All right. Thank you. Good morning, everyone. So I create TLDR. We implemented a new, not a new, we implemented the symmetric encryption scheme called InterMAC. And this is why and how, and whether it's useful in practice. So this is our timeline for the next 25 minutes. I'll first go over some motivation. And then I will introduce you to InterMAC lib. And we will use InterMAC lib to implement a few InterMAC-based OpenSSH cyber modes and then measure the performance with respect to or compare the performance to existing OpenSSH cyber modes. So I use cyber modes, cyber modes as equivalent to a TLS cyber suit suite. All right. And then we're done. All right. So we all know that symmetric encryption is a hugely important tool for achieving, for example, confidentiality of data. But how do we model security of a symmetric encryption scheme? AE or authenticating encryption has emerged as the defective standard that symmetric encryption schemes should satisfy to be considered for practical deployment. But AE or other standard cryptographic security notions all have the same underlying assumption. They assume that the ciphertext, when it's sent to Bob, arrives in its entirety. And we also assume that the entire ciphertext is available when decrypting. But is this a good approximation? In 2002, Belay, Kona, and Pempe proved that CBC mode in SSH is secure. But then in 2009, Albrecht, Patterson, and Watson presented a plain six recovery attack against CBC mode. So this is kind of odd. So how did this happen? When data packets traverse networks or are being processed by applications, they're subject to fragmentation. So let's assume that Alice, there's a cloud here, but it's not shown. So let's assume that Alice sends the ciphertext to Bob. This ciphertext might arrive to Bob fragmented. It's then Bob's task to decrypt using the ciphertext fragments and somehow assemble the original plaintext. So we'll call this ciphertext fragmentation. Implementations must cater for this feature, and protocols like SSH contains a length field in a packet to assist the receiver in decrypting. And it's the presence of ciphertext fragmentation that allowed Albrecht, Patterson, and Watson to mount an attack against CBC mode. On the other hand, it was the lack of catering for ciphertext fragmentation in the model of Belay, Kona, and Pempe that made it possible for them to prove that CBC mode was secure. Luckily, there is an existing framework where you can study some major encryption schemes in the presence of ciphertext fragmentation. And this was first presented by Balduriva de Gabrieli Patterson and Stam in 2012. I won't go into the technical details about this framework, but instead I'll give you a short tour of the security notions you can define in it. So you all know the standard confidentiality and integrity notions as in CCA or in Ctext. This does not cater for ciphertext fragmentation. But in the framework of PDPS, you can define even stronger confidentiality and integrity notions. You can also define more elaborate notions. And I will talk about two here. The first one is boundary hiding. So let's assume that Alice encrypts two plaintexts and gets a ciphertext C and a ciphertext C prime. And she now sends these two ciphertexts to Balduriva. What Eve will see in the network is just a long string of bytes. The boundary hiding notion BHSFCSA formalizes Eve's ability to tell which part of this ciphertext of this long string of bytes is C and which part is C prime. Protocols such as SSH tries to achieve boundary hiding by, for example, encrypting a length field or having an option of adding a random amount of padding. The second notion is a notion that captures a class of denial of service attacks. And this is best explained by an example. So this diagram up here is the cryptographic processing of an open SSH packet using an encrypt then mic cipher mode. Using such a cipher mode, you cannot encrypt a length field. But this leaves an easy attack for Eve. Eve simply flips a bit, a certain bit in a packet length field, which indicates for the receiver that the packet is way larger than it actually is. This means that the receiver will wait or stall instead of received a lot of bytes, a kind of denial of service. So you can define a notion N does SFCFA that captures this kind of denial of service like attacks. The N, the notion is parameterized by an N that indicates the largest ciphertext that Eve can produce that is not detected as a denial of service. And to have a sensible security game, this N needs to be smaller than the maximum packet size often imposed by implementations. So it's these four notions that we want to consider in the BDPS framework. And it's these notions that we believe that the Mexican encryption schemes should satisfy but be considered for a practical deployment. So right-hand notions were defined by BDPS. They also defined the confidentiality notion, but a technical flaw was later fixed by Albreg de Gabelli, myself, and Panason in 2016. And we also introduced an integrity notion. So the question is now whether there exists a scheme that meets all these security notions simultaneously. And such a scheme exists, and it's called InterMAC. And it's constructed in the following way. So giving a message, first split the message into equal size chunks of size N minus 1, where we count in bytes. Then encode each chunk with a byte 0, except for the last chunk which is being encoded with a byte 1. Then encrypt each chunk that gives a ciphertext. And then compute a MAC over each ciphertext where we also include a chunk and a message counter. The chunk counter is incremented for each chunk, and it's being reset for each new message. And the message counter is incremented for each message. Then concatenate all the chunk ciphertext and MAC text to produce the final ciphertext. Decrypting is done chunk-wise and in an online manner. So when a receiver has received this first part, goes ahead checking the MAC, or verifying the MAC, then decrypting, and then inspecting the last byte. If the last byte is a 1, we have recovered the original plaintext. This was presented by BDPS, and as I said, it achieves all four security notions simultaneously. I won't bore you with the proof, but it sort of goes a bit of intuition. The intuition is pretty simple. For example, for boundary hiding, if the encryption scheme is $CPA, this means that it produces random looking ciphertext. And the MAC acts as a PRF. It should somehow be that these individual parts here look random, and then the whole thing also looks random. And we can also note that we have the knowledge of server security for each time we meet one of these MACs. All right, so the rest of the time, I will spend talking about how we implemented the scheme and we tested the security. Oh, sorry. And we measured its performance. But before implementing, we need to fix the limitation in the scheme. Because right now, the scheme only encrypts messages that are multiple of n minus 1. It's pretty simple to fix. We just add some padding. And the padding we apply is the following. If the last byte of the message is 0, we add 1s. If the last byte is not 0, we just add 0s. So pretty simple. To avoid having to add padding, even though the message is a multiple of n minus 1, we modify the encoding of the last chunk to indicate whether we need padding or not. We'd also do some kind of detective change and remove the encrypted MAC and introduce non-space-a-e schemes. When doing this change, you need to be conscious about correctly integrating or adding in the chunk and message counter, and also generating the nonce correctly. So given these modifications, we implemented InterMAC and created InterMAC Lib. So InterMAC Lib is a C implementation. And the overall aim was to make it safe to use. So safe to use is kind of a vague term. But we used it to mean that it should be very easy to use for a developer. And it should be hard for them to make any mistakes. So we have a very small API where you can just initialize an instance of InterMAC Lib, or you can encrypt or decrypt. And that's it. We also implemented complete nonce management. So a user cannot even input an nonce or cannot modify any nonces at all. We also have a feature of algorithm agility, which means that if you want to use a specific non-space-a-e scheme as the internal encryption function, you can implement that. And right now, we support ASTCM and Chachapulli as the internal A-e schemes. Let's look more closely at how we manage nonces. So when initializing an InterMAC Lib instance, the first thing I use in my stew, or one of the things you must do, is to choose a non-space-a-e scheme to use. This affects the way that nonces are handled. During initialization, we also initialize the chunk and message counters as zero. And we will use the chunk and message counters to generate a nonce. So the first part of the nonce is the 32 least significant bits of the chunk counter. And the second part is the 64 least significant bits of the message counter. And clearly, if the chunk counter and message counter are updated as I described earlier, the nonce will be different for every time we use it. Okay, so how do we do nonces when InterMAC is using Chachapulli? The Chachapulli A-e construction, close to follows, I would see 75, 39. And this construction needs a nonce that is 69, sorry, 96 bits long. Let's say the exact nonce we just created so we can directly use that. Also note that a block counter has nothing to do at all with InterMAC. It's something that is used internally in a Chachapulli. However, you must be very careful of updating it correctly, otherwise you lose all security. Okay, so how do we do nonces when InterMAC is using ASTCM? It's pretty simple, we just initialize ASTCM with the nonce for each chunk. This can maybe be pretty costly, but InterMAC uses OpenSSL to provide ASTCM support and we haven't really found a way to avoid this per chunk initialization. All right, so now we have a library that implements InterMAC. And we have a library that uses different internal nonce-based schemes. So we took the library and implemented InterMAC-based CypherModes in OpenSSH. And the reason for choosing OpenSSH is that we believe that by the design of the protocol actually wants the security notions I presented before. Well, currently none of the CypherModes actually satisfies them. At least not all of them simultaneously. So we implemented a song for them. And we implemented InterMAC-based CypherModes using both the AES, GCM internal AES scheme and the ChatterPoly internal AES scheme. So from now on, instead of saying InterMAC-based CypherModes I'll just refer to them as IMModes. And then here, the N indicates the chunk length and unfortunately, you cannot dynamically negotiate the internal nonce-based scheme or the chunk length when you do the handshake in SSH. So instead, you need to hard-code these into the code. There's two remarks I want to make about the implementation of these CypherModes. The first one is that when you use InterMAC actually the majority of the fields in the SSH packets becomes redundant. So maybe controversially, we removed these to save a bit of overhead. The second remark is that we implemented a standalone code path for InterMAC when it's being processed and it's doing this cryptographic processing. And the reason for that is that the packet processing in OpenSSH is quite complex and historically pretty buggy. And if you don't believe me, you can read this paper or this one or this one. That's probably so little more. Also, processing using InterMAC is quite a lot simpler than the normal CypherModes. All right, so now we have some InterMAC-based CypherModes. Now we want to see how they perform against existing modes. So we ran a test where we measured the throughput. We measured how many bytes, it's actually transmitted during the test. And we investigated how it is related to the chunk length parameter. And it's a bit of a methodology. So we wanted a somewhat realistic network setting. So we spun up an AWS instance in London and one in U.S. And we used SCP to transfer 100 megabyte file from London to U.S. And the first metric we looked at was how fast the CypherModes were to transfer this file or how much data that could send us again. The first thing I wanted to show you attention is the... Okay, you cannot see the rest of the one, but whatever. So the first CypherMode I wanted to show you attention is the bottom round here, which is AES encounter mode using HMAC and BeFi as the MAC. And this boots at a speed of approximately four and a half megabytes per second. And this compares pretty well to the best performing InterMAC IM CypherMode, which is using AES-TCM and then with a chunk length of 512. So the choice of this CypherMode might seem odd, but it's actually the most preferred CypherMode by OpenSSH servers on the Internet as we measured in 2016. And it's actually preferred by almost 60% of all OpenSSH servers. So our InterMAC CypherMode performs pretty well compared to this very popular CypherMode. The best performing CypherMode was the pure AES-TCM, which already existed in OpenSSH. But as you can see, the other IM CypherMode with AES-TCM actually performs somewhat okay compared to that. There's some different, there's a funny distribution in the performance though. If you look at the chunk length, or the mode with the chunk length 412.8 and 512 and maybe 4,096, it seems to relate to the chunk length. And this behavior is probably best described with the next chart, which shows how many bytes of CypherTakes is sent when we did this 100 megabyte transfer. So for the existing modes, there's quite a uniform amount of CypherTakes being sent. The story is very different for the IM modes. And the reason for this is the padding that we needed to apply and to make the scheme usable in practice or to allow arbitrary length encryptions. So the padding, the padding combined with the extra mag tags you need to include and the encoding actually means that we will send approximately between 10 and 35% more CypherTakes compared to the existing schemes. But the conclusion here is still that for this 100 megabyte file transfer, the IM mode still performed relatively well compared to the existing modes, but with a non-negligible expansion in the amount of CypherTakes. Even though there's the IM schemes actually used or had to apply more encryptions, I have to do more macking. All right, almost done. So I alluded you to the framework by BDPS, which provides a framework for starting to make some encryption schemes that cater for CypherTakes fragmentation. And we believe that this feature of CypherTakes fragmentation to be reflected in the security models. We then modified InterMag which satisfies all the desired security notions simultaneously and we implemented it and created InterMagLib. We then used our InterMagLib to implement IM-based CypherModes in OpenSSH, because we believe by its protocol design that it wants CypherModes that achieves the security notions. And our testing shows that you can achieve greater security by trading only a minor amount of performance. And furthermore, we believe that InterMag is the SSH packet protocol done right. Just to end with a bit of optimism, thank you for your attention. Questions? Thanks for your talk. So the 10 to 35% overhead is significant. I wonder how much of that is caused by the SCP application layer writing data not in the right multiple. And if you can fix that by having your library delay the actual encryption for a little bit to see if more data comes in. That's a good point. So the way you achieve the least amount of padding is when SCP, the data segment served by SCP to run online packet processing aligns well with the chunk length. That's right. But yeah, so this is a good example of when we design crypto, we should also consider what application is actually doing and maybe also what the network configurations are. Right. Yeah, thanks. Yeah, along those lines, did you measure the performance impact for interactive sessions where I'm just typing single byte characters? So I actually forgot to mention. These are only, we're a very small amount of tests and we plan to do more comprehensive tests and one of them is definitely to do the session based one where we hope to show that you can achieve greater security by an acceptable amount of performance sets that is not observable by user. Thanks. Any other questions? Okay, let's thank the speaker again. Thank you.


This is the man of the hour the Titans new GM and we have to start with the most important part I say because I watched your dad play in the USFL Maurice Carthon I say Carthon because that's how they said it on TV. Is that your preferred pronunciation? We can yes, that's how I pronounce it my mom pronounced it Carthon Everybody has their own that's how my mom said it would you got hired? She said Carthon. I said I think it's Carthon So I use Carthon. I use Carthon. He uses Carthon Everybody has their own version of it, but we'll roll with Carthon Titans fans say Carthon. You're doing just fine All right, so when you're looking for players, I want to throw the stat at you about your dad. Okay, so in 1985 He played 22 games for the New Jersey generals. You know this yes He played 22 games for the New Jersey generals then he walked 65 steps from the generals locker room at the Meadowlands to the Giants locker room at the Meadowlands and Then he played 23 games for the Giants So in the space of less than a year your dad played in 45 games Can you find us some players like that? I don't think those exist Thank goodness, right time limitations and but no he told me that story and I was I was unaware of that story he told me that Man probably about eight years ago He told me that story. I think he had a two-week window from when the USFL season ended versus when he signed with the Giants He told me legitimately he almost retired after you know following that season just because of the toll it took on his body And especially as a lead blocking, you know fullback who didn't touch the ball He just was a batter-in-round was an extension of the old line But yeah, that's a that's a crazy stat. What's the main trait you think you take from your dad And you carry on into your life to this day work at it. That's the first income of mine I tell people I'm an early riser got that from my dad like my my biological clock no matter what coast I'm on my I'm up by 447 and it's specifically 447 and that's just my biological clock My dad when he was coaching he was in the office by 430 You know so you know my dad worked extremely hard and that's that was him all the way through and you know I mean ultimately he had a he had a stroke in 2014 new year's day And that was a lot of you know stress from the way he worked and not allowing his body to regenerate But you know the way I'm hard-wired comes from him to you know, I'll be in here early That's just who I am. I get my best works done then but that's the quality.  I'm a GM in the NFL as of today. So it's funny my dad and I we were talking about it last night when he got in town He was like you told me the news so calm. He was like, I was just waiting on more I didn't know if you were serious and I kind of got that consistently you my wife didn't take me serious But you know like I've said before I was in the middle of an airport terminal You know so I couldn't be too excited and run I you have this vision Of you know when I get that call that I'm going to be named the GM of a place like I'm It's like draft day where you see the guys jump up and family suck. I'm in a terminal with strangers You know and so I just tried to compose myself as much as possible He was just giving me a hard time last night that I was just way too calm Found a couple quotes from you from 2012.  I'm all about winning Oh 100% even more so than in 2012 I mean now in 2012 I'm assuming at the time I was with the Rams I may have just gotten hired by the Rams So I'm playing a key part in Les's operation Um, and I tell guys in the office all the time whether you agree or disagree with the decision Let's just use uh less as example since we're talking we're here to execute Les's vision Right, so I'm just a soldier in Les's army and now I get to be the general You know of an army so it's really more it's important You know like I said to uh to win, you know, and it's even more important now that I'm the general. All right, here's another one I never run away from competition. I'm a very competitive guy I'm feeling like that came from my college days um And I know the exact time we had a log jam in the backfield and I'm not afraid of competition.  All right, you're an sec guy You're back in sec country It feels like that was part of the allure of this job was to get back here Where you know football just means a little bit more no matter what level it's at Yeah, it was it's super important and I'm gonna have to be careful find my gator flag Every home that we've lived in we have Something gator oriented in the front of our home to let everybody know we're gators now. We have a A decal on the garage. I got a little gator holding a football on the front step. I remember living in Atlanta I had a gator flag after every sec game Usually my flag was in the yard because somebody came and took it down, you know It's really huge, you know, and even from an outside of football's Perspective my wife is from Miami and it just allows us to get back on this side and be closer to family You know, our parents are getting up there in age. So it's more important You know spend more time with them and you know being here in this location allows us to do so When you finished your playing career, you went back and finished your degree and then you said I'm not going to follow my dad into coaching which is what most former players do You said I want to work in personnel. Why did personnel have that kind of intrigue for you? So what people don't know is my dad's Initial thing was he wanted to do personnel. Okay, so Parcells called him. I think this was 94 95 And offered him a job and my dad was like, okay, cool But I want to do personnel and bills like if you know bill know you're coaching and once he stamps that that's what it is And so my dad always steered me, you know, it's a personnel. So when I was in college pro day circuit comes I get a call. Hey coach a is coming. You need to meet him at the office at six You need to watch film with him. You need to hold his clipboard You need to clock his 40s. You need to and you need to talk to him about your teammates And so that was kind of ingrained in me early. And so when I was done I actually coached high school ball one year in Miami in 2007 And to see if I if I had that in me and I I learned after two weeks that coaching wasn't for me But it was a good experience to have and then once I went back to finish my degree I kind of just you know went all charges ahead You went through some stuff as a player. You got hurt in high school.  You don't get drafted. You fight your way into the nfl. You're cut multiple times the journey being I don't want to say hard, but I mean certainly there were obstacles How much of that do you carry into this job? And do you feel like Helps you in the personnel world and now as a general manager? I think it allows What these young men are going through? I think it brings a A human element to it because I've been through it my one of my mentors back home bill spots would You know his family owned a hotel and at the age of 14 He knew that that's what he wanted to do. So his father made him work Every job in the hotel until he eventually took over. So that's how I see my journey, right? I've gone through all these multiple steps And now i'm here at the pinnacle of you know, what it is we do and so I've worked every job in the building So now that allows me to identify with every scout. I've been in the locker room. I've been cut multiple times So now I can identify with those guys in the locker room. So that's kind of how I see my journey Alignment in the organization. You said that's the most important lesson you've learned that successful Organizations have alignment between personnel and coaching when you talked about that today at the press conference. I totally believed you Because you could you you sense That having seen it successfully work that it can be carried over, right?  They have a way that they want to do things and it's our job as You know personnel staff to go find those players, you know miss. Amy and the crew from the interview You know, I have this quote and they've heard me say it Hunting at the same time isn't the same as hunting together And that's what I want to do here. I want our guys to hunt together So we can go out and bring home You know the the meal so to speak and allow us to be successful One way that you brought home the meal is in san francisco You identified a lot of free agents that have fit well Into the 49ers world whether they be big name or not Why has that been a strength and and what do you feel about that? That you hope you can do here the same way again, it goes back to our staffs being married You know in the collaboration of the staff knowing what it's going to take for us to be successful Knowing what our coaches want, but then at the same time knowing the right people to bring into our culture You know one thing I can say if you look at the roster that we built In san francisco, we didn't have a lot of bad character guys We didn't have a lot of guys facing disciplinary actions because we were intentional About the type of person that we wanted to come in and fit our culture. All right last one You have been so close to winning a super bowl as a player And and as a personnel man in this league.  Yeah, that was one of the things uh, I spoke with uh the intergroup uh interview group about Is you know, it would be their first lombardy and my first lombardy and after we lost the super bowl in 2019 You know obviously devastated. I'm talking to my wife about it and my wife kind of brought up the point You know, maybe it's not your time You know, maybe your time is when you're in charge somewhere. Maybe that's your time and I truly, you know, believe You know by the grace and mercy of the most high that You know that the time will come with me in charge and specifically here with this organization We're looking forward to the ride whatever it is. Thank you so much. Thank you red carthon We're saying it proper carthon carthon. We're glad he's here. Thanks very much


Good morning. So, in the last lecture, we looked at different steps involved in catalysis and based on these steps, we wrote down the equations and synthesized the rate equation, the overall rate equation depending on which step is rate controlling. So, it can be a chemical reaction which controls the rate or it can be the adsorption or it can be the desorption. So, there are three distinct steps which are nothing but adsorption, chemical reaction, surface reaction rather and the third one is desorption. And we have written down an algorithm as to which one of these steps like controls overall rate and based on that how to derive the rate equation. Now, the adsorption step can be controlling, it can be a reaction that can control overall rate or it can be the desorption that can control overall rate. And the rate equation that we get is of the form say r is equal to some rate constants into concentrations, the bulk concentrations, it can be K dash C B for a reaction A in equilibrium with B. And in the denominator, we are going to get an expression something like this. This is one particular rate equation depending on what assumption I make probably the rate of reaction on the surface to be controlling. So, this is one rate equation. I am just showing you the form the rate equation. It can be slightly different depending on which step controls the overall rate. So, if you just closely observe this particular rate equation what you see is in the numerator you have something similar to what you would have otherwise got when the reaction was not catalyzed by the solid catalyst. That means it was a normal reaction taking place in a homogeneous medium. So, this is an additional thing that comes in and what do you have here? These are the adsorption constants or probably desorption whatever something to do with the adsorption and desorption. So, the denominator that you are seeing here is because of the adsorption and desorption of the species or the components involved in the reaction because there is a competition for the empty or active sides for the reaction to take place. So, in a way these adsorbed sides are going to or the already occupied sides are going to be unavailable for the reaction to take place. So, this is how we interpret the overall rate equation that we get. But then of course, this may change depending on the assumption that we make which step is the rate controlling step. So, we looked at different steps to arrive at this particular rate equation how to synthesize this rate equation. But that is not enough because once we get this rate equation what is the guarantee that the rate equation is valid. So, in order to know whether the assumption that we have made are correct or not or the equation that we have got is right or not as I told in the last lecture we need to look at the experimental data. We need to generate experimental data in laboratory and see whether the data fits very well in this rate equation. If it does not then we have to go back and make another assumption and come up with different rate equation and see whether the data fits or data falls in line with the rate equation that we have got. So, in this today's lecture what we are going to do is like we are going to look at a simple example and we are going to look at how experimental data can be used to see the authenticity or validity rather of the rate equation that we are talking about. So, for that we are going to consider one example. So, let us go for a real example where like for example let us say for instance we have a dehydrogenation reaction say cyclohexane hydrogenates to give benzene and hydrogen. Now, this is cyclohexane let me call this as C benzene B and of course hydrogen. So, I am going to denote my concentrations with C B and H 2 as I mean I go ahead. So, let us first synthesize the rate law with certain assumption and then we look at a possible experimental data and see how we select finally the rate equation. So, now I do not want to really go through each and every step quite systematically like what we have done before, but quickly come up with the rate equations for three different cases when first the adsorption controls, secondly the reaction controls and the third is the desorption control. So, let me first start with adsorption controlling before that let me write down the mechanism. So, I have C plus S giving C S adsorption then C S gives B S plus hydrogen. Now, it is very important step look at a way I have made an assumption here chemical reaction. What is happening here is the hydrogen is generated, but it is coming out immediately it is not occupying the sides. One can make some different assumption here saying that C S plus S gives B S plus H 2 S that means H is H 2 or hydrogen is on the in the adsorbed condition. Right now I am making an assumption which is quite similar to a layer ideal assumption where one of the components is not in adsorbed state it does not have affinity for the catalyst as far as adsorption is concerned and as and when it is formed it goes to the bulk. So, that is the meaning of this particular step chemical reaction. So, this particular component is always present in bulk there is no adsorbed hydrogen. Now, you may ask me why I have made this assumption right now there is no basis for this I have just made this assumption I am going to go ahead and see whether this is valid or not based on the experiments. If it is not valid then I will come back and make some other assumption. So, all possibilities are there, but I am just trying to tell you the meaning of or significance of this particular step that is chemical reaction and then the desorption that is B S gives B plus S right. Now, you know how to derive the rate equation if one of the steps is rate controlling. So, in the last lecture we looked at simple homogeneous sorry not homogeneous simple isomerization reaction A giving B reversible and we said that chemical reaction was controlling and we looked at the rate equation. Today let us derive the rate equation for adsorption controlling and then quickly write equations for the chemical reaction and desorption and then look at the validity of these equations based on the experimental data. So, first we will spend some time deriving the equation for the case when adsorption controls. So, C plus S giving C S alright. So, rate of adsorption is equal to overall rate that is my assumption rate determining step rate controlling step slowest step is equal to k adsorption right into C C into C S minus k A dash C S C S right. So, this is the rate equation that I have got and depending on the stoichiometric coefficient I will have a sign here like what you do in writing any normal rate equation. So, I am writing a general rate equation here. So, let us go ahead now in this particular equation we have bulk concentration C C, but apart from bulk concentration we have empty sites concentration vacant site concentrations and a concentration of adsorption species and these are the concentration which are unknown and they will come from equilibrium steps. Now, what are the equilibrium steps here the first step is of course the reaction now reaction is not the slowest step like what we did before in this particular case adsorption is slowest step desorption and reaction they are they are equilibrium control or they are at equilibrium instantaneous. So, the reaction is C S giving B S plus hydrogen. So, adsorption controls the overall rate and these concentration which are C S and C C S which are some which are the concentrations we do not know they will be obtained from the equilibrium state and which are the equilibrium steps here the reaction chemical reaction which is now instantaneous earlier in our earlier example we consider reaction to be controlling here adsorption controls. So, reaction is instantaneous which is at equilibrium. So, the reaction and desorption these two steps are going to give me this bulk sorry the adsorption species concentrations and empty site concentrations. So, this is reaction equation C S in equilibrium with B S plus hydrogen. Now, if you write equilibrium equation for this it is K e q capital K e q is equal to C B S C H 2 divided by C C S. Now, we can express C B S in terms of C C S here. So, let me just write equation 1 here I am going to use equation 1 later to get value of C C S. Similarly, I write desorption which is again an instantaneous process where you have B S that is benzene is going to get desorbed. So, this is adsorbed benzene benzene in bulk and empty site. So, again same thing I can write K capital K D is equal to C B C S divided by C B S. So, C B S is equal to capital K D dash C B C S equation 2. Now, using equation 1 and equation 2 equation 1 and equation 2 I am going to substitute for C B S and C C S in the main equation that is obtained by adsorption to be the rate controlling step right. So, what I need is C C S. Now, how do I get C C S from this equation C C S is equal to some constant into C B S C H 2 right and from equation 2 you have C B S is equal to this. So, what it means is C C S is equal to again K E Q dash K D dash C B C H 2 C S. So, I have got the expression for C C S which is to be substituted in the main rate equation the overall rate of the reaction this is the one C C C this is where I need to substitute for it. So, if I do that what I get is R 0 that is overall rate is equal to rate of adsorption is equal to K A C C C S minus K A dash K E Q dash K D dash C B C H 2 C S. Now, in this I do not know only C S rest all our bulk concentrations. By the way now it all depends on what kind of reaction that I am talking about whether it is a gas phase reaction or liquid phase reaction for the gas phase reaction normally the concentration is expressed in terms of partial pressures if for the liquid phase reaction it is the concentration. So, right now I am just expressing all these in terms of concentrations, but if you are talking about gas phase reaction you can convert it to partial pressure. So, P by R T is equal to concentration fine all right. Now, you have this equation further site balance C S plus C C S plus C B S cyclohexane adsorbed benzene adsorbed is equal to C T. Now, C S what is C C S? C C S is equal to and C B S is equal to which we have already derived is equal to C T right. So, C S is equal to C T divided by 1 plus let me write again some constant here C B C H 2 plus K B C B right. So, this is I have just clubbed all this in digital constants in K double dash right. So, I have got expression for C S look at expression slightly different look at this term which was not there before. Now, it is adsorption controlling that is why you have different terms appearing in the denominator try and observe the differences. Now, I have got the expression for C S I will substitute for C S in the main rate equation that I have got. So, R O is equal to K forward. Now, I am just giving you the final form you can because this is all of obtained by clubbing all those constants right this K F which is function of K A and C T and so on. C C that is concentration of cyclohexane minus K F dash or one can say K reverse K backward K B rather C B C H 2 divided by 1 plus right. So, this is the final expression when adsorption controls right adsorption controls adsorption is a rate determining step adsorption is a rate determining step. Now, this is one expression that I have got suppose I do the same exercise, but then now with reaction to be controlling reaction to be controlling then I will get a different rate equation then I will get a different rate equation and you can do this exercise on your own I am going to do it here I will just write the rate equation the form of the rate equation it is going to be R overall is equal to R reaction is equal to sum K into C C minus oh sorry C C minus K dash C B into C hydrogen divided by it is quite similar to what we have done before K C C C plus K B C B this is when the reaction controls. So, look at a difference this was when adsorption is controlling look at a difference the numerator is not changing much, but the denominator there is a difference there is a difference and we have to exploit this difference to know based on the experimental data which one is the right equation, but it is not yet over now I am going to look at the case when desorption controls case when desorption controls R o is equal to R d is equal to. So, desorption controlling R 0 is equal to R d is equal to sum K C C again the numerator look at this numerator is not changing C B into C hydrogen divided by I will have a very peculiar expression here you have C hydrogen plus K dash C C plus K double dash C C and C hydrogen. Now, you can derive this equation on your own I am just writing the final form the equation because I am going to make use of this equation later to see whether this is the right equation based on the experimental data or not. So, this is when desorption controls. So, I have three expressions when these three different steps are controlling steps the first one is when adsorption controls that is this equation then reaction controls this particular equation and when desorption controls this equation. These are three different expressions that I have got now I want to know whether which equation is valid in a real situation. So, how do I know this as I said before I need to do experiments in laboratory. Now, think of a situation you want to do experiments in laboratory for a solid catalyzed reaction you have cyclohexane getting converted to benzene and hydrogen. So, let us say it is a vapor phase reaction I am using any hydrogenation catalyst say nickel adsorbed on alumina whatever say palladium adsorbed on alumina as a support right the polar support and reaction takes place right. Now, how do I do the experiments in laboratory how do I get a rate value for the rate they are different types of reactors which are used in laboratory for solid catalyzed reactions. So, it is quite similar to what we have already learned for simple reactions normal reactions here we have a differential reactor. So, for example, you have a tube in which you have packed the catalyst and a flow is taking place. Now, this particular length is very small what does it mean that means there are no gradients along the length. So, it is like a differential that is why it is called a differential reactor though it is a tubular reactor in that the catalyst volume that I am I am using is very small. So, that the concentration does not change drastically. So, if that particular volume acts like a CSTR where the concentration is uniform the it has an advantage because it is going to get me the rate equation for the particular concentration the concentration change is not much. So, I can assume the concentration to be the feed concentration at which the reaction is taking place anyway that will be clearer later. So, this is a tubular reactor with difference or one can have a stored reactor a continuous reactor one can have a fluidized bed reactor. But of course, typically like such reactors are not so common in laboratory this is the most common reactor used in laboratory one can have. So, if so much you are dealing with liquid phase the reactants and products are in liquid phase where the catalyst is solid in that case again I will have a slurry reactor similar reactor. Now, these are continuous, but I I may have a batch reactor right and, but then I will have a concerned performance equation for the batch reactor and can evaluate the value of rate can evaluate the value of rate. So, this is how I can perform experiment in laboratory for solid catalyzed reaction. Now, in all these reactors I can use solid quite comfortably. In this case in the fluidized bed case we have to make sure that this particular flow rate should be good enough or high enough to fluidize the catalyst. So, all these reactors are going to give you some performance based on which you can calculate the rate the fluidized bed and the CSTR type of slurry reactors are integral reactors. And by differentiating the conversion versus catalyst loading plot we can obtain the instantaneous rates and can calculate the reaction rate constants. Also it should be noted that the differential reactor has a limitation as I said before that the conversion should be as small as possible. So, if the conversion goes beyond a particular limit then probably the differential reactor will not work like a differential reactor and the analysis will be incorrect. Now, let us get back to our example of cyclohexane dehydrogenation to give benzene and hydrogen. So, in this case suppose I am using a tubular reactor here cyclohexane will go inside and a mixture of cyclohexane plus benzene plus hydrogen will come out. You may use some inert also here of course, which inert is to be used and all will look at it because inert may have some effect inert may get adsorbed on the catalyst surface we will look at it separately. But then we can do this experiment here to determine the rate. So, let us say I determine the rate for this particular reactor by changing the concentration of cyclohexane in a field. Now, it is not pure cyclohexane it you have a inert and let us make an assumption that inert does not adsorbed on the surface it is like hydrogen. Hydrogen of course is not inert here it is a product, but like hydrogen there is another inert say argon or say helium or nitrogen for that matter which does not have any affinity for the catalyst. So, what I am doing is in this particular reactor I am passing cyclohexane along with that I have some inert also going inside. So, it gives me an opportunity to play with the concentration of cyclohexane. So, I can do or I can perform experiments at different concentrations of cyclohexane in the field and get a rate for every concentration. So, if I do this experiment if I do this experiment what I get is at different partial pressures or different concentrations of C C what is the rate? What is the rate of the reaction? This is the rate that I observe this is the rate that that is the overall rate that we have derived the rate equation for and this particular rate should fit very well in the rate equation that I am going to get. Now, C C verses are overall are observed let us let us say I am getting a train like this I am getting a train like this this is the experimental data. Now, I need to know whether any of those equations that I have derived falls in line with this experimental data or not. So, what is that nature here? Initially it is a bit linear and then the slope is decreasing and it is getting saturated after certain concentration of cyclohexane in the field which is sufficiently high there is no change in the rate it is almost becoming constant. So, that is the general observation that I have. So, this is the experimental data observed I am going to get back to this I am going to get back let us observe our earlier rate equations that I have derived. Now, I have the equation when adsorption controls what do I see here? Even if I change the concentration in the feed what is the rate? Now, there is a finite value for this whatever is the concentration that I am doing the experiment with. So, there is some finite value here what is the feed concentration of benzene? Zero reaction is taking place at the feed concentration remember what I said the concentration is not changing much here. So, whatever rate that I am calculating here is at the feed concentration. So, suppose the concentration of cyclohexane entering is 50 percent the extent of reaction is not much it is going to come out with say 48, 49 percent concentration of cyclohexane. Some benzene will form some hydrogen will form which will be very small depending on the extent of reaction, but this extent is small. So, that for all practical purposes I can assume that the reaction takes place at a concentration corresponding to the feed concentration. This is a very important assumption and it is valid provided this catalyst length that is length that I am using is relatively small. So, what happens here? C c is some finite value say 50 percent concentration units of course, C b is negligible 0 C h 2 0 C b 0 C h 2 0. What does it mean? R 0 is equal to k f C c and what does it say? C b is equal to it says that the rate is directly proportional to the concentration of cyclohexane. So, what train will I get? I will get a train which is linear with respect to cyclohexane concentration in the feed. So, it is going to be like this C c versus R overall when adsorption controls this is a train that I would get. Is that a train that I have got in the experiment? No, my train was like this. So, what does it mean? It means that the rate equation that I have got by assuming that adsorption to be controlling is not valid here is not valid here. So, let us look at desorption controlling now. Let us look at the rate equation when we get when we assume that the desorption controls the overall rate. So, what is the rate equation that I have got? This is the desorption rate equation. What do I see here? C c is finite C b is 0 I do not have benzene in the feed I do not have hydrogen in the feed this is also 0 this is 0 this is 0 this is finite. So, R 0 or R observed overall is equal to k divided by k dash because C by C c by C c that we get cancelled. So, rate is constant in the desorption controls rate is independent of C c is a feed is pure cyclo pure in the sense there are no products present it will be only cyclohexane in some inert medium. So, if I change C c and observe the rate and in desorption controls what is the train that I expect? I expect that. So, there is no change when desorption controls I hope this is clear as the concentration changes the rate is unaffected because of this because of this or rather this it is a desorption of benzene that is controlling. So, the rate does not depend on cyclohexane concentration at all. So, is that my observation my fixed bed reactor or other that tubular reactor differential reactor that I am using what is the observation there? I am seeing that the rate changes in this fashion and what I should get is this not true or does not fall in line. So, what it means is from the experimental data that I have got neither that sorption is controlling alone nor the desorption is controlling alone. So, let us look at the reaction now or let us look at the expression when the reaction controls the overall performance or reaction is the slowest step. So, what is the expression that we have got the expression that we have got is this when the reaction controls. In this particular case again for the feed when there is no benzene and hydrogen present this term becomes 0 this term becomes 0. This term becomes 0, but then I get rate equation r overall is equal to k c c divided by 1 plus k c c c. Now, this is a very typical rate equation for very small values of c c when c c is very small r 0 is this is very small r 0 is k into c c. For small values of c c r 0 is proportional to c c is proportional to c c and what happens if c c is very large if c c is very large then this becomes very large compared to 1. So, r 0 for c c to be very large r 0 is constant the very typical rate equation that we have got where in the nature would change depending on the concentration of cyclohexane in the field at very low concentrations the rate is proportional to c c and at large concentrations the rate remains constant or it attains saturation it attains saturation. So, how will it look like on the graph of c c versus rate it looks like this c c versus rate overall or observed it is almost constant here it is linear here and there will be some range where it is neither linear nor constant and this is when it is a reaction surface reaction controlling the overall rate and remember the data experimental data that I had shown you before falls in line with what one would get by assuming surface reaction to be controlling. So, it is like you have initially linear and then going to saturation. So, what it means is the rate equation in this particular case for this experimental data is the rate equation when chemical reaction the surface chemical reaction controls overall rate at least the nature is quite similar. Now what we do now later now nature is similar means I will try and fit this particular model or this particular rate equation in the data that I have got and get the values of parameters. Now which are these parameters the parameters are adsorption constants rate constants. So, I get a rate constants I get adsorption constants by fitting this data. So, if you have this particular equation. So, this is something that I obtain for certain values of the parameters. Now which are these parameters k the adsorption constants then k f k backward or k f dash whatever all these are rate constants these two and these are adsorption constants. If the surface reaction controls the overall rate then the k by k dash is related to the overall equilibrium constant of the reaction and that can be determined independently. So, the unknown parameters are only three those are k c k b and small k that is the forward reaction rate constant. So, these will be determined such a way that my data fits well in this particular equation. So, once you do that if the data fits well over a large range of c c then you have got a right rate equation. And this data fitting is normally done by the many methods of regression is something called as least square method where what we do is you have the experimental data you have the experimental data and you have the curve for certain values of parameters k k f k f dash right. And for every point there is some difference we try and minimize this difference. And there is a function called summation of the square of these errors why square because sometimes it is quite possible that this difference is positive at one point this difference is negative at certain point. So, if we just go on adding the differences they may compensate for each other and say that find the overall total error is 0 but that is not correct. So, we have to square it so that the magnitude is always positive. So, that I get a better best fit fine. So, that is nothing but parameter estimation or regression analysis it is called as. So, least square method is the best method to or that is a commonly used method to estimate the parameters. And there are mathematical techniques optimization techniques which are used to calculate these values of parameters. Now, after doing all this does it mean that you have got a right right equation which will work very well. Now, there is one thing that we are not really looked at what is that you have just looked at the effect of cyclohexane concentration in the feed. We are not looked at the effect of benzene concentration we are not looked at the effect of hydrogen concentration. There are some constant associated with benzene adsorption right in the denominator you would be able to determine them only if you have benzene present in the feed. So, after doing all this you have to generate more data by doing some experiments with either hydrogen in the feed benzene in the feed with certain proportion not negligible now or both of them in the feed. So, doing all this you generate as much data as possible and see whether your rate equation fits or rather it falls in line with the experimental data and that is the complete exercise. Now, I have just shown you three different rate equations here and said that fine possibly the reaction controlling mechanism may work, but in real situation things are likely to be more complicated where I need to make certain other assumptions to get different types of rate equations. As I said before now I am assuming a single site adsorption they can be dual site adsorption I am assuming hydrogen to be present in bulk only, but valadium may have affinity for hydrogen and it has. So, I cannot just make that assumption. So, I have to assume that there is another adsorption step that is hydrogen. So, hydrogen is first present in adsorption state and then there is a desorption from not just benzene, but hydrogen as well. So, it will become more and more complicated, but then if you want the right rate equation you have to bear with this complications and finally when you get a right equation then only you can use it for the reactor design. Because ultimate aim is to have a proper reactor design for the given reaction of interest and that will be valid over the range of concentration that I am studying. Before we go ahead to the reactor design let us look at the effect of temperature. Now, I am not talked about temperature at all. Now, let us look at the rate equation that we get when the reaction controls the overall rate. Now, this is the reaction rate. Now, in this rate expression I have several parameters. This is one parameter, this is another parameter and these are adsorption related parameters. Now, how will the temperature affect these parameters? Now, k is the rate constant, k dash is the rate constant for the reversible reaction. Now, these two are likely to be highly sensitive to temperature. Of course, it will have adsorption constant also, but it has the rate constant or these two will have the rate constants as well in them. See, I have got this by clubbing or other putting together many constants. There is the capacity C t also coming in picture, but in that if you remember you have forward rate constant and backward rate constant also in this and those are the constants which are highly sensitive to temperature. That is why these constants are highly sensitive to temperature. What about these constants? These are only based on adsorption. There is no rate constant here. So, adsorption constants relate you to or compare to rate constants. These two are less sensitive to temperature, less sensitive to temperature, but it is quite likely that because of temperature or high temperature, adsorption will get affected. Now, you all know that if I increase the temperature, adsorption is less favored and desorption is favored. So, the adsorption constants will go down and you may make an assumption that at very high temperature, the denominator which is related to adsorption, these two terms are negligible. What happens? If these two terms are negligible, then you have in a denominator just one. So, this boils down to a normal rate equation which is similar to homogeneous catalysis. So, at high temperature you may see such effects. You can even change the temperature and generate data at different temperatures to get the hidden parameters here, which are the hidden parameters activation energy, frequency factor and so on. Right now, the parameters that I was talking about was only like forward rate constant, backward rate constant and adsorption constant, but then if you generate a data at different temperatures, you can get the activation energies also. Something that we have already done at different temperatures, you have 1 by T versus l n k or l n r 0 that is initial rate. You have a plot like this. The slope gives you the activation energy and the intercept gives you frequency factor, where the k that is the rate constant is nothing but A into exponential minus E by or delta E by r T. So, I can get activation energy and frequency factor for both the rate constants forward and backwards. Now, both the examples that we have considered so far, first isomerization and now dehydrogenation, both the examples have considered the reactions to be reversible, but not necessary that the reaction has to be reversible. The reaction can be reversible as well. So, the same expression also the same methodology or procedure can be applied to irreversible reactions and one can obtain the rate equation. You can do that as a homework. Just take an example of A going to B, but now it is not reversible. It is just irreversible reaction. Fine. So, you have a rate equation. After we get a rate equation, what we do? We need to design the reactor. So, I have told you the procedure to get a rate equation. Synthesize the rate expression for certain assumptions and then validate it against the experimental data. This is the entire procedure that I need to perform in laboratory before I design the chemical reactor. And remember, we have not considered the external mass transfer effects. We have not considered the intraparticle diffusion that will come later anyway. But right now I am just assuming that these are negligible and I have the rate equation. Now, how do I design a reactor? Now, the procedure is not much different from what we have done before. What do I do? Earlier I have a CSTR. I have a CSTR.  You have F A 0, F A here. And then in R A, I have minus K C A for irreversible first order reaction A going to B. And by knowing F A 0, for the required conversion, I get a volume. That is my reactor design. Or for the given volume, I calculate the outlet concentration that gives me the conversion. So, this is the exercise that I do. Now, what is going to change for heterogeneous reactions?  And in most of the cases, instead of V, we have W equal to 0. Now, what is W? Here it is the volume. Here it is the weight of the catalyst. Why? Because the rate now is minus possibly K C A divided by 1 plus K A C A plus K B C B something like this. If the reaction is reversible, you will have one more term here. So, instead of this, I have this. This is one difference. Another difference is instead of volume, I have W here. Why? Because the rate constant here is per unit weight of the catalyst. The rate constant here is per unit volume of the reactor. Here the rate constant is per unit weight of the catalyst. So, this is the only difference. Otherwise, the entire exercise that I have done, what are problems I have solved for homogenous reactor? Similar procedure will be adopted for heterogeneous reaction. Remember, volume is replaced by the weight of the catalyst and the normal rate equation which is probably of the power lot type is replaced by either Langmuir-Wood or Euler area, whatever mechanism depending on whatever assumption I make. It is slightly complicated rate equation, but the methodology or procedure remains same. Thank you.


Seeking our lady's intercession, let us pray. Hail Mary, full of grace, the Lord is with thee. Blessed art thou among women, and blessed is the fruit of thy womb, Jesus. Holy Mary, mother of God, pray for us sinners. Nound at the hour of our death, amen. Good morning, brothers and sisters. So on Wednesday, Ash Wednesday, this past week, I began what would be ultimately a seven part series of homilies on the last words of Jesus. The last words of Jesus are not the seven last individual words of our Lord, but the seven last statements that he made while he was dying on the cross. And the church has always recognized that these words are probably the most significant in the entire Bible. For many reasons, they were the last words of the gospel before his death, and they cost him so much to say. I'll remind you again that crucifixion, being the cruelest of deaths, the cruelest of tortures, did not kill somebody through bleeding them to death, right? They died of asphyxiation, which means they basically choked on their own spittle. The reason was being nailed to the cross by their wrists and feet, they could not breathe unless they pulled up on their hands and pushed up on their feet. That meant they would cause themselves great agony just to get one breath, and they had to do this for as long as they were up on that cross. And it would usually last days, if not weeks, depending on the strength of the individual. Now if it was that painful just to take a breath, imagine how much more painful it would have been to get enough breath to speak. So it cost our Lord so much to say these last things to his people before he died, which means each one of them is very precious. On Wednesday, I spoke to you about the first of his seven last words. Father, forgive them, they know not what they do. And if you want, you can go back and listen to that sermon, that homily, I'm sure it was recorded and put on the website or something. But today we hear the second statement Jesus made while dying on his cross. Amen, I say to you, today you will be with me in paradise. Again, these are some of the most consoling words from the cross. Amen, I say to you, today you will be with me in paradise. I think each one of us as believers in Christ can say that we want these words to apply to us. We want Jesus to say these words to us, especially when we're nearing death. On our deathbed, don't worry, I pronounce to you, today you will be with me in paradise. What consoling words. And whatever comes after that point, you know you won't be worried or afraid because you are assured of your salvation. But to understand these words of our Lord, we have to understand to whom he spoke them because Jesus did not speak them to everyone present at his crucifixion. He certainly wasn't speaking to the crowds, not even his own mother, technically. He was speaking to Saint Dismas, the good thief we call him. You will call that Jesus was crucified with two others, two thieves or revolutionaries, one on his right, the other on his left. And these criminals were having a little debate, so to speak. The first one turns to Jesus and says to him, if you are the Messiah, if you are the Christ, come down from that cross and take us with you. And then Dismas turns to the other thief and says, you and I deserve to be up here, but this man is innocent. And then he says to Jesus, Lord, remember me when you come into your kingdom.  The fathers of the church tell us that these two thieves represent every human being on earth. You are either one or the other. There's no third option. You fit into one of those two categories. And so it's important that we understand them, how they're similar and how they're different, and why Jesus gave such a consoling message to one and spoke nothing to the other. Now we don't know what happened to the bad thief. We don't know if he went to hell or he got to purgatory. We don't know. The gospel doesn't tell us, but we do know the good thief went to heaven that day. So let's look at their similarities first because they have a lot in common. They're both public sinners and criminals. They both committed the same offenses in society and against God. They're both being punished in the exact same way for those offenses. They both know what they did and they know that they deserve what they're getting. So up to this point, there's little to no difference between the two men. The only thing we know that's different between them is what is revealed by their words. You see the bad thief, the one whose name we will never know, he reveals his mind and heart to us in what he says to the Lord. He says to him, if you are the Christ, if you are the Messiah, come down from your cross and take us with you.   He knows he deserves to be punished, but he doesn't wanna suffer for his crime. He refuses to take responsibility for what he has done. He doesn't believe in Jesus as the Messiah. If he did, he'd understand that the Christ had to come and suffer and die. He wants to avoid suffering by any means necessary even when it is deserved, let alone for love's sake. That's all that he cares about. And such an individual cannot be saved for one simple reason. They deny their own sin. They refuse to repent. Now St. Dismas is the opposite. He's the same kind of sinner. He's no better, no worse in regards to the sins that he commits. But what he says to the Lord reveals everything we need to know about him and his faith. First he says, this man is innocent, Jesus is innocent, and we deserve to be up here. So he takes responsibility for his sins. He says, I deserve the pain and the suffering that I'm enduring because of my sins. But he doesn't stop there. He doesn't just take responsibility. He professes faith in Jesus Christ and he asks for mercy. Lord, he says, remember me when you come into your kingdom. Now for a Jew you have to understand, when you ask someone to remember you, especially in any spiritual context, that's not just like, oh yeah, I'll think about you later. Remembrance for a Jew is one of the most powerful concepts. It's to make present. It's basically saying, bring me to your kingdom. He's basically asking that. So he's saying, save me Lord, save me. And so after taking responsibility for his sins, accepting his cross, not running from it, not fleeing from it, he asks for mercy. And that's when Jesus says, today you will be with me in paradise. That's really all it takes to be saved. So many times Christians think they need to be sinless to be saved. That's ludicrous. Where in the gospel does it teach that? Christ came for the sinner. Doesn't mean we don't strive for sinlessness, but you could sin until the day you die, not that you should, but I'm just saying, you know, whatever sins you struggle with, whatever weaknesses and failings, you could continue to fall until the day you die, but as long as you imitate St. Dismas, you accept whatever suffering and punishment and pain comes to you in your life. You endure patiently, humbly, realizing it's for your sins and you ask Jesus for mercy. You're fine. You are good to go. You will be saved. How often throughout the gospels does Jesus say, if you would be my disciple, you must pick up your cross daily and follow me. Our daily crosses are always because of our sins. And many people come to me at various times and they say to me, Father Miller, you know, I'm dealing with this cross in my life, but it's not my fault. You know, so and so just keeps doing this and I'm not doing anything wrong or I was driving down the road and, you know, minding my own business, safe driver, you know, going the speed limit, seatbelt on, and somebody just hit me and now my car is damaged. Why am I suffering this cross? I didn't do anything to cause it. And I say, yeah, not in that moment, not in that instance, but just think over the years of your life with all of the sins that you've gotten away with and haven't been punished sufficiently for. God is just giving you an opportunity. He's like, I'll let you get away with it then, but now it's time to pay. There's always a reason why we suffer. And the fathers are very clear. The majority of the time, it is because we are making up for past sin. That's why those are the crosses that God sends us. But even if, as I've said many times before, we have already made up for all of our sins, even if we've already done that. Well, like Jesus, we now have to suffer out of love for the sins of others. I'm Jesus wasn't guilty of any sin. He didn't have to make up for his own sin, but he's still suffered crosses, not his own now, but the crosses of others, our crosses. So to deny the cross is to deny Jesus Christ himself. To deny Christ is to be damned. But salvation is really not as difficult as people make it out to be. Jesus knows how hard it is to turn from sin. That's why he's so merciful. So let us imitate Saint Dismas, especially during this time of Lent, but throughout our whole lives. Let us accept patiently the crosses that come to us each day as just punishment for our sins, even repeating the words of Dismas. You and I, we deserve this. I deserve this. But, Lord, remember me when you come into your kingdom. Jesus, forgive me of my sins. And then we will be consoled for on that day, our Lord will turn to us and say, don't worry, everything's going to be fine. This day, you will be with me in paradise. In the name of the Father and of the Son of the Holy Spirit.