Why is the notation for partial derivatives so prevalent in thermodynamics?

[ehrenfest]

Why in the world is the notation $$\left(\frac{\partial T}{\partial V}\right)_S$$ so ubiquitous in thermodynamics when it means exactly the same thing as $$\left(\frac{\partial T}{\partial V}\right)$$, that is, the partial of T with respect to V. The definition of a partial is that all the other variables are held constant! Is the problem just that physicists need a constant reminder of what a partial derivative is or what?

 

[Mapes]

It is not the same thing. In general,

$$\left(\frac{\partial T}{\partial V}\right)_S\neq \left(\frac{\partial T}{\partial V}\right)_P \neq \left(\frac{\partial T}{\partial V}\right)_U \neq\left(\frac{\partial T}{\partial V}\right)_N \neq \left(\frac{\partial T}{\partial V}\right)_\mu$$

and so on. The subscript is not redundant.

 

[Mapes]

Maybe I should provide some more detail. You may be used to working with systems in which the variables are all independent, but this is not generally the case, and is certainly not the case in most thermodynamic systems. Let me give an example: consider the relationships

$$a=bc$$

$$d=a-1$$

It should be clear that

$$\left(\frac{\partial a}{\partial b}\right)_c=c$$

is not the same as

$$\left(\frac{\partial a}{\partial b}\right)_d=0$$

In other words, yes, everyone needs a constant reminder of what the partial derivative is if they care about getting the right answer.

 

[ehrenfest]

That is mind-blowing! I will have to think about that.