Showing Partial Derivatives Commute?
In Thermodynamics,
Free energy is defined as F=E-TS and is a function of T,V,N:
F ==F(T,V,N)
The equations of state are
1. ∂F/∂T= -S
2. ∂F/∂V = -p
3. ∂F/∂N = μ
I need to use the fact that partial derivatives commute to show that
a) (∂p(T,V,N)/∂T) = (∂S(T,V,N)/∂V)
b) (∂μ(T,V,N)/∂T) = -(∂S/∂N)
c) (∂μ/∂V) = -(∂p/∂N)
I sort of know what commute means in terms of addition but not really in terms of partial derivatives. I was going to just show that for each question they are equal but this doesn't seem right. Does anyone have an example of a proof of partial derivative commutations?
Comments
Look at the bottom of page 2 of the following link to see how to use the communative property with partial derivatives
http://snowball.millersville.edu/~adecaria/ESCI341...
Next, you are right. Show that they are equal. I think its that simple (I could be wrong).
Ill do the first one for you.
a) (∂p(T,V,N)/∂T) = (∂S(T,V,N)/∂V)
Look at the first line to the known state functions
1. ∂F/∂T= -S
Plug S in so that
a) (∂p(T,V,N)/∂T) = (∂S(T,V,N)/∂V)
becomes
a) (∂p(T,V,N)/∂T) = -(∂/∂V)(∂F/∂T)
So that is what you are actually trying to show is true
From the communative property
(∂/∂V)(∂F/∂T) = (∂/∂T)(∂F/∂V)
Now look at the second state function your are given:
∂F/∂V = -p
Do you see ∂F/∂V anywhere in the commutative property above??? If you do then plug "-P" in its place.Now the commutative equation that your wrote looks exactly like the thing you are trying to prove.
Properties Of Partial Derivatives
You don't know what the commutativity of the mixed partial derivatives means but you want a proof of this fact. You should begin by finding out what this commutaivity means, then learn how the fact is used, then finally seek out the proof. Before completing these prelininaries you wouldn't be seriously ready for a proof. The theorem you want is Clairaut's theorem, and it states that if a function F(x, y) is defined on an open disk D and the partial derivatives Fxy, Fyx exist on D and are continuous there, then Fxy = Fyx. See Tom Apostol's Calculus for a proof or any other reputable text on advanced calculus. Do beware that proofs of this sort belong to Real Analysis and are not usually given in the easy going loose style of texts on the mathematical methods of physics. However, the really good rexts (Morse and Feshbach, Couranr and Hilbert, Whittaker and Watson) give rigorous proofs.