 So, one other thing we can do relatively simply now that we have this definition for the Helmholtz energy is to obtain a thermodynamic connection formula that will allow us to connect the thermodynamic quantity of the Helmholtz energy to a more statistical mechanical view of the Helmholtz energy or what would it be if we think about partition functions and Boltzmann probabilities. So, that's particularly easy. At this point we have enough thermodynamic connection formulas for other quantities like the energy and the entropy that we can obtain the thermodynamic connection formula for the Helmholtz energy by using what we've had previously. So, we know the thermodynamic connection formula for the energy is kT squared derivative of log Q, log of the partition function with respect to temperature, that's U. I can subtract from that T times S and we also have a thermodynamic connection formula for the entropy. That was k log Q plus kT d log Q dt. So, those are both results we've had before and combining these results simplifies quite a bit. This term kT squared d log Q dt is exactly the negative of this term, kT times T makes kT squared d log Q dt. So, there's cancellation between the second term in parentheses and the term out front and all that we're left with is this term negative kT times the log of Q. So, Helmholtz energy is minus kT log Q. That's our Helmholtz. It's a thermodynamic connection formula for the Helmholtz energy. If we know what the partition function is, as we do for example for a particle in a box or for an ideal gas or a rigid rotor or whatever, then we can use that expression for the partition function to obtain the Helmholtz energy of that particular system. So, it's fairly striking how simple the form of this partition, this thermodynamic connection formula is, simpler than the one for the energy, simpler than the one for the entropy. In particular, it doesn't involve any derivatives of Q or log Q with respect to anything as our other thermodynamic connection formulas have. The reason this expression is so simple, as we've seen the natural variables for the Helmholtz energy are the temperature and the volume. It turns out the natural variables, we haven't talked about it in this way, but the natural variables of the partition function Q are also temperature and volume. What I mean by that, if we remember what the partition function is, that's the sum of Boltzmann factors. The sum of e to the minus energy divided by kT. Summed over all the states the system can have, energies, the energy, if we think about a system like a particle in a box for example, the energies will depend on the volume. If I change the volume of the system, any quantum mechanical system can come find a sum volume. That volume affects the energies. If I change the volume, it affects the energy levels. The energies depend on the volume. The temperature also shows up in this expression, but the most natural thermodynamic variable dependence of the partition function is on these two variables, v and t. Since Q is a natural function of v and t, A is also a natural function of v and t, that means the relationship between them is relatively simple. The thermodynamic connection formulas for these other quantities like entropy or internal energy are not quite as simple because it involves changing from a function of t and v to something that's a natural function of s and t, for example. Or s and, yes, s and t, I know, s and v. So we have this thermodynamic connection formula. What we'll do next is move on and talk about a fourth flavor of the energy. We've talked so far about the energy and the enthalpy and the Helmholtz free energy. One more form of the energy that's more important and even more convenient than the Helmholtz free energy. So that's what we'll do next.