Great. We have now derived, from first principles, partition functions for an ideal gas, an ideal monotomic gas. And I think that's a, it's a pretty exciting concept, what we've managed to do. We've managed to go from first principles to explaining macroscopic properties. And I'm excited about that. I hope you're excited but what I want to do here is continue to work with that ensemble partition function and look at the properties we derive from it. So we derive for that monatomic ideal gas that the ensemble partition function is equal to a molecular partition function to the nth power, divided by n factorial. This is valid for an ideal gas. That means non interacting particles. That's what allows us to take a molecular, or in this case, atomic partition function, and raise it to the nth power. And the particles being indistinguishable, we divide by N factorial. Now, the individual components of that monatomic ideal gas molecular partition function are translational components that depends on volume and temperature. And an electronic component, and for purposes of this lecture, we'll consider it to include the degeneracy of the ground state, that's G1 and we'll consider the possibility that there is a first excited state that may be accessible. So we'll include the degeneracy of the second state g2e to the minus beta e2, that would be the next term appearing in the, partition function for the electronic component. So, let's look first at the internal energy of the ideal gas. And you'll recall that internal energy is equal to k, Boltzmann's constant, T squared. Partial log of the ensemble partition function with respect to T. That constant number of particles and volume. And if I make the substitution, then, that the ensemble partition function for my indistinguishable non interacting atoms, in this case is little q, the atomic partition function, to the nth power over n factorial. I can substitute that in for capital Q. I don't need to keep the N factorial term. It has no dependence on temperature. So it won't be there. But I'll then have NkT squared, partial log q, little q, partial T, at constant volume, right. And so the N that was there as a exponent, when I have a logarithm, will come out and be a multiplier. And finally, I can make the insertion what is this little q, it's just what I had in the last slide, namely it's the product of a translational partition function and an electronic partition function, when the gas in question is montaomic. And so, if I write that out, that is insert all the individual terms, first I will have the translational component. And that involves 2pi, the mass of the atom, Boltzmann's constant times temperature over Planck's constant square at all to the 3 2nd power times volume. And followed by the electronic partition function where we're keeping the first two terms and if there were more terms, of course we could add them but, for now, let's just keep an eye on the first two terms. And when I then go on to look at the relevant partial derivative I'll bring down the three halves power, then it's partial partial T log of a whole bunch of things. They will all separate out because I'm taking a log rhythm. The only one that will depend on T is T itself, so I'll get partial partial T log T, that's just 1 over T, and that gives rise to this term then. 3 halves N, so the 3 halves, Nkt squared, over T from this piece, 3 halves, Nkt, and so that is the derivative associated with the translational partition function. Now, what about the derivative associated with the electronic contribution to the overall partition function? So, I think I'm going to let you work on a piece of that derivative, and then we'll come back and see it in more detail. Alright, so obviously the degeneracy g1 has no dependence on temperature, and hence it doesn't appear in a derivative. This term on the other hand, the second term in the overall partition function expansion, does, and if you adopt the chain rule, you'll get a 1 over q electronic and then this is the piece I asked you to work on, N, degeneracy of the second, of the first excited state, so the second term in the partition function, epsilon two, e to the minus beta epsilon two, and had there been more terms we would have had to be continue with this differentiation. Now generally that contribution from the electronic partition function is small, right this depends on the second term actually having some relevance. Had we really been working with something entirely in the ground stage. It doesn't depend on temperature. There would be no contribution overall to the internal energy. And in that circumstance when that term is indeed small we would have that the internal energy is simply this first term. 3 halves NkT. So internal energy dominated by the translational contribution because the fraction in excited electronic states is usually very small at low at every day temperatures is what I mean by low. Is a room temperature here we are at a temperature that is not all that hot. Roughly 300 kelvin. So we may as well actually put some numbers on that though just to get a feel for the importance of that electronic term, so let me take two example gases, monatomic ideal gases. One would be a gas of lithium atoms and one would be a gas of fluorine atoms, so those are mildly unusual gases. You might have trouble actually assembling a flask full of lithium atom gas and fluorine atom gas, but that's okay, this is a demonstration that's a thought experiment. So we'll just work with the data, which are known for these atoms. And in particular, what's known for lithium is that its first excited state is quite high in energy relative to its ground state. 14,900 let's round to 4, reciprocal centimeters. And that excited state has a degenerisity of 2. For fluorine on the other hand, there's actually a rather low line first excited state. It's only 404 reciprocal centimeters above the ground state. And it too has a degenerisity of 2. And in order to put specific numbers on things. Let's actually pick values for the number of particles, and the temperature. And in particular, let's take Avagadro's number of particles. So we'll be working with a mole. And, let's use standard room temperature, 298.15 Kelvin. And I'll express my energy in kilojoules. And so this low lying electronic state for the Florine atom means that at room temperature, so if, if you think of room temperature multiplied times the universal gas constant, gives you a feel for sort of what's the ambient energy that's just floating around in the room available for harvesting. If you were to express that in wave numbers, it would be about 200 wave numbers, more or less. And so here we have a excited state, a first excited state, that's only about double that up in energy. So it's relatively accessible given the thermal bath in which these fluorine atoms acting like an ideal gas would be residing. And, the electronic partition function instead of being one, just remember one refers to everything being in the ground state is got some population of the first exited state, so it's 1.142. So, if we now go down we evaluate the contribution of the internal energy from translation, that's 3 halves, N is Avogadro's number times Boltzmann's constant, will be r. So we'll get 3 halves, rT, and in kilojoules, 3 halves rT for this temperature is 3.719. And for fluorine, of course, it doesn't matter what the atom is here, there's no dependence on the atom, it's just a constant. So it's 3.719 for fluorine as well. If we go to lithium, and we plug in to this expression, so q electronic for lithium is effectively 1, so that's what in the denominator here. 14903.66, and then we express kT in the relevant units of reciprocal centimetres as well. You discover that the contribution that's made to the internal energy is 2.081 times 10 to the minus 26. So that is 26 orders of magnitude smaller than the contribution from translation, which is to say it just doesn't matter at all. On the other hand, if we plug in the appropriate values for Fluorine, with its much lower lying excited state, there is a non-negligible contribution. It's 1.204. So, it's about a third again as much as what's being derived from translation. And that's An interesting phenomena associated with that low laying electronic state. Now what about the heat capacity? So that's an interesting property of a gas. how much heat can it store and how much will added heat cause its temperature to rise? So, given this expression for the internal energy and, recalling that the heat capacity at constant volume is the partial derivative with the internal energy, with respect to temperature, we then need to differentiate this expression, with respect to temperature, and when I do that, well, the first term here is pretty trivial. So, when I take three halves NkT, and I differentiate with respect to T, I get 3 halves Nk, that's straightforward. The next term, not quite as friendly. We've got temperature dependence and the argument of the exponential, the partition function itself has temperature dependence, so you would need to use the quotient rule here to complete the full differentiation. I'm going to express one of the terms that would come out of doing that quotient, and that is the differential of the thing on top times 1 over q electronic, and so that will give this, and I will let you work that out for yourself, if you just love doing differentiation. There will be an additional term but I've basically run out of room on this slide, and I'm not going to write down that additional term. It is there. and again if you love differentiation I'll let you do that for yourself. You can see that if there were ongoing terms because of a second excited state, a third excited state, this all gets a little bit messy relatively quickly. But what I want to do is focus on, the importance of a low-lying excited state, or lack thereof to the heat capacity. And so let's continue to work with our slightly unusual ideal gases of lithium atoms and fluorine atoms. And now again I'll keep track of the translational part and the electronic part, and I'll actually keep that extra term that's not being show in the slide. So if you try to work these numbers out for yourself, that second one is there, but again, I'm a little short on space, so I won't write it out. So continuing to work with one mole and at room temperature, this value 3 halves N times k that's just three halves r. And so in these units of joules per kelvin that's 12.47 and again that's independent to the nature of the gas, there's no excited state energy appearing in the translational part. If we plug in this large number to this exponential relative to the ambient kT, we get 5.02 times 10 to the minus 27th. Again a completely neglegeiable controbution to the heat capacity. However in the case of fluorine, when we put in the much smaller 404 wave numbers, you end up with a contribution from the electronic partition function, of 5.193. And so that's not quite, but it's close to half again, as much, as is coming from translation. And I'd like to make clear what, what that means. Remember what the heat capacity describes. It says, how much heat would I need to put into my ideal gas in order to raise its temperature by 1 degree. And that requires us to think about what is temperature, when I clasp that flask full of gas, why does it feel warm or cold to me. And so temperature is a measure of the kinetic energy of the molecules, so there zooming around and bumping into the walls of the flask and that causes the wall of flasks to also have some motion, and it's that motion, in a sense, that we sense as temperature. So, if I put heat into the system, in the case of lithium, the only place it has to go is into the translations. It can't populate anything else. And as a result, it takes this much, 12.47 to raise the temperature by a degree. On the other hand, what happens in the fluorine system? When I put heat into it, it's got two options. Just like the lithium, it has a translational component. And it can put energy, heat in this case, into the translational modes. But it also has available to it relatively low lying electronic excited states. Those don't contribute to the electronic energy. Rather, it's just a place where the heat can be stored by increasing the population of those excited states. And as a result, instead of taking 12.47 joules worth of heat It takes, hm, looks like 18.4 roughly. So more heat to raise the temperature 1 degree because some of that heat isn't going in to what I perceive as temperature of the kinetic energy of the molecules, instead it's going into populating those excited states. Of course I can harvest that out again later but the point is that energy is being stored in a place that doesn't contribute to the rise in temperature. So it's an interesting phenomenon and generally associated with availability of states. Okay maybe the last thing to do is work with one more property, namely the pressure. So here we have the monatomic ideal gas, and recall the pressure is kT, partial log q partial V, so I'll do exactly what I did before. I'll replace q with molecular partition function of the nth power over N factorial. Again, and the N factorial doesn't depend on V, so I'll just lose that, and this equality is held. And now I insert for the molecular partition function, the product of the translational and electronic partition functions. And, when I go and expand using the actual electronic, sorry here's the electronic, and translational partition functions. There is only one thing that depends on V, and that is V itself. So, I just have partial, partial V of log V. Well that's pretty straight forward, so I get 1 over V and NkT over V is the ultimate result. P is equal to NkT over V and of course that is the ideal gas law. PV equals nRT if n to number of moles. Little n, that is, number of moles as opposed to capital N, number of particles. So let's just summarize, then what we've actually derived for this partition function for a monatomic ideal gas. Where these are the individual components, the translational component of the molecular partition function, and the electronic component. If we restrict ourselves to those cases where the first excited state, the first electronic excited state, that is, is too high in energy to really contribute, so not unlike the somewhat exotic fluorine atom, but more like other systems, then the energy is 3 halves NkT. The heat capacity is 3 halves Nk and the pressure is NkT over V. And we more typically for convenience work with molar quantities and molar units, in which case we'll have U bar, the molar internal energy is 3 halves RT. The molar constant volume heat capacity is 3 halves R and finally the pressure is equal to RT divided by the molar volume. Alright, well, that takes care of the monatomic ideal gas. There are not that many interesting monatomic gases, Lithium and fluorine, as I mentioned, are extremely exotic if they're monatomic. There are the noble gases, helium, neon, argon, they get more exotic as they get heavier, but let's move on to the next thing that occurs after a monatomic ideal gas and that is an ideal diatomic gas. More things will happen in the diatomic, and so it will take us more than one lecture to get through them all but we'll start with the first half of the ideal diatomic gas next. [NOISE]