersey eight hudgens and lions 97 and 99 Best harmony when they sing to the bottom George Pickens 22 30 crazy looking RPA here Two monster pickings so far in this break See I knew we would get people like on the opposite ends here Yeah, all the Fast and Furious movies are pretty much hot garbage. The first one is not hot garbage But it's not that good Vaughn Miller 1099 for the bro for the Broncos I Get why they make them They're just dumb action. I get it. That's fine Way blank Patrick mohomes a second 10 of 10 that is sick Yeah, that's a style people like it illegal. That's that's pretty much that Dak to 60 for the Cowboys Jersey card You don't like Morbius everybody loves Morbius. That's a lie. It's a fact that everybody loves it Magma Holmes actually mark Brunel 250 on the Jags sky high signatures. We go Jags Yeah, saw movies are overrated First I was good had a really good twist Everything else after that was unnecessary. It was just like those people that like to watch torture 75 Ricky Nico Collins Texans grease haul Matt Corral to 25 Ricky dual patch Panthers Yeah, John McSkin movies Jalen Tolbert to 125 rookie patch auto for the Cowboys Talk about Tarantino drives you nuts. Yeah, like his movies are in movies are really good, but Some people go bananas over those movies Kirk Cousins 54 60 Vikings. I don't know. Maybe we're sort of I'm not like a super fan of like anything. 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Oh nice numbering here six of ten Nice neon blacklight RPA Kyle Pitts Falcons There we go Falcons. Yeah, you know like that a lot Michael Parsons To 35 Cowboys rise above round Delmore cardinals to 99 Elijah Moore to 35 for a triple relic jets The neon black lights are fire Michael Vick Odyssey pulling tonight sky-high signatures to 50 Michael Vick Falcons. Oh Yeah, I know over friends for sure Yeah Also better meme better memes nowadays from Seinfeld all those Kramer memes a few months ago Jalen Darden rookie bucks to 99 Barry Sanders Lyons to 149 Justin Jefferson to 99 Sunday spectacle Jersey card Vikings and Great resort to 99 rookie auto for the bills. I don't even know if I have a favorite TV show. I don't ever ever finish like one of the Hi, I've never I've never seen the wire. I've never seen the sopranos ever seen Breaking Bad. I've never seen The whole thing of Seinfeld their friends or you know any iconic TV show. I don't think I've ever seen the whole thing of Not a show guy Brian Cranston's legit though. I'll say that now come the middle Seinfeld Breaking Bad Trailer Park Boys. Oh, man. She'll ever was a great though J.J. Watts of 40 Cardinals Or shod Bateman Ravens Malik Willis to 60 your key triple relic Titans and Aiden Hutchinson 15 of 99 RPA lions Yeah, my wife's watching y'all so and I catch it here and there that show is really good Yeah, I watched a good bit of shameless shameless a salad a Micasicki neon splatter die cut for the Dolphins a one of 10 Yeah curvy enthusiasm one of my favorites to Rookie aura for the Titans Malik Willis numbered 99 Sean Watson to 99 Browns Jersey card and Fred Warner full spectrum auto for the 49ers both Niners on that one To 149 I haven't seen the new curvy yet though. That's the one thing. I haven't seen the new curb. Is it good? Is it just Larry David going? You know, is that because I like that Ray Lewis Ravens to 75 Kevin Harris Rookie Pats Bears Willis Jones jr. 11 of 60 monumental rookie patch Danny Gray one out of two crazy nebula Radiant rookie patch auto here with a giant 49er on it Wow, nice Michael Thomas on the psychedelic three of five for the Saints More Chase silver bangles Michael Vic to 99 max impact Jersey Falcons Eagles rookie out on the Kobe Dean 57 of 60 12 15 up in here. Damn back to 2021 here We've got Tyler Boyd to 99 bangles Tyler Wallace Ravens rookie to 149 To do at well to 15 rookie dual patch Rams I hope we see Sheila one of these Dwayne bow to 60 auto for the Chiefs second Dwayne bow Xavier and Howard of the Dolphins number to 75 herbie chargers to 50 Offensive player you're probably Kenneth Walker, right? He's heavily favored 75 rookie dual Jersey Jalen waddle Dolphins And jet Zach Wilson a 75 on the RPA There you go jets DK Metcalf 4 of 30 for the Seahawks Rookie aura dolphins Jalen waddle to 25 the Vonte Smith to 75 rookie Jersey Card Eagles And lions Deandre Swift 42 of 99 sky-high signatures you go lions Jacob Harris rookie for the Rams number to 35 Titans Derek Henry is 60 Yeah, dude, they've been here just about as long as we have Yeah, the 35 rookie triple relic is Kellan mon Vikings and Chris Evans to 60 rookie auto bangles They must be jack-o-lantern. Oh, yeah Yeah Yeah, we had a wheel like a packed house our shop was literally like like shoulder-to-shoulder Like not like not many places. We had like open up part of the back of it Yeah, extra tables and everything pretty serious If you're local you need to check you I think it's like every Friday for the most part We have like a show that we are attending at a 15 actually Harris yours and even then we might still do it But we've been doing like meta two tournaments and like MTG tournaments and stuff like shop prizes and give giveaways You can you can learn just come hang out It's awesome, man to 15 Austin Eckler patch card chargers. We'll teach you. We'll teach you whatever you want to know Whoa Philist Jones jr. RPA bears to 125 Yeah, Blake you missed out dog. I Did the I did the pizza giveaway? I did the pizza giveaway and then uh, oh, we also we also had pizza for everybody You could just come and hang out eat pizza. No, honestly, that's it. You could just leave Jerome Ford rookie Browns is 75 Kind of surprises Why would we tell you because it's a surprise? Who free-knuckle sandwiches marina to 60 jersey-card dolphins a free-knuckle sandwiches from Sean. Yeah, yeah, yeah Free insults mustard and ketchup on yeah free insults to yep to 60 Kenny Pickett RPA 19 of 60 there for the Steelers Steelers have crushed it and it's all been the 2022 all the 2022 for the Steelers a Meta zoo is a zoo that is meta Bonta Adams Raiders are 25. I might even come say hey if you guys come in Maybe probably not but maybe rookie aura John met you the third Texans four of eight Nice neon splatter rookie there Derek cart to five on the psychedelic partial swoosh patch for the Raiders and Jags full spectrum auto James Robinson neon marble to four I believe Yeah, three or four. It's part of the multiverse. It's all part of the multiverse Terry McLauren to 15 commanders orange die-cut Antonio Gibson for the commanders Kyle Hamilton to 99 recc patch Ravens and Danny greater 35 on the RPA Niners That is is a trading card game Yeah, I hope we can start doing trade nights. We don't have enough room for a big trade night though We're gonna get there though. We're gonna get there. We got a gaming shop opening in like two months I haven't seen it. I haven't seen it. I'm excited. They're ready They gotta have space like tournaments like that That's how gaming that's how gaming goes. I couldn't believe it. It's a 99 Big Ben Steelers Terry McLauren commanders to 149 Jacobi Myers to 25 patch card Patriots and refrigerator Perry auto psychedelic to a five bears Raider brainer the Raiders. I don't know what's good. I don't know what Brady gonna do Go Miami He was heading that way before all this stuff went down Joe makes and bangles to 60 Got Herbert Chargers to 50 Davis Mills to 60 monumental rookie Jersey Texans. This recap is gonna take like 10 minutes Zach Wilson to 15 on the RPA. There you go jets Call duty and Madden tournaments go on. They still happy. Oh, yeah, for sure Madden has just been garbage for 10 years now. So they still go strong. Oh, they still go out. It's a strong people drop a ton of money on big time Gronk for the Bucks to 99 Vikings killing Mond rookie to 10 Finally got one of our buddies that goes crazy on Madden ultimate team who spends a lot of money every year to admit that Madden is garbage. Yeah 75 high voltage Jersey car Joe Burrow. I Saw a tweet and it said Cuz this is the like he passed away. So it's like the Madden edition like, you know, they did like it mad at 23 is dedicated to The life of John Madden. He must have really hated that dude Cliff Harris or like a love letter to him. It was the hate of that guy to 125 in the auto for the Cowboys Mafia It's a 25 for the Texans Khalil Mac Bears to 75 Ron Delmore Cardinals to 99 rookie Jersey And all of us were he auto Jalen Phillips to 15 Do the great like a big opening for Oh, yeah A party Right, yeah, including every so it's just huge yeah, juju Smith Schuster to 60 for the chiefs Best sports video game ever or just like franchise The franchise is still mad and is the best but there's garbage That was awesome. Yeah, love me some feet to 60 and 20 o gauge Jersey for the Chargers Beelish Jones jr. On the Nebula RPA Bears One of two NBA Street dude, I love the NBA Street. It was a EA NASCAR 2002 for me I was kidding. I loved it. That was fun champ Bailey to 50 for the commanders there And I felt blitz is good Frank Gore To 75 Justin Herbert Chargers. I mean Mario Kart's pretty good if we're gonna count that as sports because that's racing, right? We count that if I go hit a baseball and I'll be the show be great. Yeah, I'll be the show is awesome I'm just terrible at it Play the field really well Christian Watson aspiring RPA Packers number to 50 To be determined Brian. I mean this is the last box of 2022 so McBride rookie 1099 Cardinals And the NBA Jam. Yeah, it's classic. It's a 25 Christian McCaffrey Panthers Mac Jones is 60 patch card for the Pats Rex Ryan auto tough 30 for the Jets Mario super sluggers Current Mario golf is fun. I mean we all right the original we sports That's a game That literally everybody has played so that could be argued as like the best sports game of all time Mike Gisicki to 75 dolphins talk about the og we sports it came with every single damn console It's like built into it James Robinson CD Lampton 99 Jersey card Cowboys and Tyrion Davis price to 125 RPA Niners one more box of the break everybody. I've cracked my neck like 35 times Never play a Tecmo Bowl Yeah, dude, how good the graphics on that game Yeah Tecmo Bowl just say you like going blind. Yeah, I was tearing at the game to 75s avian Howard dolphins John and Taylor Colts number to 50 Mike Jones rookie patch to 99 I Know what Atari is This pong is but are we counting pong in the sports game and Michael Carter radio radio rookie RPA for the Jets to 50 Yes Tecmo Bowl and in 8k ultra hybrid HD Quantum LED if it's not then I'm not even looking at it. I don't even consider it The Army Brown rookie Washington to 35 Lawrence Taylor retro spectacle Nice insert there for what a giant 25 Rashad Bateman rookie triple relic Ravens Yeah, what's the frame rate? How many giggers? What's the What's the storage on the console if it is it's less than a terabyte that I'm not even looking at it Joe Montana champion signatures 49ers to a five Get that magged Dude is duck hunter count is hunting a sport is duck hunter frugger frugger. That's not sport Have to draw the line somewhere. I guess Kellan Mondrick evikings to 99 my home's chiefs to 149 Jottenham Taylor to 60 jersey card for the Colts Golden tee in the arcade is good. It's a 30 RPA purple Kenan a long woo Vikings Long woo Vikings again here to 15 terrace Marshall jr. Panthers to 75 to 60 almond Ross St. Brown rookie jersey card and And on the back is an almond Ross St. Brown RPA for the Lions. That's nasty Good hood on the back Lions 80 of 99 Guys Sean's gonna get 40 lashes All right three randoms here 40 lashes Gonna have Sam do it We finished late tonight that break took forever Okay two points cards 1500 and 600 top two spots in that random are gonna get to everybody you're gonna get points top spot 15 second spot six After that we've got Colts chiefs Alec Pierce Skymore on the dual patch out of the 30 Again Colts chiefs then we have Broncos Jets Michael Carter Chavante Williams to five on the psychedelic patch out of Broncos Jets three times on everything Good luck everybody we've got a recap after this three times top two get points here one two three John D gets the 1500 Brad K gets the 600 the Pierce and the Moore One two and three going to the Colts and the Williams and the Carter One two and three Jets so many hits. That's just the relics right there Dude, this is more than like a Sunday Monday has Okay relics Ron Dale Moore Jacoby Myers Derek Carr psychedelic Elijah Moore Cortland Sutton Zach Wilson TJ Hawkinson Tyler Boyd Jacoby Myers Trevor Lawrence Nico Collins Alec Pierce Josh Allen Tori Holtz Jordy Nelson DJ Chark Mac Jones Anthony Schwartz Michael Vic Skymore Traylon Berks Jerry Rice Joe Burrow Devontae Smith Jamar Chase Brandon cooks Christian McCaffrey Dalvin Cook Sam Howell and Drake London. I'm in Ross St. Brown Jonathan Taylor Rashad Bateman Mac Jones CD Lam Mac Jones Justin Herbert Antonio Gates Joe Burrow Davis Mills Kyle Hamilton and Dan Marino Austin Eckler Kellan Mond Devontae Smith Jalen waddle tutu Atwell Michael Vic Milish Jones Jr. Deshaun Watson Mleek Willis Justin Jefferson Justin Jefferson Jacoby. I was Justin Herbert Jacoby Myers Antonio Gibson Matt Corral Bradley Chubb Jamar Chase Randy Moss Dak Prescott Aden Hutchinson Jalen Darden Elijah Moore Brett Favre I was a non-reli quitted an insert retrospect and then our Andre Stevenson Autos you've got Deandre Swift Mark Brunel Daz Nussan Dave Casper Thailand Wallace Brian Robinson Jr. Harry Carson Charles Woodson Ram's Redemption RPA 2-2 I will expires in June Rangore This was a dual case of 2022 and 2021 got Davis Mills Pierre Strong Jr. Trent McDuffie Breece Hull Elijah Moore Anthony Schwartz Redemption Brown's RPA Bob Greasy Davis Mills Simeon Rice Tony Rathman Jake Ferguson Sam Howell Odafe Owe Steve Largent Kyle Trask JC Jackson Vilish Jones Jr. Michael Vic. I went Colts on the duel here Alec Pierce Skymore went jets on the Javante Williams and Michael Carter. I'm a Ross St. Brown Chris Evans Zach Wilson Dwayne bow Nicoby Dean Fred Warner Aiden Hutchinson Greg Rousseau Michael Vic again Dwayne bow Kyle Phillips Jalen Tolbert Kenny Nuangwu Michael Carter Tyrion Davis Price Rex Ryan Christian Watson Vilish Jones Jr. Nebula Jalen Phillips a Cliff Harris Zach Wilson again refrigerator Perry to five Danny Gray James Robinson as Jags Jags and Vilish Jones Jr. again We had a Pickens purple RPA. That's a nice one there for the Steelers Justin Fields Redemption Bears Get that one back. Nebula patch card to two Harris and Smith Vikings Gold rookie monumental memorabilia Nike swoosh George Pickens one of one It's gold on the checklist. It's gold final though Crystal Lave on the RPA Saints number to 35 We've got champion signatures Joe Montana Niners to five Danny Gray Niners on the Nebula RPA there a Neon black light Kyle Pitts Falcons Mahomes wavelength 10 And Kenny Pickett Steelers RPA 60 I don't do it. Thanks again everybody


Hi everyone, Joe for jaspyscasebreaks.com coming at you with a two-box break of 2020-2021 Panini Mosaic UEFA Euro Soccer. This is a random country break number 20.  So big thanks everybody here. Let's double you up And there's all the countries right there Let's roll it. Let's randomize it three and a five eight times for names and countries one two three four five six seven and eighth and final time That's Brian down to Brian and three and a five eight times for the teams one two three four five six seven and Eighth and final time Turkey down to Austria All right Brian. You got Turkey a Brian Croft with Turkey Brian Vormberg with Croatia Matthew with Finland Brian C with Belgium Steve Wright with Portugal Brian V With Poland Brian C with Hungary you on with Scotland Brian C with Switzerland Matthew with North Macedonia Steve with England Brian Croft you got Germany Sweden Ukraine Wales You on with Denmark Steve with Italy Matthew with Spain Brian Croft with France Matthew shear with Russia and Slovakia See with the Netherlands Matthew with the Czech Republic and Brian Croft with Austria Let's sort alphabetically by country And we're going to pause the video and allow for a little bit of trading time while I go run and grab those two boxes Be right back All right, welcome back No deals were done here in two box break number 20 on a Tuesday Thanks everyone for getting in I appreciate it here the two boxes that I randomly grabbed and we posted another tube So if you want to run this back Definitely be happy to do so Great tournament overall. I thought A great final two I feel like England should have won it though I had a little financial interest in Italy. It worked out but uh But England should have I mean they were at home He scored early and then just got a little too conservative Italy became more and more Took more and more control of the game In the end they deserved it But England's got a pretty young team. So I think it'll be interesting to see what What they look like in the uh in the world cup Those young players have a little more experience under their belt All right Good luck everybody You know guys like Jayden Sancho Remember all card ship So this one will go to Steve Wright in England Of course, there's rookie parallels are always nice to have that Stanislav Lobotka For Slovakia Matthew Shira The MLB all-star game on in the background as well Jack Grealish also for England some Cristiano Ronaldo For Portugal That'll go to Steve Wright He ended up winning the uh golden boot had the most goals in the tournament And we've got a blue parallel here 22 out of 99 that's Lucas uh Maso Pust For the Czech Republic It's going to go to Matthew Shira The next little stack here There's a Rui Patricio Ruben Diaz red also for Portugal Equal Williams for Wales There's an Mbappe It's a rough tournament for France as well But they also have a good collection of youngsters like Mbappe That'll look good in the next major tournament And be good for the hobby. There's a Mason Greenwood Silver also for England Where's our autograph? Oh, there it is And it's going to be Pedro Neto Portugal It'll be for Steve Wright A couple rookie parallels there We've got a silver rookie for Belgium Yari Versaurean Brian Croft Belgium And then Austria's Kristoff Baumgartner Also for Brian Next box Rex, what's up? I wonder if... I don't think I've... I've been looking for Pedri on the checklist But I don't know if he's in this set Let's go to cardboardconnection.com really quick It's going well Rex Got an all-star game on in the background Doing a nice little soccer break here These are going pretty well I think his name is Pedro González Lopez No, the only Pedro's in here are Pedro Netos So maybe in another future set Look out for Pedri He plays for Spain What club team does he play for? He won Young Player... Oh, he's a mid-central midfielder for Barcelona Pedri Won Young Player of the Tournament So that's someone to look out for See you Jesse Nice, Ryan Get that on the schedule once this is done You can read that Mets are interested in Chris, Brian I'm sure a lot of teams are interested in Chris, Brian Dodgers are interested in Kimbrell That's interesting I feel like they're okay with the relief pitching I thought if anything, they would go for maybe a start Maybe they do want to take care of the back end of that rotation Or the bullpendo It's Cristiano Ronaldo I don't know if I'd be that happy with Kimbrell He's having a great season this year But what if he reverts back to last year's Kimbrell? That's always a concern Especially in the playoffs And I don't know I don't think the Dodgers would give up very much for Craig Kimbrell There's David Seiger for Hungary And that'd be for Brian I want to say Craig Kimbrell's in the last year of his deal too, right? No, he's got 2022, so he's got another year Oh, the Dodgers might actually do that But they traditionally, unless it's for like a Mookie Betz type They traditionally Oh, here's our Otto It's Julian Brandt Germany's forward Yeah, Dodgers traditionally don't give up a lot of A lot of big-name youngsters For middle relievers, you know what I mean? Yeah, exactly, Adam Kimbrell is one of the most nerve-wracking Pictures to watch in a playoff game There's Ivan Obliakov For Russia There's Yav Filish There's Rodry, not Pedry, but Rodry Right, yeah, well that's what I've been saying since the beginning Of the season, Rex They got to move the likes of They got to move as many players as possible So they could rebuild that farm system And get some young players on that squad And kind of have them all be able to grow together All right, we've got a couple rookies here For France, it's Eduardo Camovinha Red rookie for Brian Croft And France And Brian also has Belgium We'll get this Boyata And you guys to look out for in the The next major tournament here There's Bukayo Saka Rough penalty kick for him But again, he's super young He'll learn Learn from the experience There's Mosaic, Jack Grealish There's Timo Werner And we've got a couple red rookies here A couple fun inserts there And that is that Soiree for Finland That's for Matthew Shira And we've got a Dmitrievsky For North Macedonia That's going to be for Matthew Shira as well All card ship And I've said this before in these videos It's worth kind of looking up A lot of these players Because you may be surprised at some of the secondary market value Especially when the next tournament rolls around Or when the next soccer season rolls around later this summer Slashfall I'm Joe for jaspyscasebrakes.com We've got two more in the store We can run it back tonight We'll see you next time for the next one