Episode 17 of Math 1050, College Algebra. I'm Dennis Allison and I teach mathematics here at Utah Valley State College. If you remember in the previous episode, we were talking about exponential functions, that is functions of the form f of x equals a to the x, where a is a constant, bigger than zero. And we looked at a special function, e to the x, where e has a very specific value. Let's see, who can remind us, what is the value of e approximately? 2.718? 2.718. And you know, if you needed to know e to a further, to a longer decimal expansion, you could find it on your calculator. But let me just tell you what the first few digits are. It's not that difficult to remember. If you expand e further, it's 2.7, and then it's 1828, 1828, whoops, 1828. And then 459045. I don't know why you would need to know this, I don't know the number out that far, but the hard parts remember the 2.7. After that, we have the number 1828 twice. That's the year Andrew Jackson was elected president. And then 459045, you can remember that by an isosceles right triangle, where the angles are 45 degrees, 90 and 45 degrees. So now you know e to 16 significant digits. And the graph of e to the x, while we're looking at this here, the graph of e to the x, which we're going to be using today looks like this, the graph of e to the x has three target points. And what we do is we start off from the origin, we go up one, and if I go one to the right from the origin, I go up e, or 2.7, roughly. And if I go to the left one unit, I go up one over e, and you remember how much one over e is approximately? One over e, if you calculate it on your calculator, it's about 0.36, or roughly a third.  And then this is the graph of the function e to the x, f of x equals e to the x. And this function is called the natural exponential function. And sometimes you see it abbreviated as exp of x, the so-called natural exponential function. Well, let's look at the list of objectives for today's class. We want to look at several applications of exponential functions, and these are all involved with compound interest. Now, we want to talk about compounding over n periods per year, like per month or per day or per quarter. And then we also want to look at something called continuous compounding, where interest is being added in a continuous fashion into the account. We'll also look at population growth, which is sort of a variation of the continuous compounding. And then finally in today's lesson, we're going to look at an introduction to logarithms, and the ideas for logarithms continue into the next episode. Okay, well, let's begin with compound interest. You know, if you have a certain amount of money that you put into an account, that's referred to as the principal. So I'm going to let p represent the principal in a banking account. And let's say r represents the rate of interest. That would be the annual rate of interest. And then if I were to put a principal of p dollars into an account that has a percentage rate of r, then the interest that I would earn in a year would be r times p. So for example, class, if you were to put $100 in a bank account that paid 5% interest, and the interest was put in at the end of the year, how much money would be in your account at the end of the year? $100, 5% interest? $105? $105, exactly. You see, what you do is you take the amount of principal that you started with, and you add on the interest that's added to that account. And that would be the principal plus r times the principal which is p times 1 plus r. Okay, now with that idea, let me derive a few formulas that we want to use to talk about compounding interest. Suppose that a at t represents the amount in the account after, let's say, t years. Okay, so if I say a at zero, that would be the amount in the account initially before you have any compounding that takes place. So a evaluated at zero would just be p, whatever the principal is that you put in the account. Excuse me. Okay, now after one year, after one year, I'll call that a at one, the amount in the account would be p plus rp, rp being the interest that's added to the account, and that would be p times 1 plus r. Okay, so if the interest is added at the end of the year, then you would have p times 1 plus r dollars in the account. After two years, well, let's see, at the beginning of that second year, you would begin with the amount that you carried over, p times 1 plus r, plus you would earn interest on that at a rate of r, p times 1 plus r. So this is your interest rate, and this is the new principal that you're collecting interest on. And if I factor out a common factor there, let's see, I could factor out a p and 1 plus r. And if I factor that out, let's see, if I factor that out, there'd be a 1 left over here, and there would be an r left over there.  At the end of three years, how much money would be in the account? Well, I would begin the third year with this amount of money, p times 1 plus r carried over from the previous year, plus I would get an interest rate of r times that, p times 1 plus r to the second power. So if I factor out the common factor, just like I did before, this would be p times 1 plus r squared. And when I factor that out, there'd be a 1 left over here, and there'll be an r left over there. And this reduces to be 1 plus r to the third power. Now, I think you can see then, that as a general rule, if I were to leave money in this account for, let's say t years, rather than zero, one, two, or three, you notice there's a pattern in these answers, p, p times 1 plus r, p times 1 plus r squared, p times 1 plus r cubed. And you notice that power is the same as the number of years. So after t years, this would be p times 1 plus r to the t power. Now, let me just take an example of how I would use that formula in a specific problem. Suppose that we were to deposit, let's say $1,000 in an account that pays 5% interest, and this money is, this interest rate is compounded annually. That is, the interest is added into the account at the end of each year. That's not the way most banks would normally compound money, but suppose that were the case. And so we might ask the question, how much is in the account at the end of five years? At the end of five years. Well, I can use the formula that I just arrived to work this problem. The formula said p of t is, oops, a of t is p times 1 plus r to the t power. So in this case, a at five would be 1,000, because that was the initial principle, times 1 plus 0.05, because that's the interest rate to the fifth power, because we're talking about five years. So this is gonna be 1,000 times 1.05. Maybe somebody can get out their calculator and be calculating this for us, so I don't have to stop to do this. This will be 1,000 times 1.05 to the fifth power. Stephen, it looks like maybe you have an answer for that. What did you get? $1,276 and 28 cents. And 28 cents. Now, did you round up or round down when you said 28 cents? I rounded down. You rounded down, okay? Now, you know what? At a bank, I don't believe they would ever round up, because they would say you haven't earned that 28th cent there, so it's really 27 point something cents. So I think they would round that down, and they would call that a 27. So at a bank that's always rounded down because technically you haven't earned that extra full penny, you'd have to ask your individual bank to see, but we're talking about small potatoes here, but it looks like at the end of five years, you'd have $1,276 and 27 cents in that case. Okay, well now from this formula, I wanna look at some other formulas that go with it that will be useful when we compound numbers. Suppose that rather than adding the interest at the end of the year, suppose we divide up the year like in the months, for example, and we add interest at the end of every month so that your interest gets in there a little bit earlier and then the interest begins to draw interest, so that would be called compounding monthly. Most savings accounts that you would see at a savings loan or at a bank are compounded daily, so actually the interest is added at the end of the day and that way the sooner the interest gets in, the faster the money compounds and so the more money you could actually earn. Now the formula we just had said, whoops, I keep writing a P there, A of T, the amount at time T is P times one plus R to the T power. Now this was for money that's being compounded annually, but suppose we have in compounding periods in a year. For example, what if the money's being compounded per month? There would be 12 compounding periods per year. Here's how we adjust the formula. The amount in the account now will be the initial principal times the interest rate per period. Well, if there were in periods in a year, you're getting an interest rate of R over in per period. So for example, if it were 5% per year, how much would that be per month? It would be 5% over 12. But then on the other hand, the power changes because instead of having T years, now we wanna look at how many compounding periods there are and in T years there would be in times T compounding periods. So for example, in T years there would be 12 times T months. So you divide by 12 or divide by N inside, but you multiply by N in the exponent. Now this actually tends to give a slightly larger answer. Suppose that we were to take that same principle of $1,000, Stephen, maybe you could get your calculator out again for us here. And suppose we were going to have our money compounded at 5% per year, 5% per year, but it's compounded monthly and we're gonna leave the money in the account for five years. I think you'll find this ends up being a little bit more than what it was before. The way I'll calculate the amount in the account after five years will be 1,000 times one plus 0.05 divided by 12 because this money is being compounded monthly. And then in five years there are 12 times five months. There are that many compounding periods. So this gives me 1,000 times one plus 0.05 over 12 to the 60th power. And Stephen, how much do you get this time? $1283 and since we're not always rounding down 35 cents. And 35 cents. Okay, so we're gonna round down there. And before I think we had $1,270 something dollars or was it 81, I've forgotten, but it was slightly less than this, not much, not much. Now how would we have done this differently if we had been compounding daily, which is actually what most banks do? How would I change this expression up here? You changed the 12 to 365. Right, we'd put 365 there, which makes this interest rate per day even smaller than what it was per month. But then what would we put up here? 365. 365 times five. So now we have a bigger period and you would find out that this ends up being slightly higher, like it might go to $1,284, $1,285, wouldn't be much more, but slightly more. Now let's go to the next graphic and we'll see two of these three formulas listed. There are three formulas for computing interest. First of all, the interest compounded annually and our formula is A equals P times one plus R to the T power. Interest compounded N times per year and A equals P times one plus R over N to the NT power. And now finally, here's a new formula for continuously compounded interest. Now let's come back to the green screen and let me explain what that means. Rather than being compounded per month, per week, per day, even per hour or even per minute, what if money was being continuously added to your account? It would be very small, but that's called continuous compounding. And then the formula changes and the formula becomes this. A equals P times E to the RT. Now this is where E is that same number, 2.718 approximately. We gave a longer decimal expansion for that and you notice that the interest rate is now moved into the exponent. Now to explain how this happens, you really have to know a little bit about calculus. So what we're gonna do is take this formula for granted and let's see how much would be $1,000 placed in an account that pays 5% interest annually, but it's compounded continuously and we leave it for five years. Now last time we had $1,283.35, that's what we got. Let me just write that down, $1,283.35. That's what we saw when we were compounding it daily. So this time the amount in the account after five years will be 1,000 times E to the 0.05, that's the interest rate, times five, that's the value of T. And this is 1,000 times E to the 0.25 power. That's the 1 fourth power. And when you multiply that out, Stephen, what did you get? $1,284.02. And 2 cents. So by continuous compounding, we have increased this by a total of 67 cents. Now you might say, well Dennis, of all these compounding schemes, it sounds like continuous compounding is the fastest growing scheme of all, but most banks don't do this because you see if you had an account where money was compounded continuously, then you'd have to have a time clock so that if you went to close your account, they'd have to know at what time of day you actually closed it because every moment there's interest being added to the account. So what they do is they just run all the files at the end of the day and upgrade them with the newly added interest at the end of the day. And you see it only makes a 67 cents difference over a period of five years. So there's a very slight increase in going to continuous compounding and it would be too much bother, I think, for a bank to have to determine at what time you actually made a deposit or made a withdrawal. Let's go to the next graphic and we'll see the results of three different schemes for some other numbers. I just computed these before class and I'll show you the answers and you could check these out on your own. The example says verify the value of a $1,000 investment that should be singular there at 4% after three years. Now that's 4% annually but the money may be compounded by these three different schemes. Then it also says find the equivalent annual interest rate. I'll show you how to do that in just a moment. Now in part A, if the money is compounded annually you get $1,124.86 and the equivalent annual interest rate is 4% because this is compounded annually. If the money were compounded monthly you would get $1,127.27 and it says the equivalent annual interest rate is 4.074. Let me show you how that number was just derived. If we go back to our original formula that says A equals P times one plus R to the T power this was for an annual compounding. Now under this scheme where we were compounding monthly our $1,000 grew to $1,127.27 but it started off at $1,000. Now what would be the annually compounded interest rate that would have produced this amount after three years? So what I'm gonna do is solve this equation for R and that's the percentage interest rate that you saw on the screen. It's gonna be slightly more than 4% because I'm acting now as if this were just annual compounding. So I would divide by 1,000 and I would get 1.127.27 equals one plus R to the third power. And then to solve for R I'd take the cube root of both sides. So I'd take the cube root of 1.127.27 equals one plus R and then if I solve for R I would subtract one from this answer. So on my calculator I'd compute the cube root of 1.127.27 and then I would subtract one. Now if you do that you'll find out that R is approximately 0.04074 R 4.074% Now what that means is rather than having your money invested at 4% compounded monthly for three years if you invested it annually, excuse me, if you invested it in a scheme that compounded it annually it would, the interest rate would have to go up to 4.074% in order for it to grow to that same amount. So this is the equivalent annual interest rate if it were compounded annually. Now let's go back to that graphic one more time. And it says in part C that if you were compounding continuously the money would grow to $1,127.49 that's what, only a gain of 22 cents after three years. What would be the equivalent annual interest rate if it were being compounded annually rather than continuously and it would be slightly higher 4.081%. Now in order to get that figure we have to know something about logarithms that will come up later on in this episode in the next episode. So we'll just have to take that number as it's given and move on. Okay, let's go on to the next graphic and look at an example. In this problem it says assume that the population of a city grows exponentially at an annual rate of 2.4%. Now by the way, when I say it grows exponentially what I mean by that, that sort of a label growing exponentially means that the population of a city grows continuously. That is, people are added to the population. It could be any moment of the day. I mean people aren't born at the end of the day or people aren't born at noon only and the same way for people who die within the population or move away. People may move into the town at any given time or they move out of town theoretically at any given time. So we say it grows exponentially meaning I'm gonna use my natural exponential function to model this. So as a continuously compounded problem and it says its population in the year 2003 is 286,000. Now if this continues, predict its population in the year 2025. Okay, well let's take that information and put it up here. We're assuming the population grows exponentially. Now what that means mathematically for us is that the population at time t, and here I will call it p because it stands for population, equals the initial population, which I'll call p sub zero. That's the population in the year 2003 times e to the r t power. You see this looks like a continuously compounded interest problem except now we're talking about money growing, we're talking about population growing rather than money in an account growing. So the initial population was 286,000. So I'm gonna substitute 286,000 right here for p sub zero times e to the, let's see now the interest rate or the rate of growth was 2.4%. That'll be 0.024% times t because we haven't been given the t yet. Now the question was if this is the population in the year 2003, what's the population in the year 2025 if this continues? Well how many years lapses between those two times? 22 years, so what I'm gonna do is calculate p at 22. So if I calculate p after 22 years that'll be 286,000 times e to the 0.024 times 22 power. Now let me just show you how I would calculate that now on a calculator. Okay now if we multiply that out that's 286,000 times the exponential function and I can find the exponential function it's written right above this key that says ln on my calculator. So if I push second then you'll see it's e raised to the power of and I'll open parentheses 0.024 times 22. Close parentheses and I'll enter. And this looks like it's around 485,000. I don't think we should take it so literally that we should round it off to the nearest person. I'll say it's 285,000. So this is approximately, whoops, 485,000 it should be. 485,000 and you know to be really more reasonable about it I think we should say it's about half a million is what it is. So it's about half a million people. So we have a town here with a population of 286,000 people and it's growing at a rate of 2.4% per year it's growing exponentially. And so at the end of 22 years we'll have about half a million people in this town. Okay, let me go to a graphic, not to a graphic but to a newspaper article that was in the Salt Lake Tribune and let me show you what this looks like. If you can read this, can you zoom down on this? I want to show in particular the graph that's over on the right-hand side. Now this is a newspaper article that was printed on Thursday, December 28th. The year was 2000 so this is a rather old cutout from the newspaper that I made. And you know it gives the population of Utah back here in the year 1900 and the entire state had a population of about 277,000. Not too far off from what that example had in it initially that we just looked at. By the year 1910 the population was about 377,000 and now you notice that it just keeps climbing every year, there's a big jump between 1940 and 1950 from 552,000 to 696,000. Anybody have an explanation of why it would jump so much during the 1940s? Baby boomers, yeah, right after World War II there were probably a lot of children born right then and you notice from that point on this begins to rise rather dramatically and if you look at this, shall we call this graph overall, it looks a little bit like an exponential function. The population is a little bit different from an exponential function. The population is growing more or less exponentially. I don't think we should expect that an exponential function would calculate all of these numbers precisely but it does look like the state has been growing exponentially more or less. And in the year 2000 the population of the state this was estimated at about 2,160,000. Now you see this is in a newspaper article from December 28th the year 2000 so they didn't have an accurate estimate of the population at that time. Okay, so it does look like populations may grow more or less exponentially. Okay, let's go to another example. This one also comes from the newspaper and this was in the sports section of the Salt Lake Tribune just this spring. Now I'm talking to you in the spring of 2003 and this is during spring training and this is an article about the New York Yankees. You know the title of the article says the Yankees reach another milestone this time for salary. Let me just read a little bit to you and there's a mathematics problem here about continuously compounded interest. Now the article says even before the start of the season the New York Yankees were smashing barriers. New York set a record with $138 million payroll last year that would have been 2002 according to final tabulation by the commissioner's office and is on the verge of becoming the first team to reach $750 million. Now it says the Yankees 2003 payroll stands at $149.2 million that's this year for 22 signed players likely to be on the opening day roster plus an injured pitcher, John Lieber according to contract. Okay, well let's put some of this information down on the green screen and let's predict what their team salary will be next year. It says in the year 2002 the team salary was $138 million and this year in 2003, now the season hasn't actually started yet the team salary is $149.5 no, .2 it was million dollars. Now let's assume exponential growth in the team salaries and the question is will the team salary next year in 2004 for the New York Yankees? So we want to make a prediction on this. Well, I'm going to take 2002 as my base year I'm going to call that t equals zero 2003 I'll call that t equals one one year later and 2004 I'll call that t equals two. Now if salaries are growing exponentially that would say that the salary at time t should be s of zero that would be the starting salary in the year zero times e to the rt power. Trouble is we don't know what r is. So I'm going to say that s at t is equal to s at zero now in the year zero that was 138 million I'll just put 138 there we'll have to remember this is in millions times e to the rt power and I'm wanting to calculate the team salary in the year t equals two so s at two equals well what I do know is that s at one is equal to 149.2 million dollars and that would be 138 times e to the r times one power I'm putting in a one for t there and so e to the r is 149.2 over 138 that's what e to the r is so what is s at two let's see I think maybe I can squeeze this in right here what is s at two that is an s there well that will be 138 times e to the r times two power which is 138 times e to the r squared you notice I'm using properties of exponents here I'm putting r on the inside and the two on the outside and if I raise e to the r and square it then that's e to the two r power and I just found out that e to the r is this ratio so this is 138 times 149.2 over 138 squared now what I'll need to do is to calculate that on my calculator Stephen you have your calculator out already have you calculated that many chance okay and I'll be calculating here those of you at home try calculating it and I'll work it over here on the side and I get 161.3 approximately 161.3 is that what you got Stephen okay so of course this is measured in millions so the New York Yankees have not topped 150 million dollar team salary but they're right on the verge of it and we would predict that next year if things continue to grow exponentially it'll be 161.3 now of course in the real world there are a number of factors that affect salaries and what the team salary will be you know there may be a major player who retires an expensive player who retires or is traded or it may be that they pick up a player with a really big contract and so this figure could be a little bit higher or lower but this is our prediction based on the evidence that we have here okay so much for exponential functions and compound interest let's move now to another topic and this is this is a completely different topic it's an introduction to logarithms let me begin on the green screen and tell you that if you write an expression such as 5 squared equals 25 I would say this is written in what I'll call exponential form that's because I'm using an exponent namely the 2 and I put the exponent up in the air we're all accustomed to this and so 5 squared is 25 now there is another form of writing this called logarithmic form and what you do is you write log base 5 that's the base that was over here so now I write that as a subscript and I put a 25 after it the log base 5 of 25 equals 2 now this is referred to as logarithmic form and it turns out that in this notation some problems are much easier to solve this idea was invented by a guy named Napierre in the 17th century and you see what's happened is what used to be the exponent is now written on the ground here on the right-hand side the logarithm is equal to the exponent the number that was the old base becomes a subscript and the number which used to be the exponential value 25 is now written inside the logarithm so we read this as the log base 5 of 25 equals 2 let me just write one more like this suppose I were to say 2 to the 5th power is 32 I'm using the same numbers but I've reversed their positions 2 to the 5th power is 32 this is written in exponential form how would I write that in logarithmic form well I write the log base now the base is whatever the base was over here base 2 but I write that as a subscript and the exponent is going to be put over here on the right-hand side the logarithm always equals the exponent so the 32 has to go in here now what if I reverse this process this time I'm going to write something in logarithmic form I want you to tell me what is this exponential equivalent suppose we say the log base 4 of 1 fourth is negative 1 how would I write that in exponential form 4 to the negative 1 power 4 to the negative 1 power is 1 fourth and that's true 4 to the negative 1 power is 1 fourth you notice that whatever is the subscript I call that the base, log base 4 that becomes the base of the exponential and the negative 1 the logarithm equals the negative 1 that becomes the exponent so 4 to the negative 1 power equals the quantity inside 1 fourth so the number inside here couldn't be anything other than 1 fourth or this would not have been true let's go to the next graphic and we'll see some other examples of this expressed on the screen on the left-hand side we have the exponential form of an expression 4 to the third power 64 and on the right-hand side we have the logarithmic equivalent of that the log base 4 of 64 equals 3 and then as a second example we have 2 to the negative 1 is a half and on the right-hand side the logarithmic equivalent of that says log base 2 of 1 half equals the exponent and so the exponent is negative 1 now, right below this is a very important statement that we will refer to frequently during this material for this next exam you want to remember that a logarithm is an exponent you see the logarithms on the right-hand side always equal to the exponent they equal to 3 the second one equals negative 1 so the logarithm is an exponent because it's equal to the exponent so whatever in doubt what the logarithmic value is you set it equal to the exponent okay, in this next example it asks us to do the same thing again with a few more problems write the following exponential expressions in logarithmic form and this gives us some practice in recognizing this notation and knowing where the individual pieces go let me write those three problems down right here on my green board and then I'm going to ask people here in the classroom how to write each of these in logarithmic form part A was that 5 cubed is 125 part B is that 0.2 to the zero power is 1 and the last one is 9 to the negative 1 half power is 1 third now first of all would you agree with all three of these statements 5 cubed is 125 I think so 5 times 5 times 5 is 125 any non-zero number to the zero power is 1 so 0.2 to the zero power is 1 and 9 to the negative a half well now see what does the one half power mean the square root of 1 over the square root of yeah exactly so the one half means take the square root and the negative means flip it over so we're supposed to take the square root of 9 and then invert it so the square root of 9 is 3 and we invert it and get a third well let's see Matt what is the logarithmic equivalent of 5 to the third power is 125 a log base 5 of 125 equals 3 that's exactly right now these two things mean the same thing Sam what would be the equivalent of 0.2 to the zero power is 1 it would be log base 0.2 to the 1 equals 0 okay I would say log base 0.2 of 1 equals 0 so if you take this base raise it to that power then you'll get 1 last case Susan what would you say is the logarithmic statement for C log base 9 of 1 third log base 9 of 1 third I'll put parenthesis around that equals negative one half equals negative one half that's exactly right okay we have another example after this that's a little bit different so let's go to the next graphic okay this time let's define the value of the variable in each one of the following logarithmic expressions let me write each one of these on the screen here for the class to see as well as the people at home log base 4 of x equals 2 the question is what is x and then in the next one it says the log base 0.01 of 100 equals y so the question is what is y and in part C of the log base B of 6 equals one half this is a B looks sort of like a 6 doesn't it so let me try to emphasize that's the letter B okay coming back to the classroom to figure out what x is here what I would do is I would convert this to exponential form because you and I are much more comfortable with exponential form because we've just introduced the logarithmic notation can anyone tell me what this says in exponential form 4 squared equals x exactly you see here's the base 4 the logarithm is equal to the exponent so 2 is the exponent and that has to equal the third number x so x is 4 squared so what that tells me is that x is equal to 16 yeah that's the answer to this logarithmic expression or actually equation it's a logarithmic equation okay in the next case let's see Matt can you tell me what is the what is the exponential equivalent of this 0.01 yeah rate to the y equals 100 equals 100 now let's just go over that you remember y is the exponent because the logarithm equals the exponent so the y goes in the air this is the base so 0.01 is the base and this is equal to 100 now this may look rather complicated but think of a way to a simpler way to write this so we can solve for y 1 over 100 to the y power equals 100 right 0.01 means 1 over 100 to the y power equals 100 now what do you do to 1 over 100 to get 100 well you have to flip it over and to flip it over what should the exponent be negative 1 so y is going to be negative 1 yeah in the next case you didn't think of that let me show you another route you could have gone taken to solve this we could have written the inside as 100 to the negative 1 power yeah because 1 over 100 means 100 to the negative 1 power now if I multiply exponents together this says 100 to the negative y power equals 100 and by the way this is 100 to the first power and therefore these exponents should be alike so y should be 1 and therefore y is negative 1 now this is a little bit longer tack to get to the answer but if you find this more comfortable or more agreeable to you you're welcome to follow that approach what we did back here the shorter way was to just say what we need to do is invert the 1 over 100 to get 100 so the exponent that will do that is the exponent negative 1 and y was negative 1 okay so either way you look at it this answer is y equals negative 1 okay now we have a third problem here log base B of 6 is a half Steven how would you write this exponentially B to the 1 half power equals 6 B to the 1 half power equals 6 exactly because this is the base B we always take that subscript to be the base this is the exponent over here so it's B to the 1 half power and the number inside is the value of the exponential expression so this says the square root of B is 6 and therefore B must be what must be 36 must be 36 so B equals 36 is the answer by the way the base that I put here for the logarithm obviously it doesn't have to be an integer because we have a case here where I used a decimal but the base always has to be a positive number so that's the only restriction that we place on that logarithm okay now you can find some logarithmic values on your calculator and let's go to the next graphic to describe the two that are available on your calculator the two logarithms available on your calculator call common logarithms and natural logarithms now the logarithm log base 10 of a number say log base 10 of x this sort of a logarithm is called a common logarithm and it's actually the older of the two logarithms that we're going to discuss here log base 10 or common logarithms go back for several hundred years and because they're so quote common this is normally abbreviated as just log x so if you don't see a subscript on the log then that's normally taken to be log base 10 so in that first sentence it says the logarithm log base 10 of x is a common logarithm and is abbreviated as log x with no subscript at all now the other logarithm that's rather common is the log base e you remember e is the number 2.718 approximately and the logarithm log base e of x is called a natural logarithm and it's abbreviated as ln of x for natural logarithm now you may say Dennis, I don't see anything natural at all about the number e but if you recall when we had the function e to the x we call that the natural exponential function and now in this case log base e we call the natural logarithmic function and they both have abbreviated names and this one has an abbreviated name because it's used so often now if you look on your calculator the last sentence here says your calculator is designated for logarithms and you'll have a key that says L-O-G and a key that says ln now let me just show you on this on this graphing calculator come back to the green screen if you can zoom in close enough over in the left hand column I have two keys one of them says L-O-G one of them says ln and let me just show you how these work suppose I want to take the log of 100 and I think if I put it over here on the side I can write just beside it the log of 100 now what this means is log base 10 but if there's no base shown there we just assume it's a 10 even if it's not expressed there now if I were to put a 10 here what exponent would you put on 10 to get 100 right so that answers 2 so the log of 100 should be 2 let's try this on the calculator and see what happens I'm going to press the log button and I'm going to enter 100 and then push enter and we get a 2 over here on the right hand side okay let's take another example what if I were to take the log of 0.1 now let's see to figure out what this answer would be remember this means base 10 what exponent would I put on base 10 to get 1 tenth for the answer negative 1, yeah you see if you think of this as a fraction this is the fraction 1 over 10 10 to the negative 1 power is 1 tenth or 0.1 let's try that here on the calculator and I'm going to take the log of 0.1 and when I enter it it's exactly right okay now the other key on this calculator I want you to notice is the natural log key now what if I take ln ln of 1 I'm wondering what that would be well let's see now you know there's actually a base here the base is e 2.718 approximately I'm going to put on base e to get 1 for the answer 0 yeah I think that would be 0 does everybody agree with that because see what we're thinking is an exponential form e to the 0 power is equal to 1 that's what I'm thinking now of course the e isn't shown but when you see ln it's assumed to be base e let's try computing this I'm going to push the natural log button the natural log of 1 equals so if I push enter and I get a zero right over there and let's take one more okay this one's kind of tricky what would be ln of the quantity e to the 3rd power does anyone have a guess as to what that answer would be 3 that would be 3 okay Stephen tell us how you came up with that answer well you want to find out what what power you raise e to to get e to the 3rd and well you're kind of given the answer there so the question is what were you going to say well to get e to the 3rd from e simply raise it to the 3rd power so the answer is 3 so we have the answer right there in front of us so the question is going back to the green screen if we were to take this base e which generally of course is not shown if you were to take that base e what would the power be well the power should be 3 now I'm going to try that here on the green screen and I'm actually going to substitute in the approximate value of e so I should get an approximate value for 3 so I'm going to take the natural log of the quantity see how much was e again who can tell us 2.7 2.7 okay approximately that's the number we've been using and I'm going to raise that to the 3rd power so I have 2.718 raise to the 3rd power close parenthesis now do you think I'm going to get an answer of exactly 3 no no because this isn't exactly e is this number a little bit too big or a little bit too small for e a little bit too small so anyone want to take a wild guess as to what the answer we're actually going to see on the screen when I press enter 2.97 2.97 okay so Sam's going out of limb 2.97 anybody want to go for 2.98 2.99 okay Matt says 2.99 but I notice he kind of glanced at his calculator so we'll see if he's already worked this let's see 2.99688 et cetera okay now let's do this one more time natural log of the quantity I'm going to put in a more accurate decimal expansion of e 2.718 now if you remember then it was 281828 and then 459045 you might say Dennis are we really supposed to remember all that no not at all but at the beginning of this episode I told you more digits in that number I'm going to raise that to the 3rd power close parenthesis and now I should get I hope 2.9999 I'm going to get a bunch of 9s I still won't get 3 because I think that number is a little bit still too small so I press enter and I got 3 why do you think it said exactly 3? calculator rounded the calculator rounded off you see this only goes out so far and it rounds off the last digit so it rounded off to 3 so that is our best estimate for that number okay let's go to the next graphic and look at this example it says evaluate the following logarithms well I'm going to do this first of all without a calculator and then I want to show you one more graphic before we leave about using your calculator okay without a calculator I'm wondering if someone can tell me I'm going to write this on the screen here what is the log of 100,000 and while we're at it let me write down the other problems what is the natural log of eq oh my goodness all I just happen to make up the same problem I had on the on my graphic so I think you'll know what that one is and for part c what is the log base 12 of 1 over 144 that's kind of a ringer it's not a natural log or a common log okay coming back to the green screen in part a when I say the log of 100,000 what base are we assuming here base 10 okay so 10 to some power is 100,000 does anyone know what power that would be it'd be the fifth power what's a quick way you can tell the power on 10 that will give you 100,000 just count the zeros yeah 1, 2, 3, 4, 5 that's 10 to the 5th so for example if this had been 100 just 100 100 has 2 zeros so 10 to the 2nd power is 100 so that answer is 5 this problem is way too familiar for us because I just happen to make it up again for the second time this answer is how much it could be 3 it could be 3 it could be if you were lucky this is your lucky day and okay now in the last case log base 12 of 1 over 144 let's see Susan do you want to take a guess at that one negative 2 is exactly right you are a winner okay negative 2 now which one of these would I have not been able to have computed on my calculator C because we don't have a button that says log base 12 we have a log base 10 we have a natural log now you might say well gee what's wrong with these cheap calculators they don't even give us a log base 12 button what are we supposed to do if we encounter a log base 12 and we don't know the answer otherwise well we will find a way in the next episode to convert any logarithm base to base 10 or base E so we can actually use these two buttons to work a base 12 problem but we haven't seen how to do this yet okay now in the last minute or so let's go to the last graphic we have one more graphic here and this title logarithms and calculators I wanted to give you an example of two different calculators and the order of the buttons that you have to push in order to do these first of all I'm going to use my graphing calculator once again and if you come back to the green screen the question was to find the log of 481 now on this TI-82 and other TI calculators similar to it if you want to take the log of 481 what you do is you enter it in that order you push the log button and then 481 and the answer is 2.6821 and back on that graphic you would have seen this is the answer I had written out I would have guessed there was going to be a 2 in front because this number is more than 100 481 is more than 100 and the log of 100 is 2 so this would be a little bit more than 2 2.68 now if I go to a different TI calculator just happens this is another TI calculator this is a little TI-30 that I brought from home and on this calculator I have a log button and a natural log button written not in the left-hand column but on the third row and if I want to take the log of 481 on this calculator I enter 481 and after I push the logarithm and I get 2.6821 et cetera now you notice there was a slight pause before that number came up let me go back and do that again I'm going to enter 481 and when I push the log button there's a slight pause before the number appears there you go it was only a fraction of a second but that's because the calculator has in it a little sub-program where it's actually making some arithmetic computations to come up with that number it doesn't have this answer memorized it actually has to compute it and that little pause is telling you it's sort of like thinking just for a moment you might say thinking in quotes before it can give you that answer now on the other hand if I say 2 times 2 look how fast the answer comes up 2 times 2 equals bingo 4 because there's only one calculation there 2 times 2 is 4 but for the logarithm value there was a slight pause because there's more going on than meets the eye and you'll hear about that in a calculus class I'll see you next time for episode 18 where we will continue talking about logarithms see you then


Now, let us lay the framework for the study of thermodynamics by beginning our second topic, basic ideas and definitions. See, thermodynamics is a science. I will shorten it to thermo now because thermodynamics is in my opinion too long a word, you know mechanics, statics, and physics are much shorter words, but thermodynamics does not remain in an isolated box. It is a proper part of science, may be a very proper part of physics. So, it works with other branches of science, physics, chemistry, solid mechanics, whatever you have you. You cannot develop thermodynamics in isolation with other branches of science and other branches of physics. You cannot study thermodynamics unless you study something in mechanics, something in fluid mechanics, at least the basic stuff, something in geometry, which is also in my opinion a part of physics on the water line of mathematics and physics. So, although thermodynamics defines its own ideas, thermodynamics inherits ideas from other branches of science, particularly other branches of physics. So, any entity in thermodynamics, a property, a definition, an idea, a concept will be called a primitive in thermodynamics. Primitives are those ideas, concepts, properties which are inherited or obtained from other branches of science, physics, chemistry, mathematics. So, these are thermodynamics says, well, these ideas are defined in say geometry or mechanics or physics or electricity and magnetism. We know what they are, let the other branches of science define them, explain us their characteristics. We will simply use them in thermodynamics as required. So, the idea is like volume, pressure, velocity, etcetera. These are all primitives in thermodynamics. We will not define a volume in thermodynamics. We will not define pressure in thermodynamics. In physics, pressure is defined as force per unit area. We will say that, look, physics says pressure is force per unit area. We will use it like that. It is a primitive for us, no contest. It is defined well by some other branch of science and we will use it. There are some ideas which are not really primitive, but are on the borderline of primitive. For example, the ideas of work and the idea of energy, these are defined in other branches of physics. For example, work done when a force is applied against a spring and the spring gets compressed, very properly defined in mechanics. So, the idea of work is a primitive, but thermodynamics will formalize this. Similarly, the idea of energy is also defined in branches of physics. For example, you have a mass and with respect to certain datum, it is at a height in a gravitational field. So, mass at a height has a potential energy that is defined in mechanics. We inherit that and consider that to be a primitive in thermodynamics. There are other components of energy. For example, kinetic energy of a particle or a body moving with some velocity v for an extended body rotating along some axis. This has a kinetic energy of translation or rotation associated with it. A fluid flowing has a kinetic energy associated with it. That is primitive, but will also be formalized in thermodynamics. Work is a primitive which will be formalized by thermodynamics. Energy is a primitive which will be, let me say, will be redefined by thermodynamics. But it is necessary for us to be very clear about what are the primitives in thermodynamics because we are not going to spend time on primitives. We will say this is a primitive. That means it is defined by some other branch of science, typically physics and is used in thermodynamics. Then the second things are primises. Primises is a formal names for basic assumptions or the big heavy world laws. For example, you take geometry. All of us know about what is meant by Euclidean geometry. Now, the Euclidean geometry is based on five postulates of Euclid. Characteristic of a point, characteristic of a straight line, parallelism, etc. Characteristic of a point, there is no proof. These are basing assumptions on which Euclidean geometry is based. Similarly, the idea of mass, the characteristics of mass is a basic assumption in mechanics. And particularly the laws of Newton. In classical mechanics, there is no proof for the laws of Newton. They are basic assumptions of classical mechanics. The whole of classical mechanics is based on that. So, these are primises which are basic laws. For us, there would be a number of primises. Our primises in thermodynamics, there are many. Three of them are famous as the laws, zeroth, first and second in some order. But there are other primises in thermodynamics, which for some reason have not yet been given the state or the status of laws. But we will look at them as we proceed. For example, one primise is any isolated system tends towards equilibrium. This is a primise in thermodynamics. We say that if you wait long enough and do not disturb a system keeping it isolated, we will find that it will reach a state of equilibrium. This is a primise. We have not yet defined what is a system, what is isolated, what is equilibrium formally. But since all of you have a background in thermodynamics, you at least partly understand what I am talking about. There are certain other primises which are known as state postulates. This is one of them. Another one which we will come to is a primise which we are not conscious about. But which says that the state of a thermodynamic system can be defined using only primitive properties. This also is a state postulate. It is a postulate, it is an assumption, it is a primise. For some reason, it has not been given the status of a law of thermodynamics. But unless we make that assumption, we cannot proceed significantly in our study of thermodynamics. A significant part of our study of thermodynamics will be the study of the primises of thermodynamics. Once that is done, what remains are derivations. Derivations are things which can be logically and using standard mathematical tools and some definitions derived from primises using appropriate set of primitives. For example, the idea of energy as a state function is a derivation. Similarly, the definition of entropy and its derivation is a characteristic, is a derivation. Similarly, the open system form of first law of thermodynamics, what we call the steady flow or steady state energy equation is a derivation. Then, the formula you have for availability, exergy, etcetera, etcetera. These are all derivations. Property relations, Clausius-Clappelon relations, these are all derivations. That is one part of thermodynamics, but the real understanding comes in appreciating the primises and undertaking the basic derivations as required for thermodynamics. Now, we come to definitions in thermodynamics. They say that a picture or a sketch is worth a thousand words. But, if we start using pictorial language, I think people will get very, very confused. So, we have to use words and we have to give very specific meanings to those words. We have to define things like a system, we have to define a thing like a property, we have to define a concept called as a Freud interaction. For all these, we have to provide a definition and we will be using definitions of essentially two kinds. Definition we say is an explanation of something, so that something can be very compactly said in a word or two. So, if you ask to define a human being, definition of a human being will require something like 30, 40, 50, may be more words depending on the detail you want to provide in the definition. But, once that is understood, everybody understands and we can make out whether I am a human or not, whether this pen is a human or not. So, we will be having definitions of essentially two kinds. One is a definition which is a short form. For example, we will come sooner or later to this definition. We will say a process, a thermodynamic process will be defined as I will have this habit of writing three lines indicating that what I am writing is a definition is nothing but a change of state. That means that in any sentence, in any discussion, wherever I see a word process, I can replace it by the expanded form change of state and the meaning should not change and will not change. Similarly, wherever I see change of state, if I have defined that as a process, I can replace change of state by a process. These are absolutely equivalent, just the way 2 is equivalent to 1 plus 1 in mathematics. So, you can say 2 is defined as 1 plus 1. So, wherever in mathematics you see 2, you can replace it by 1 plus 1 or wherever you see 1 plus 1, you can replace it by 2. So, these are just short forms. The second one and perhaps the more important one is a operationary definition. Now, an operationary definition as the word goes defines something, a property, a characteristic, a procedure whatever by defining a set of operations to either take a decision or to make a measurement and quantify. For example, black, I say the color of this pen is black. What is the operational definition? I say that in any form of light if I look at it, I find nothing reflected from it and hence I call it black. It absorbs almost everything which comes onto it is black. So, those who have studied heat transfer would have seen the definition of a black surface or a black body. It is defined as a body which absorbs any radiation of any form, any intensity, any direction, any wavelength which is incident on it. So, the operational definition is this. Bombard that body or that surface by radiation of various kinds, various intensities, various wavelengths, various sources. If it absorbs each and every one of it, then it is a black. Otherwise, it is not black. That is one type of operational definition. Another type of operational definition is to specify a set of operations which will do a measurement and quantify something. For example, mass. How do you define mass? An operational definition of a mass would be take a 2-pan balance and a standard set of masses. Mass of a body is to be measured, put it in pan A, put the appropriate set of subset of the standard masses on the other pan so that the balance balances. The value of the masses in pan B is the mass of the substance in pan A. That is the operational definition. Operational definition of a length would be take a standard measuring scale. If I want to measure the length of this table or width of this table, align 0 to 1 edge, find out the mark at which the other edge is at the top. That would be the operational definition. Operational definition has to be specified using things which are already defined or which are primitives in thermodynamics. In particular, we want to see to it that we will not have any circular definitions. I write this and underline it maybe twice particularly because I have found during my study of thermodynamics and during my teaching of thermodynamics. Continue to find even now that those who do very well in thermodynamics, those who are excellent thermal engineers tend to understand some concepts of thermodynamics in terms of circular definitions. They tend to define heat in terms of temperature and temperature in terms of heat. One assignment for you today which you do not have to submit is just go back to your study of thermodynamics and see whether you can define or we can understand the idea of heat transfer without any reference to temperature. If you can do that, then you can define temperature in terms of heat no problem because you have defined heat first without a reference to temperature. So, while defining temperature you can refer to heat or can you define temperature without any reference to heat transfer. If you can do that, then you can define heat in terms of temperature. But quite often you will find that many people, those who have become successful thermal engineers also and many textbooks in thermodynamics and some very good textbooks also fall in this trap. They tend to define temperature in terms of heat and heat in terms of temperature that is a circular definition and a typical circular definition is something like this. You ask what is x and you will say x is x is x is y plus 3 by 2, fair enough. Then you say now explain what is y and you will say y is 2x minus 3. What have you done? You have defined x in terms of y and you have defined y in terms of x. You know some a relation between x and y, but you cannot get x in terms of either the value of y or a value of x. There is a book by North. I like to refer to that in the index of which it is on computer programming. In the index of which if you try to see circular definition in C it will say C definition circular, fair enough. Then you go to D in the index. Among in definition, among various types you have a circular definition and for circular definition, definition circular he says C circular definition. All that you go is go in a circle that is exactly the meaning of a circular definition. It makes you appreciate what circular definition is without telling you exactly what circular definition is. We do not want to get into any such trap and hence we are going to essentially look at, so we are going to look at operational definitions. Now let us start our study of thermodynamics by defining our first definition and that is we will now define a thermodynamic. We are going to have an operational definition. So we are going to say that how do we decide whether a given entity is a thermodynamic system or not. If you look up the books you will find thermodynamics entity is a region of space in which we are interested. It may contain mass, it may contain energy, it may contain both. It will have some boundaries, some real, maybe some imaginary, some flexible, some inflexible, all sorts of things. For us a thermodynamic system will simply be a region of space of our interest. This is important because a thermodynamic system is never dictated to us by nature. It is we who define a thermodynamic system. So it is a region of space of our interest, the way we define it and of which the boundaries by fully defined we mean that it must be a completely enclosed space. That is all and fully defined means that at any instant of time we know what the location of the boundary is. So that means any properly bounded zone of space, whatever be its content may be from pure vacuum to very densely packed stuff is a candidate for a thermodynamic system. Only when the boundaries are defined a system is defined. If the boundaries are not defined the definition of a system is not complete. So the importance of a system is that boundaries must be very defined and that automatically means that it is a region of space. Sometimes we say that it is absolutely necessary for us to know what is inside a system but from a thermodynamic point of view it is not necessary for us to really know what is inside the system. If we know something about it it is okay but if we know nothing about it also that is okay. From a basic or pure thermodynamic point of view we do not have to know what is inside a system. So is this an operational definition? I think it fits the candidature of an operational definition because we can define whether a given entity is thermodynamic system or not. All we have to see is whether the boundaries are properly defined. If it is so then it is a thermodynamic system. If it is not well it is not a thermodynamic system. Now a question almost always comes up is the sun a thermodynamic system. The answer is yes if we can define the boundaries of the sun and say that it is a sphere or an ellipsoid with this center and this coordinates this radius or this axis then well it is a thermodynamic system. Then the question is the solar system or thermodynamic system or not. Then do not just say the solar system define that region of space in the Milky Way galaxy which you call the solar system define its boundaries and you have a system. It is important to know that the boundaries may be physical as the boundary of this surface of paper or if you have a glass of water the inside of the glass of water it is a boundary or it could be an imaginary not in a mathematical sense. It could be purely defined at some surface which you can imagine with some coordinates but which is not a physical surface. It could be physical it could be defined the boundaries could be flexible they could change shape and size. It is also possible that the boundaries may be continuous like a single drop of water or the boundaries could in some cases even be discontinuous. You take the drop of water and maybe make it into two small drops of water. You have two disjoint boundaries but it is possible to consider that as a single thermodynamic system. Continuing with our discussion on the boundaries which came out of the idea of what a thermodynamic system is and I think you have already appreciated the importance of boundaries because unless a boundary is there and it completely encloses a thermodynamic system or completely encloses a region we cannot call that region a thermodynamic system. Now what type of boundaries can these be? I have already listed here some boundaries are physical some boundaries can be defined some boundaries can be flexible some boundaries can be inflexible solid. Let me take some illustrations being mechanical engineers we are used to engines. We know our IC engine and refrigeration compressors are cylinder piston arrangements the whole steam engine many of our reciprocating pumps are cylinder piston arrangement. So you typically have a cylinder generally closed and a system. Now let us say that the innards of the cylinder the inner surface of its head the cylindrical part and the top surface of the piston. These are boundaries which enclose a region of space and so this is our system it may contain a liquid a gas or whatever. Then during some interaction the piston may move up and down what happens to the boundaries? First all the boundaries are physical boundaries you can see them you can touch them can characterize them real surfaces real physical surfaces. You can disassemble it feel the top of the piston feel the inside of the cylinder head and the surfaces. As the piston moves up and down the cylinder head boundary remains fixed no change in it. The top part of the piston boundary this boundary moves up and down different locations at different times. The side boundaries are sometimes they are short if the piston goes down here the piston head boundary moves here the side boundaries extend. So these are extensible boundaries these are illustrations of physical boundaries some fixed some moving some extensible. Sometimes we will take a system you take a section of a pipe through which some fluid is flowing and we say that look we are interested in what happens to the fluid as it moves between this plane and this plane. And we can define the system as the following boundaries the inner surface of the tube from plane A to plane B and the surface of plane A as it cuts through the tube the surface of plane B as it cuts to the tube. The boundaries A and B can be sketched on a drawing but you cannot feel them we have put them there as dotted lines. So these are illustration the inner surface of a tube is a physical boundary this as well as this these are imaginary or defined boundaries. During a process I may move this boundary B further in and out but at any time I must know where my boundary B is. Similarly A I can move in and out but at any time I must know where that boundary is. The tube may be flexible this may be a physical boundary but it is a rubber tube I can pump in high pressure water and it can expand it could be flexible the arteries and veins in our body are illustrations of such flexible boundaries. And I can say plane A is the exit of my heart and plane B is the inlet to my lungs or something like that. So this is an illustration of an imaginary boundary. Another illustration of a system consisting of purely imaginary boundary is this. In fluid mechanics we derive the so called continuity equation a differential equation representing the conservation of mass. In heat transfer we derive the differential equation of conduction the differential equation representing a combination of Fourier's law of heat conduction and the conservation of energy of the first law of thermodynamics. To do this we say imagine a control volume of size delta x by delta y by delta z. Take any good book on thermodynamics fluid mechanics or heat transfer or even on solid mechanics where differential equations of equilibrium are. We say let this be delta x, let this be delta y, let the third dimension be delta z somewhere in our domain of interest. Now we are defining six surfaces enclosing a small system. This is a thermodynamic system but it has all boundaries are defined none of them is a physical boundary. You say let at some point let this be some point x, y, z. So you can locate all the six surfaces and you determine how much mass comes in from which surface, how much mass goes out from which surface and write a mass balance and derive a differential equation. This is an illustration of a system containing made up of purely imaginary or defined boundaries but at any time we know what is the location of each and every boundary but I cannot touch the boundary. I cannot touch my finger to it. If I try to put a finger finger will go through because there is no real boundary associated with it. Now boundaries are important not only from the point of view of defining a system. They are also important from the point of view of what happens to the system. If you change the boundaries of a system you are creating a different system. Take for example a simple break. You have a drum I am showing only a part of a drum which is rotating and there is a break shoe which is being pressed against it and I want to study the interaction between the shoe and the drum. There is a boundary between the shoe and the drum and I can define my system say of the shoe with a boundary just inside the shoe and the other boundary of the physical extent of the shoe. This is one system. Then I can have a system in which the boundary is exactly the interface of the shoe and the drum and the third boundary which is slightly inside the drum. So, these three boundaries define three different systems and later on maybe we will come back to this as an exercise or a homework. The interaction between the drum and the shoe will be discussed and you will realize that as you change the location of the boundary maybe just by a few millimeters in one case inside the drum in another case inside the shoe you will find that an interaction which you are confident to be work interaction turns out to be purely a heat interaction just because you have changed the boundary. But remember there is no contradiction here. You have changed the boundary. So, you have changed the system. So, you have changed everything. So, if an interaction of the work type now turns out to be an interaction of the heat type there should be no surprises in it. We will come to this later. This was just an illustration of something which is going to come up later. Now let us stop the thing here. We have defined what is meant by a system. System has to be enclosed in boundaries of some type or the other physical or purely defined fixed or flexible or extensible etcetera enclosed in boundaries. We know what the importance of boundaries are. So, a system and its boundaries are one and the same boundaries define a system system is defined by its boundaries. So, enclosed space defined by boundaries. The next thing is what typically we talk about is classification of systems. Now in thermodynamics we will never talk about just one system. In thermodynamics we will always talk about a system A and a system B almost always.  There is some sort of a transaction. They talk to each other. They play with each other. There is a give and take. We will define an interaction later. But there is some sort of an interaction a transaction you may call it. Something is given something is taken. Many text books define a system and the surroundings. Surroundings are defined as something which is outside the system. But let us not be so vague. For us the surroundings is also a system. So, its boundaries also have to be defined properly. So, the surroundings is as much a system in its own right as is our main system. Actually it would be better to talk about a system A which is interacting with system B. Because remember that if we say that surrounding is whatever is outside the system and leave it without defining its boundaries then we are going to get into trouble. Because then we have not defined the surroundings as a proper thermodynamic systems. And whatever we define for our system and the laws of thermodynamics are applicable only to systems which are properly defined. They are not applicable to systems which are not properly defined thermodynamic systems. So, although we may be using the word surroundings we will define it as a system B which is next to system A and is interacting with it. So, keep that in mind because many text books will keep the definition of surroundings vague. We will not leave it as vague. We will say that the surroundings is a system in its own right very properly defined. And hence all its boundaries will also have to be defined. We may call this as the primary system which is of primary interest to us. This is a secondary system in which we are not that much interested. Maybe we do not want that much detailed study of the system, but it is a system in its own right although we may call it a surrounding. Now in the classification the classification is usually say we have a closed system or we have an open system or we have an isolated system. I have a problem here because a system by itself with its boundary defined cannot be closed open or isolated. This classification should come slightly later because this classification pertains to systems undergoing a process. We have not yet defined the process, but a system is just thereby itself not interacting with any other system. Then there is no question of it is being closed or open or isolated. A system by itself is something which is enclosed in boundaries. We do not ask what is happening to it. Only when we talk of a process we talk about what is happening to it. So, actually we should push this definition after a study of processes, but most textbooks will talk about this classification just after the definition of system.  All this thing has to do with interactions. Again we have not defined a process. So, we have not talked about interaction, but let us get over it. We have said the science of thermodynamics talks about energy and energy and energy transfer between systems. So, unless there is an energy transfer, thermodynamics does not take much interest in that. So, what we will do is we will say that we will put three columns or two columns energy transfer mass flow. If we say energy transfer is allowed, that means system A can do either work or transfer heat to system B. Again work heat we have to define. We are just overtaking that just to complete this definition, but no mass flow is allowed across the boundaries of the system. Then the name given to that system is a closed system. If energy transfer is allowed, mass transfer also does take place. Then we call the system an open system. If on the other hand no energy transfer is allowed and no mass transfer also takes place, then we call that system an isolated system. This is the classification. Now there are some names, slightly different names which are also used. For example the closed system, there is no mass flow across the system boundary. So, the mass of the system remains fixed. So, quite often a closed system is also known as a control mass indicating that the mass is controlled or mass is constricted to remain within the system. Unfortunately, for some reason the open system has a very common name called the control volume. This is unfortunate because the control volume and the short form CV is so common that quite often in textbooks, the subscript CV is used to represent an open system. The unfortunate thing is in a control mass, a closed system mass is fixed. It does not change because there is no mass flow across the boundaries. But in an open system, the volume need not remain fixed. The volume can go down, can decrease, increase. So, the volume is no way fixed. It may be controlled in some way, but it is no way fixed. So, do not be under the idea and do not let your students absorb the definition of an open system as a control volume where they appreciate control volume as a fixed volume. It is not a fixed volume. And for that we again take the illustration of our common illustration of a cylinder piston arrangement. Again I will show a cylinder and a piston and let me say it is part of a pump or an engine or a compressor which has two valves, valve 1, valve 2, piston P. And let us say that again our system is whatever is within this. I say that both the valves are closed. What is the type of the system? Our system will be a closed system. Now if say one valve is open, then do you say it is an open system? The answer is open system only if fluid is flowing in or out of that. Otherwise just because a valve is open does not automatically make it an open system. Take this illustration. I am thirsty. So I see this water bottle. Show me. So I am thirsty. I see this water bottle. Now I am teaching thermodynamics. So I look at it and say is this a system? And I say yes. I may define the system as the inner surface of the bottle including the inner surface of the cap. So it may contain the inside partly water, partly air or I may define the system as water itself. So what is the system? The surface of the water that is wherever it is in touch with the inside of the bottle that surface plus that top surface of the water. I am neglecting the evaporation of water. So if I move it, the top surface and even the side surfaces are either extensible, flexible. They can be as complex as they like. If I neglect evaporation, it is a closed system. Again let me say that my system is water and air above it. I open the cap. Just opening the cap does not make it an open system. But if I blow air through it or during the process of drinking water, something has come out of my system. So I have allowed of mass flow rate. I have noticed that there was a mass outflow from this. System A, I was system B. There was a mass transfer, transfer of mass, transport of mass. So during that transaction, it was an open system. And just now if I keep it here and neglect that small air movement, then it will back to be a closed system. The thing to remember here is system by itself cannot be open or closed. We have to look at the process which we will soon come to and then decide whether it is an open system or a closed system. So remember this that only when you look at the process of a system, can we assert that a system is open or closed or isolated? Now how do we describe a system? We describe a system, we do this in a thermodynamic fashion. So how do we define a system thermodynamically? See we are interested in the following. We are interested in the change of the system in what way has it changed? That bottle which was my system earlier had lots of water. Now it has lesser amount of water. So there is some change in the system. Maybe initially the water is cold. I hold it tight in my hand, warming it up. Maybe the temperature had changed. We have not defined temperature. So do not be in a rush to assume that I have defined temperature. We will do it much later in the course. So something has changed in the system. So we describe the system by defining the state of a system. So that means we have to define what is meant by the state of a system. Now we come to the second important definition, the state of a system. And the state of its system will bring it to the third important definition, the property of a system. The state of a system, the operational definition of the state of a system is this. The state of a system is defined when we do the following, two-step process. List all relevant characteristics of a system. And I will jump the gun slightly. Each relevant characteristic, we will shorten this to one word property. Each relevant characteristic is each property. So let us take an illustration and let us say that the inner surface of this is our system. It contains a gas and we are going to do some funny things with the gas. So we want to know what happens to the gas. So we will list relevant characteristics that is properties. Now maybe we will say volume is a relevant characteristic. Maybe mass is a relevant characteristic. Pressure may be a relevant characteristic. We have not defined, but we know something about temperature or relevant characteristic. Now the shape of the cylinder, whether it is spherical or whether it is cylindrical, is it a relevant characteristic? We do not know. It all depends on what we are going to do with it. The cylinder may be lined inside with may be rubber or may be gas or glass or it may be painted inside with some anti-corrosive paint. The type of paint, the type of lining, is it going to be a relevant characteristic? We do not know. It depends on what we are going to do with the gas. So all that we appreciate is if we start looking at the detail, we can come up with a large long list of relevant characteristics, a large list of characteristics. Not all of them may be relevant. Relevant characteristics will be only those which are of use to us in which we expect to see any difference or change during our play or process of the system. So how to decide on the relevance? Thermodynamics does not tell us anything about it. It is only out of experience that we decide. For example, we know from our experience that generally volume, mass, pressure and temperature, these are the four relevant characteristics. The shape of the cylinder or thickness of the cylinder, type of inner lining, whether like the bottle which I showed you, whether the shape is serrated or plain, may be that is not a relevant characteristic. So the first thing is this. First item in definition is list of properties. The second step is to quantify the properties. That means we must do an experiment or we must do some measurement to provide numerical values to each of these properties. Maybe we can use some geometric tricks to measure the volume. Maybe we can use a weighing machine or a mass balance to measure the mass. We can fix up at the top of the bottle or the cylinder, a pressure gauge to measure the pressure. We can insert a thermometer to measure the temperature and maybe we will come up with results like volume is 1.2 meter cube, mass is 20 kg, pressure is 6.9 bar, temperature is 32 degrees C. So what we have here, we have a list of relevant characteristics that is properties and its quantification. That means to each of these properties we have assigned a numerical value. That means we have quantified, a quantity is associated with it and be sure each quantity will have a numerical value and the unit of measurement associated with it. When it comes to solving problems, we will again emphasize this various units and their significance. So when we do this, we say that now we have defined state of our quantity. So we have defined state of our quantity. And while doing this, our third thing, so our definition was, definition 1 was system and along with that we understood the importance of boundaries. The second definition and third definition more or less went together, state of a system, property or properties of a system. Property of a system again emphasize is a relevant characteristic. Now let us discuss something more about the properties. Just now we have seen and for simple gaseous thing, we have listed out volume, mass, pressure, temperature as properties. Later on we will come across energy E, enthalpy H, entropy S, maybe kinetic energy, thermal energy. These are all typical properties that we come across in thermodynamics. So let us look at the classification of properties. You will notice that some of these properties volume, mass, pressure are defined in other branches of physics. That means these are primitives. So one way of classification is primitive properties. These are defined in other branches of physics and they are just inherited by thermodynamics, physics or chemistry or geometry whatever. So volume comes out of geometry, mass comes out of mechanics, pressure comes out of mechanics, fluid mechanics whatever and later on velocity, kinetic energy, potential energy, electrical energy all these things are primitive properties. Property can be primitive, a property can be basic thermodynamic. A basic thermodynamic property is a property which is defined using the laws of thermodynamics. We will soon see that the three laws of thermodynamics which we are going to study, zeroth, first and second. Each one of these laws will help us or will force on us a basic thermodynamic property. Zeroth law will define temperature, first law will define energy, second law will define entropy and these three and only these three are basic thermodynamic property. Out of which E has a special status because energy is defined not only in thermodynamics but in other branches of physics. So the thermodynamic part of this later to be known as the thermal energy you can consider as a basic thermodynamic property. But the fact remains that there are just three basic thermodynamic properties, one for every law of thermodynamics. Zeroth law helps us with temperature, first law helps us with energy, second law helps us with entropy and the third one is derived thermodynamic property. These are properties which are combination of properties and they are useful because these are useful short forms. Perhaps the most famous of this is enthalpy. We will find that the combination of property u plus pv comes up so often in thermodynamics particularly when we look at fluid systems or open thermodynamic systems that we find it useful to define one letter for it and call it enthalpy h. Later on you will even have for example the Gibbs function, this is also a derived thermodynamic property. There are properties which are common in other branches. For example cp sorry partial of h with respect to t at constant p similarly cv partial of u with respect to t and v. The so called specific heats and unfortunate name we will discuss that later. But these are derived thermodynamic properties and such derived properties are common in other branch of physics. For example stress strain are useful properties or useful characteristics in solid mechanics. Their ratio the Young's modulus is a derived property.