 And welcome to module 23 of chemical kinetics and transition state theory. So, now we have a final formula for transition state rate. Today, we will look at a specific example on how numbers are exactly evaluated. We will look at the example that we have looked at a little bit before anyway, which is h plus h v r going to h 2 plus v r, ok. All right. So, let us study this problem. In your assignments, you will be solving more such problems for practice. So, this is the formula. That is what we need to evaluate. So, the required quantities that we needed and we discussed this in the last module. So, it is a quick recap. If we need all the masses of h and v r that is readily available, we will need the activation energy of the reaction. We will need to find whether the reactants and transition state is linear or non-linear and what are the corresponding frequencies. We will finally need to look at the electronic degeneracy of the reactants and transition state. So, let us start doing that, ok. So, the masses we know 1 gram per mole inverse and 80 gram per mole inverse, ok. That I have just looked up the periodic table that is all. Activation energy of this problem is known as 5 kilo joules per mole. So, we are going to use that. All right. So, let us start asking about the structural information. So, I have a reactant is h and h v r. So, let us start with h. So, the first question I am really trying to ask is, let us say somebody has given you this reaction, somebody experimenter has done this reaction and you are supposed to find the rate constant, ok. So, you have to go to your lab and first accumulate all the data. So, the question is what all data do you need to find, right? So, that is what we are doing. So, for hydrogen atom, what all data will you accumulate is the question. Mass we know, is there any other information you need for hydrogen atom. So, take a moment and think about it, pause the video if you want and think about this question, ok. So, hope that you have thought about it. So, the question that you have to answer is, well for each reactant we have to figure out whether it is linear or non-linear, we have to find the moment of inertia correspondingly and we have to find the frequencies. So, is h atom linear or non-linear? Nothing, h atom is a point. So, it has no rotational partition function, it has no vibrational partition function, ok. So, for h atom is simply Q translational into Q electronic and atom cannot rotate or vibrate, get that concept absolutely clear, ok and atom is simply a point. What about HBr now though? For that do we need any more information other than mass? Yes, we do. So, is HBr linear or non-linear? Well, of course it is linear. HBr is simply HBr, you have only 2 points and 2 points can only form a line. So, this is certainly linear. So, we need its moment of inertia and how many frequencies do I need for HBr? Only one. Why? Well, one can see that I have only one frequency here. It can really just do this kind of a stretch, right. But if you want to be more mathematical, you have 3n minus 5 frequencies where n here is 2. So, I want one frequency for the HBr molecule. What about the transition state? How many, what do I need for that? So, let us start, we are going to assume actually to a good approximation that the transition state is linear. So, this is an information I am giving you. So, then I need one moment of inertia for the transition state. What else do you need? You need frequencies. How many frequencies is the question now? Do this calculation very carefully. Think you have a linear transition state of 3 atoms. How many frequencies should you get? So, pause the video and get this answer right. So, let us discuss. It is linear. So, total frequencies is 3n minus 5 by 5 2 rotations for a linear molecule and 3 translations. So, a linear molecule can rotate in 2 ways like this or like this and it can translate in all 3 directions. So, I have 3n minus 5, n is 3, 3 atoms. So, I get 4 frequencies. However, one frequency is imaginary at the transition state that we do not include in our calculation. So, required frequencies is for transition state is always 1 less. So, I should get 3 frequencies corresponding to the transition state out of 4. So, here is the table where I have provided you all the information. So, I did my calculations I one way or another by experiment or otherwise I gathered all the data. In a future module we are going to discuss how this data is exactly tabulated. How in practice do you get it? For the purpose of this course we whenever data is needed we will provide you with data. We will not ask you to do the actual experiment. So, we have HBr is by definition linear and we have assumed transition state to be linear. I have provided you the moment of inertia both of one moment of inertia because they are both linear. I have provided you one frequency of HBr and 3 frequencies for HBr in the transition state. H of course has a degeneracy of 2, electronic degeneracy because H can be in the positive spin direction or negative spin direction alpha or beta. HBr has degeneracy of 1 and transition state I have provided you it is a radical actually it has a degeneracy of 2, electronic degeneracy of 2. So, this is the formula I need I have all the formulas with me I will use these formulas and I will have to calculate basically all the components I have to calculate kT over H the three partition functions and this exponential. So, that is what we are going to do now. So, let us start with Q H naught. So, the required data I need for Q H naught is simply the electronic degeneracy and the mass Q H naught is Q translational naught into Q electronic which is nothing but Q translation divided by volume into Q electronic. Q electronic is nothing but the degeneracy. So, this is nothing but 2 which is Q electronic into Q translation over volume. So, volume will cancel and I will get 1 over H cube. I have always to be very very careful with units I will be using throughout Si units only. So, the mass I need is in kilograms that is the Si unit ok. What I have is 1 gram per mole we have done this a few times 1 gram per mole into 1 kilogram per 1000 gram into 1 mole divided by alagadro number. So, this is equal to 1.7 into 10 to the power of minus 27 kilograms ok. So, that is the mass I need to put here. So, this is then equal to 2 Q H cube, root pi mass is 1.7 into 10 to the power of minus 27 kilograms Kb is 1.38 into 10 to the power of minus 23 into temperature. I am calculating everything at room temperature only and the unit of K into T will be kilogram meter square per second square ok. So, let us just quickly look at our units. In the numerator I have square root of kilogram into kilogram meter square per second square divided by in the numerator I have kilogram square meter 6 second square second cube sorry I have done something wrong. There should be a cube here my apologies. I am supposed to take the cube of this. So, there is a 3 half that I had forgotten that is why my units you see the importance of looking at units that is how I figured out I have made a mistake because I looked at units. So, now if I look at it this becomes a kilogram cube meter cube per second cube divided by I also had kilogram cube here meter 6 by second cube kilogram cube cancels this cancels. So, I am left with 1 over meters cube. So, rest is just plugging in the numbers I have to just calculate this very carefully on a calculator and so if I calculate that I have done that I get 19.8 into 10 to the power of 29 meter that is my units meter minus 3. So, first thing is you get something in the order of 10 to the power of 30 that is what you want. Hydrogen being very light it usually is slightly smaller usually you get into the power of 31 but fine hydrogen is light you get a factor of little bit lesser. So, you get something in the order of 10 to the power of 30 everything is in sensible my units are correct. Next looking at HBr the data that I need will be these following we will need the mass we will need moment of inertia and the frequency and actually for HBr we already have calculated all the components in a previous module. So, I am not going to redo it today you can look back at the previous module you can look at module 17 where we actually plugged in the numbers the same way as for hydrogen what we have been doing and we got for translational 10 to the power of 32 roughly rotation is roughly of the order of 25 vibration is very close to 1 remember we are using the vibrational answer this answer and not the classical answer. So, that quantum answer we did in module 19 and the electronic is 1 which is the electronic degeneracy of HBr. So, if we multiply all these numbers together I take this into this into this into this I will get this ok. So, that is the partition function for HBr. So, I have done with this I have done with this. Now, moving forward to the transition state we will compute all the 3 partition functions translational rotational and vibrational. So, let us start with the translational partition function. So, the translational partition function is given here and so, we first of all divide by volume as always. So, the translational partition function not is q translational by volume. So, the volume will cancel. So, I will be left with 1 over h cube 2 pi m kT to the power of 3 half. So, first we need to calculate this m m is the total mass. So, m will be mass of HBr, but what we have to be careful about is units. So, this we have to convert in kilograms and moles is 10 to the power of 23. So, mole cancels with mole gram cancels with gram and I get I have written the answer here with me have calculated this already. So, this is simply punching it on a calculator. So, finally, I get qT r naught again we have to be just very very careful with units 2 pi mass is 13 into 10 to the power of minus 20 kilograms into 1.38 into 10 to the power of minus 23 kB is units of kilogram meter square per second square Kelvin into 300 Kelvin to the power of 3 half divided by h cube h is 6.6 into 10 to the power of minus 34 kilogram meter square per second cube. So, first thing that I will leave it to you this time is to verify that the units work out correctly. So, what is the final unit you should get for qT r meter to the power of minus 3. So, make sure that is the unit you will actually get here and all other kilograms second Kelvin's all everything else is going to cancel. So, that I leave it up to you we have done that a few times now and the rest the numbers if I punch in I get 7.32 into 10 to the power of 32 meter minus 3. So, once again in the ballpark of 10 to the power of 31 and 32. So, it makes sense the rotational version we will use the linear formula. So, this is equal to 8 pi square kB is 1.38 into 10 to the power of minus 23 kilogram meter square per second square Kelvin always be careful with units into moment of inertia is given to be this divided by h square I have to just write everything very carefully and I will be good. So, the first thing is all units must cancel rotation and vibration partition function are dimensionless. So, kilograms into kilograms is kilograms square cancelling with kilograms square in the denominator meter to the power of 4 here meter square square 4 Kelvin cancels with Kelvin here second square cancels with second square and that becomes plugging in numbers which comes out to be this vibration. I will need the frequencies. So, the frequencies note are provided in wave numbers what we need again are all units in SI units all quantities in SI units. So, I have to go from centimeter inverse to second inverse frequencies units are second inverse. So, again the formula I keep in mind is this one. So, this is in centimeter inverse my apologies omega equal to 2 pi C into nu bar this is in second inverse this is in centimeter inverse. So, I have to multiply this by 2 pi into speed of light. So, you notice centimeter will cancel and this comes out to be 4.4 I have written it in my notes here and 26 my bad 460 I also multiply with 2 pi C and I get this is simply punching in numbers on a calculator. So, then what I do is I have to calculate the vibrational partition function will basically be 1 over 1 minus beta h bar omega 2 and omega 2 actually appears twice it is degenerate. So, that is how you calculate vibrational partition function you multiply them together and we are using the quantum version once more quantum version is more accurate in the classical version. So, you can plug all these numbers in you have the omega once in a second inverse. So, omega 1 basically will go here and this will go here and you know what is your KB your T everything you plug in properly and you can find this is equal to 1.27 again dimensionless. So, we have found all the various components now we have found a translational rotational vibrational and the electronic is simply the electronic degeneracy which is 2. So, I take all of these and multiply them together to get this number here. So, I have found the component for translational state as well. Finally, we have to calculate the pre this exponential factor and KT over h. So, KT over h comes here and that is your exponential. So, the exponential is simply minus EA 5 kilojoules per mole I will multiply it by 1000 joules per kilojoule because and KB instead of using KB I can use R if I want to use per mole unit and temperature I am using is 300 Kelvin. So, Kelvin cancels with Kelvin kilojoules cancels with kilojoule joules cancels with joule mole cancels with mole it is all dimensionless which is what I want and then you plug it in a calculator and you get finally is KT over h. So, this is 1.38 kilogram meter square per second square Kelvin temperature is 300 Kelvin Kelvin cancels with Kelvin kilogram cancels with kilogram meter square cancels and this second cancels one of the seconds. So, I get this is equal to 6.3 into 10 to the power of 12 second inverse. So, finally, is the time to plug in all the numbers together. So, I have KT over h which I have found to be this we have found the partition functions for transition state we found it equal to this and for h we found this for HBr we found this. So, I have just plugged in the numbers from the previous slides and exponential I have found to be 0.13. I multiply them all together notice that one of the meter cube cancels. So, I am left with meter cube second inverse and simply punching in into a calculator. So, I get rate in the order of roughly 10 to the power of minus 18 meter cube second inverse again things you should like keep sense of that is how rates will look like in this unit. So, the final thing is let us just calculate this in the units of liter mole inverse second inverse. So, this will be equal to 5.88 into 1000 liters in 1 meter cube into the Avogadro number per mole. So, you can do this calculation and this comes out to be roughly I have not written this in my notes. So, I am doing the calculation you can confirm this roughly 3.4 into 10 to the power of minus 9 liter mole inverse second inverse I have forgotten to write this actually in my notes but no issues this is simply plugging it into a calculator you can do that and tell me whether I have got it right or not. So, the rates generally are of the order of 10 to the power of minus 9 which is what I expect plus 9 actually I am sorry I apologise this should be plus but this is simply punching it on a calculator you can do that better than I can and get that number all right. So, we end here today and we have looked at exact calculation of how to calculate it for a specific example which is of H plus HBR with a linear transition state with the you see you can use the same procedure to calculate it for any calculation and get a transition state estimate. So, the first step is always identifying the parameters you need you have to figure out whether the structures are linear or non-linear, calculate the appropriate amount of moment of inertia, calculate the appropriate frequencies always remember that for transition state I require one frequency less and then plug in into the formulas and after you plug into all the formulas then you have to be just careful with units and get the final answer. Thank you very much.