Partial derivatives vs total derivatives in thermodynamics

Question

The specific heat of a system is defined as

$$C_z = T \left( \frac{\partial S}{\partial T} \right)_{z=\text{const}}.\tag{1}$$

Sometimes however, I find the same definition, but with total derivatives instead of partial derivatives:

$$C_z = T \left( \frac{d S}{d T} \right)_{z=\text{const}}.\tag{2}$$

How can this be and what is the difference? Also, in class we calculated the specific heat of a superconductor from a given formula for the entropy. While we started off with the definition with partial derivatives, somewhere in the process the total derivatives started popping up out of nowhere. When a student asked why that is the teacher said something along the lines of "the partial derivative in the definition means the 'partial derivative in the thermodynamic sense'" and said that it's somehow equivalent to the total derivative, which I didn't understand.

So... what's the difference?

Answer 1

First of all: you and the people in your course are most certainly not the only students encountering this problem. In my 2nd and 3rd university year I had it myself. The reason is that physicists use abusive notation. And they do it a lot. Mathematicians have less of a problem with these things. In this answer I will try to use intuitive terms and be precise at the same time.

First a comment on parameter spaces. It's important to be clear on which one you are dealing with to see the difference between partial and total derivative. For example you will probably work with systems that have (p,V,T,N) (usual notation). Or for simplicity let's take constant particle number N and have some equation of state (e.g. ideal gas). Then the state of your system is completely determined by (p,V). But through the equation of state they are linked to T, so you could also express it in terms of (T,V) or (p,T).

Now we are ready for the derivatives. A partial derivative is a derivative along a certain, specified direction in the parameter space. For example in your case that direction is the z=constant line. Note that the direction has to be fully determined, so when your parameter space is n-dimensional you have to specify n-1 variables (i.e. one free variable which is the parameter along that direction). The first expression for the heat capacity you have above is exactly that IF z=const. determines a direction (i.e. it works in the 2-dimensional (P,V) parameter space we had as an example). One further thing that we will need to compare to the total derivative: the partial derivative along a certain direction is a function of the position in parameter space. I.e. in our example case: $ C_z = f(p,V) $

So what is the total derivative then? In fact it is a completely different object since it is not a function of position in parameter space only. Instead it also depends on the direction you are differentiating along. So it is better to think of it as the expansion (again using the example of heat capacity):

$ dQ = \left( \frac{\partial Q}{\partial p} \right)_{V=const} dp + \left( \frac{\partial Q}{\partial V} \right)_{p=const} dV $

So to find dC you need to specify how much you move in dp and dV direction. So this is in fact quite a complicated object. How is it useful? One can find relations between different total derivatives. E.g. dividing by dp above give:

$ \frac{dQ}{dp} = \left( \frac{\partial Q}{\partial p} \right)_{V=const} + \left( \frac{\partial Q}{\partial V} \right)_{p=const} \frac{dV}{dp} $

This relates the derivative $ \frac{dQ}{dp} $ to $ \frac{dV}{dp} $. So if you know the latter (which is saying you know the direction) you can calculate the former. One can then also read off different partial derivative relations, e.g. setting $ \frac{dV}{dp} = \left( \frac{\partial V}{\partial p} \right)_{q=const} $ (note that this is not a generally true expression, it is rather choosing the q=const. direction) gives an expression for $ \left( \frac{\partial Q}{\partial p} \right)_{q=const} $

I have now explained the principles. It will be useful to go through all your definitions and see which one is which. You will probably find that most explicit definitions are actually partial derivatives and the total ones are only used to relate to each other. In your example above a total derivative doesn't even make sense.

Answer 2

I consulted wikipedia, I found the following definition $$df=\frac{\partial f}{\partial x_{1}}dx_{1}+...+\frac{\partial f}{\partial x_{n}}dx_{n}$$ it is written after "In the one-dimensional case this becomes" $$df=\frac{ df}{dx}dx$$ and it's not true, the writing is:$$df=\frac{\partial f}{\partial x}dx$$ by omitting the index (1).

and I consulted several math books(*), the differential is defined by $$df=f'dx$$

so we can write: $$\frac{df}{dx}=\frac{\partial f}{\partial x}=f'$$

when you master the tool.

(*)-Higher mathematics course volume I, V.Smirnov.

-Differential and integral calculus volume I, N.Piskounov.

-Mathematical analysis volume I, G.Chilov.

-Mathematical analysis, A.Kartachev, B.Rojdestvenski