 So, we have this definition for the Helmholtz energy or the Helmholtz free energy and one thing we can immediately obtain from that is the fundamental equation for the Helmholtz energy. So, starting with the definition for A, the Helmholtz energy, we can take the differential. Differential of A is DA. On the right side of the equation, U becomes DU and then the differential of T times S is T times DS and S times DT. But of course, we have a fundamental equation for DU that tells us exactly what DU is equal to. DU is equal to TDS minus PDV, so if I then subtract the TDS term and the SDT term, we see that some cancellation is going to occur. TDS cancels this negative TDS and all we're left with is the simple form differential of the Helmholtz free energy is, I'll swap the order of these two, negative SDT minus PDV. So, very similar to the fundamental equations we've seen before for the internal energy and for the enthalpy. The difference here is in which differentials the differential of A is expressed in terms of and this tells us immediately several things. One, it tells us that the natural variables of the Helmholtz energy are T and V because this DA has this nice clean and simple form when expressed as a function of DT and DV. What that means is that if we think of A as a function of T and V, this expression is simple. If we were to think of A as a function of T and P or SNP or SNT or some other combination of variables, we could obtain a differential but it wouldn't be quite as simple. So natural variables for Helmholtz energy are temperature and volume. The other thing we can see immediately from this fundamental equation, so I'll put the fundamental equation in a box because that's one we'll come back to over and over, thinking of A as a function of T and V, writing DA as some coefficient multiplied by DT and some coefficient multiplied by DV, this term, negative S, is equal to the rate at which A changes when we change the temperature, DA, DT, when V is held constant. So we can read directly off of this fundamental equation, negative S is equal to DA, DT at constant V, and likewise, DA, DV, when T is held constant is equal to this coefficient, negative pressure. So those are useful expressions. We can add those to our toolbox of thermodynamic derivatives. That's a list that's growing quite steadily and we'll see more of them as we go forward. If we were ever to happen to want to know how quickly the Helmholtz free energy is changing as we change the temperature or as we change the volume while holding the other one constant, these expressions tell us how to do that. So we'll make good use out of those expressions and others like them, but for now we'll move on and the next thing we'll do is to obtain the thermodynamic connection formula for the Helmholtz free energy while we're talking about the Helmholtz energy.