Types of variables you need to know to understand thermodynamics

I’m been through the first 200 pages of Dill’s Book “Molecular Driving Forces (2003)” which is all about thermodynamics and statistical mechanics, things that must be understood to have any hope of understanding cellular biophysics. There are a lot of variables to consider (with multiple names for some) and they fall into 7 non mutually exclusive types.

Here they are with a few notes about them

l. Thermodynamic State Variables: These are the classics — Entropy (S), Internal Energy (U), Helmholtz Free Energy (F), Gibbs Free Energy (G), Enthalpy (H).
All are continuous functions of their Natural Variables (see next) so they can be differentiated. Their differentials are exact.

2. Natural variable of a thermodynamic state variable — these are defined as continuous variables which when an extremum (maximum, minimum) of the state variable using them is found, the state function won’t change with time (e.g. is at equilibrium). Here they are for the 5 state functions. T is Temperature, V is Volume, N is number of molecules, S and U are what you think, and p is pressure

State Name State Function Natural Variables
Helmholtz Free Energy— F —T, V, N
Entropy —S —U, V, N
Internal Energy —U — S, V, N
Gibbs Free Energy— G —T. p, N
Enthalpy — H —S, p, N

Note that U and S are both state variables and natural variables of each other. Note also (for maximum confusion) that Helmholtz free energy is not H but F, and that H is Enthalpy not Helmholtz Free energy

3. Extensive variable –can only be characterized by how much there is of it. This includes all 5 thermodynamic state variables (F, S, U, G, H) alone with V volume, and N number of molecules.  Extensive variables are also known as degrees of freedom.

4. Intensive variable — temperature, pressure, and ratios of State and Natural variables (actually the derivative of a state variable with respect to a natural variable — temperature is actually defined this way ( partial U / partial S)

5. Control variables — these are under the experimenter’s control, and are usually kept constant. They are also known as constraints, and most are intensive (volume isn’t). Examples constant temperature, constant volume, constant pressure

6. Conjugate variables. Here we need the total differential of a state variable (which exists for all) in terms of its natural variables to under stand what is going on.

Since U is a continuous function of each of S, V, and N

we have

dU = (partial U/ partial X) dS + (partial U / partial V) dV + (partial u / partial N ) dN

= T dS – p dV – mu dN ; mu is the chemical potential

So T is conjugate to S, p is conjugate to V, and mu is conjugate to N ; note that each pair of conjugates has one intensive variable (T, p, mu) and one extensive one ( S, V, N). Clearly the derivatives ( T, p, mu) are intensive.

7. None of the above — work and heat (q)

Thermodynamics can be difficult to master unless these are clear. Another reason is that what you really want is to maximize (S) or minimize (U, H, F, G) state variables — the problem is you have no way to directly measure the two crucial ones you really want (U, S) and have to infer what they are from various derivatives and control variables. You can measure changes in S and U  between two temperatures by using heat capacities. That’s just like spectroscopy, where all you measure is the difference between energy levels, not the energy levels themselves. But it is the minimum values of U, G, H, F and maximum values of S which determine what you want to know.

There’s more to come about Dill’s book. I’ve found a few mistakes and have corresponded with him about various things that seem ambiguous (to me at least). As mentioned earlier, in grad school 44 years ago, I audited a statistical mechanics course taught by E. Bright Wilson himself. I never had the temerity to utter a word to him. How things have changed for the better, to be able to Email an author and get a response. He’s been extremely helpful and patient.