Consider the expression dS=(∂S∂T)VdT+(∂S∂V)TdV
I'm trying to understand how to derive an expression for (∂S∂V)P and how is it related to (∂S∂V)T.
I tried the following:
Method 1
i) Divide both sides by dV dSdV=(∂S∂T)VdTdV+(∂S∂V)TdVdV
ii) and at const. P
(dSdV)P=(∂S∂T)V(dTdV)P+(∂S∂V)T(dVdV)P
(dSdV)P=(∂S∂T)V(dTdV)P+(∂S∂V)T
Question 1: how does
(dSdV)P=(∂S∂T)V(dTdV)P+(∂S∂V)T
become
(∂S∂V)P=(∂S∂T)V(∂T∂V)P+(∂S∂V)T???
Method 2:
Differentiate both side wrt V, holding P const. and use product rule
∂∂V(dS)P=∂∂V((∂S∂T)VdT)P+∂∂V((∂S∂V)TdV)P
(∂dS∂V)P=((∂2S∂V∂T)V)PdT+(∂S∂T)V(∂dT∂V)P+((∂2S∂V2)T)PdV+(∂S∂V)T(∂dV∂V)P
Question 2: I got so many extra terms, and how to deal with these (∂ d blah1∂ blah2)blah3
terms?
Also, on more general grounds:
Question 3: How to partially differentiate a total differential rigorously?
Question 4: Are partial derivatives that differ in only the kept const. term identical in general?
Answer
dSdV=(∂S∂T)VdTdV+(∂S∂V)TdVdV
This doesn't make much sense, because is not a well defined expression. The differential dS=(∂S∂T)VdT+(∂S∂V)TdV
You can safely take this (in this context) as the definition of the differential expression (A). In other words, writing (A) is exactly the same as stating that S is a function of the two variables T and V.
What is then the meaning of the expression (∂S∂V)P ?
It means that you are now considering T itself as a function of P and V, call it ˜T(P,V), and effectively asking for the partial derivative of the function ˜S defined by ˜S(P,V)≡S(˜T(P,V),V)
how does (dSdV)P=(∂S∂T)V(dTdV)P+(∂S∂V)T
become (∂S∂V)P=(∂S∂T)V(∂T∂V)P+(∂S∂V)T???
They are the same thing. You have a function S of the two variables T and V. What you can do is just differentiating with respect to one or the other, obtaining: ∂S∂T(T,V)and∂S∂V(T,V)
Also related:
Determine the Dependence of S (Entropy) on V and T
What exactly is the difference between a derivative and a total derivative
(∂dS∂V)P
Please, do not ever write something like this :). A partial derivative is an operation that you can apply to (multi-variable) functions. A differential is not a (multi-variable) function, and its partial derivatives are not defined.
dS means "a little variation of the variable S", which can be caused by a corresponding variation of the parameters on which it depends. If you ask what is the variation of S while keeping some other quantity constant, you just divide dS by that quantity (say dV) and impose the constraint you want (which is the first method you mentioned). Or for a more rigorous (and more clumsy) approach, you do the partial derivatives of the ˜S defined above. The result is the same.
Are partial derivatives that differ in only the kept const. term identical in general?
No they are not. Consider the following example: let F be a function of the two variables A and B, and suppose that B is also a function of other variables, say A and C: F=F(A,B),B=B(A,C)
Note that in all of these reasonings you are always dealing with partial derivatives of functions of many variables (for example the S(T,V) above). What makes it confusing (and I remember being confused myself by this when dealing for the first time with this subject) is the fact that you implicitly, when needed, consider some variables as functions themselves of other variables (just like T that becomes ˜T(P,V) above). This feels pretty natural when you understand what is the exact meaning of the expressions, but can also be confusing at first.
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