 Welcome to module 11 of chemical kinetics and transitions state theory. Last four modules or so, we have been looking at the collision theory for reactions. And we have went through its derivation very very carefully in great detail and have derived an equation. What I want to focus on today is to take a step back and calculate. So, we will look at specific problems and actually calculate numbers out and see how this collision theory actually works to in calculating a rate constant. So, just a quick recap. This is the final formula we derive for the collision theory R a and R b are the radii of a and b. This is the average speed, mu is the reduced mass, E a is the activation energy and T is the temperature. We actually let me just make another notation which is the pre factor or collision frequency. So, just this term without the exponential you will in books or literature often see this being called the pre exponential factor from the perspective of Arrhenius equation. Or it can often called the collision frequency. One important note we made two modules ago. If a is equal to b you have to be careful the collision gets a factor of half and so will the rate. So, R a is equal to R b you will get 2 R a you will get mu as mu a a, but also you get this factor of half. So, if you do not remember it how we got this factor of half go back a couple of modules and you will see the logic for that. In the last module we had introduced the idea of reaction cross section as pi R a plus R b square into P r where P r is the reactive probability and we had went through and carefully argued how to choose this P r to get this kind of an equation. One important thing before we solve problems is that of units. Units is always very very important any number you any time calculate must always have units. So, what is the unit of k? So, this k that is written here if you look at it what is the unit right now? If R a comma R b are in let us say meters and this thing is velocity. So, velocity is let us say in meters per second then k is in units of meter cube per second. I have this meter square multiplied by meter per second. So, I get meter cube per second, but the typical units of rate experimental units of rate clean liters mole inverse second inverse. So, first thing is if you want to compare to experiment these are the units we want to calculate n. By the way often in literature you will also find dm cube mole inverse second inverse where a liter and 1 liter is equal to 1 dm cube. So, these two are identical this is equal to this. So, let us get back to the basics on what we are doing the k the way we have defined is via this equation we are calculating dna over dt as the rate constant here into na into nb. You can go back on our very basic derivation and you will see this is true. Remember typically the experimental rate is in the language of concentrations. So, we have to convert in the language of concentrations first. So, what I do is I define this Avogadro number which is 6.02 into 10 to the power of 23 mole inverse. And let me write this as we will note now that well you can first verify that this equation really follows from this that is almost trivial to prove. And then note that na over Avogadro number equal to concentration of a and nb divided by Avogadro number is concentration of b. So, I get minus d concentration of a by dt equal to k into Avogadro number into concentration of a into concentration of b in the language of concentrations. So, my new k of t becomes pi into e to the power of minus beta epsilon naught into the Avogadro number. So, if I do that the new units has become meter cube mole inverse second inverse because of Avogadro I get a mole inverse. But this still I have to be careful I have to be careful of using pen first and then the k that I have written is pi basically this thing into an Avogadro number and the units is so far meter cube mole inverse second inverse. And what do I want it is in I want it in liter into mole inverse second inverse. So, there is still a little bit of a change I need how do I do that you remember how is meter cube and liter related. So, 1 liter 1 meter cube is 1000 liters. So, k of t becomes pi r a plus r b square root 8 k t over pi mu e to the power of minus beta epsilon naught into 1000 into Avogadro number and this will be in liter mole inverse second inverse if this is in meter square this is in meter per second. If you want to use different units obviously you are free to do so, but at the end you almost convert your units carefully enough to get finally in the experimental answer and in any case whenever you write calculate a number the unit should be mentioned without a unit if I calculate a rate constant and I tell you the rate constant is 10 into 10 to the power of 7. What do I understand from it how what 10 to the power of 7 elephants. So, without that unit that number is completely meaningless always remember that. So, you get your number and you get your units right. So, in summary what we have is the original expression that we had derived here is basically in meter cube second inverse. Now, if I multiply that by 1000 into Avogadro number I will get in liter mole inverse second inverse which is the same as dm cube mole inverse second inverse and always remember if my bimolecular reaction if a is equal to b then I must multiply by a factor of half. So, with that let us start solving problems. Let us start with an easy one. If you go back several modules I had read a statement from Arrhenius. According to kinetic theory of gases which is what we are discussing now the velocity of the gas molecules changes only by about 1 6th percent of its value for each 1 degree rise in temperature. So, now actually we are in a position to test whether Arrhenius was right or not maybe he was bluffing yeah, but we have equations now we will test we will put Arrhenius to the acid test nobody is above mathematics. So, what has the velocity is what he is talking about and what we have shown that the velocity the average velocity is root 8 kT over pi mu. So, basically view velocity is proportional to root T. So, if I change temperature how much will average speed change that is the question that is what Arrhenius is commenting on. So, what I want to find is nu at T plus 1 Kelvin minus nu at T Kelvin divided by nu at T and to convert 2 percent into 100. So, this is what I have to find and show check whether this is equal to 1 over 6 or not. So, you can substitute this nu above here and what you will quickly show is this is equal to root of T plus 1 minus root of T divided by root of T into 100 and basically what Arrhenius was talking of was at room temperature. This is not always true it will be close to true, but what he was discussing is at room temperature. So, this becomes equal to root of 301 I am sorry root of 300 Kelvin divided by root 300 Kelvin. So, room temperature is 300 Kelvin or close by. So, you can work this out and show that this is equal to 1 6th percent this comes very close to 1 6th percent. So, this is a very simple numerics you can plug it into your calculator and you will you can prove this out which is roughly equal to 0.16 percent. So, now let us look at a real reaction. So, kinetic theory of gases was actually particularly successful for the reaction of H 2 plus I 2 going to 2 HI and its reverse 2 HI going to H 2 plus I 2. So, let us try to calculate a reaction for that. So, we have given you a few units this often when you look at this have a physical field as well look H 2 is going to be small H 2 is roughly of the size of a few angstrom right which is 0.1 nanometer. So, that is the size of H 2 I 2 is bigger I 2 is a huge molecule. So, you get 0.2 nanometer which is a bigger size I have given you the masses of I and oh my god this is should be hydrogen iodine is not 1 gram per mole and an activation energy has been given activation energy is nothing but E and I am asking you if collision theory is true what will be the rate constant. So, the rate constant comprises of 3 factors really sigma average speed we are going to calculate each of these 3 separately pi r a is 0.12 nanometer always be super careful with units you can do this math I have done this math this comes out equal to 0.32 nanometer square but I want to convert to meter square SI units. So, what I do is 1 meter has 10 to the power of 9 nanometers whole square ok. So, I multiply by that nanometer square will cancel with this. So, I will get 0.32 into 10 to the power of minus 18 meter square 1 component down second component new not new speed that is given by 8 kT over pi mu if you solve enough problems you will memorize all of this. But again do not go end up memorizing these equations that is not the point of this course at all. Any equations that are needed in an exam situation I will provide you all equations. So, first I have to find mu, mu remember is defined to be the reduced mass. So, this is m of H2 into m of I2 divided by m of H2 plus m of I2 you can actually go ahead and solve this but I will be a bit smarter. I will note that m of I is way more than m of H 250 times more well then what I can do is this is just being a bit smart. If I add m H2 plus m I2 well it is a little bit more than m I2 just 2 more I can as well ignore it and I notice this is equal to m of H2 which is nothing but 2 gram per mole but I want it in kilograms. I want it in SI units. So, 2 grams per mole into 1 kilograms per 1000 grams into Avogadro's number 23 moles. So, Avogadro number is mole inverse I am dividing by Avogadro number mole cancels with mole gram cancels with gram I am left with kilogram and I have simplified this again and I have written the answer beforehand with me this comes out equal to 3.3 into 10 to the power of minus 27 kilograms. So, please go ahead and verify this number I may have been making mistakes. If you make mistakes in the exam you lose points be sure that the number I am doing is right punch it in a calculator and finally, I want to calculate e to the power of minus beta EA that is equal to e to the power of minus. So, let me just write it exponential minus 171 kilo joules per mole divided by RT R is 8.31 joules per mole Kelvin into temperature 650 Kelvin. So, you know this is kilojoule this is joule. So, I better multiply by 1000 here and I make this joule. So, now all the units will cancel K cancels with K mole cancels with mole joules cancels with joules again when R is required I will give you the value of R in an exam situation do not go about memorizing and this comes out equal to 1.7 14. I have forgotten to calculate mu I calculated mu very happily but I forgot to plug it in here this is equal to 8 KT over pi mu. So, 8 what is KB? KB is 1.38 due to 10 to the power of minus 23 kilogram meter square per second square into Kelvin into 650 Kelvin divided by pi and mu I have found to be 3.3 into 10 to the power of minus 27 kilograms. So, kilogram cancels with kilogram Kelvin cancels with Kelvin. And this if you plug everything in you get 2631.6 meters per second. So, finally K is a product of sigma. Sigma we calculated to be 0.32 into 10 to the power of minus 18 meter square into nu bar 2631.6 meter per second into 8 to the power of minus beta EA and I wanted to answer in liter mole second inverse. So, I multiply by 1000 into Avogadro number. So, this is 1000 of liter per meter cube I will cancel meter cube here with meter cube I will cancel I will cancel nothing else and I will calculate this is equal to at the end of the day I have this number with me 8.6 into 10 to the power of minus 3 liters mole inverse. So, at every step you have to be very careful particularly of units there is nothing very hard here. I took this formula this one here and I am calculating each number very carefully I calculated sigma here I calculated this average speed here to calculate the average speed I needed mu, mu I calculated here. I made a little approximation you can make it more accurate and see if what I have said is right or not maybe I have made a mistake maybe this is quite far. So, check whether that is right I calculated nu and finally I calculated e to the power of minus beta EA here and then I plugged all of this in and for units I use 1000 into Avogadro number. I plug every number in I have a calculator with me I had plugged all those and I get this number. Now, I actually just wanted to provide you a little context. So, the number that we have calculated is 8.6 into 10 to the power of minus 3 liter mole inverse second inverse and the experiment that were done back in that day more than 100 years ago from now was 0.31 dm cube mole inverse minute inverse. So, I want to check whether my number matches or not well first thing I notice that this is in minute inverse not a big deal. Remember that liter is the same as dm cube I have 1 over second but I will multiply by 60 second divided by 1 minute second caps cancels I get this multiplied by 60 and I will get 0.52. So, this is my K of collision theory and this is experimented. This is not bad at all given that this is such an approximate theory it does not even have bonds. So, if I have to think of H 2 plus I 2 H has a bond and I has a bond and it does not look like a sphere at all right H 2 is not a sphere but you see that the comparison is not bad within a factor of 2 in your assignment you will be solving more problems. I will solve one more problem here. So, the dimerization of CH 3 radical pen here 2 CH 3 dot this is the reaction we are looking at at 25 degree C. Somebody had figured out the pre-exponential factor and the question is you have to find a reactive cross section. So, how do I find that? So, I note K is half that remember because here A is equal to B. So, I get a factor of half into sigma into nu that is my pre-factor. So, let me not write that as let me write that as Z. So, I have to calculate this sigma. So, sigma and the unit that they have provided is in liter mole inverse second inverse. So, I multiply this by 1000 into Avogadro number. So, sigma is 2 ZAA divided by nu bar into 1000 Avogadro number. So, I have to calculate nu bar now again is 8 KT over pi nu mu is equal to M into CH 3 into M of CH 3 divided by M of CH 3 plus M of CH 3. This is nothing but M of CH 3 divided by 2 which is close to how much I have this written down here 15. So, the mass of carbon is 12 and hydrogens are 1. So, this is 15 by 2 grams per mole into 1 by 1000 into Avogadro number which will convert it into kilograms per grams into moles just like we did in the last slide mole cancels with here grams cancels with this and this I can have found already and this is equal to mass mass here 1.24 into 10 to the power of minus 26 kilograms. So, I substitute that here I get 8 into KB everything in SI units I am not going to write all the units as explicitly as last. I remember temperature must be written in Kelvin's never forget that do not write 25 degree C here at temperature at 273 Kelvin's pi into mu everything in SI units and this comes out to be then 919 meters per second. So, I have all the factors here Z is 2.4 nu I have calculated 919 this thing was in liter mole inverse second inverse 1000 liters in meter cube into Avogadro number. So, a lot of things are going to cancel liter cancels with liter moles cancels with mole second cancels with second and I will be left with meter square 1 meter will cancel with square and these are all numbers that I can just plug in and I get that is equal to 8.7 into 10 to the power of minus 14 meter. So, we will end here. So, today we had looked at practical applications of kinetic theory and how to calculate numbers out with proper units I can promise you if you do not practice this on your own enough problems you will never get units right correctly they are tricky and it takes practice. So, with that I will stop here in assignments you can solve more problems the textbooks that I have referred to also has more problems practice. Thank you very much.