Deriving Results in Multivariable Calculus for Thermodynamics

[Zatman]

Homework Statement

The thermodynamic state of a system may be defined by setting any 2 of the variables P, V and T to be constant because they are related by f(P,V,T)=0.

Given a thermodynamic function K(V,T), prove that:

(∂K/∂T)_P = (∂K/∂T)_V + [(∂K/∂V)_T]*[(∂V/∂T)_P]

and that

(∂K/∂P)_T = [(∂K/∂V)_T]*[∂V/∂P)_T]

2. The attempt at a solution
To be honest I have written down various differentials and tried to combine them in various ways to get these results but to no avail.

Obviously an expression for dK would be useful:

dK = [(∂K/∂V)_T]dV + [(∂K/∂T)_V]dT

Since f(P,V,T) = 0, I can say that p is a function of V and T, and then:

dP = [(∂P/∂V)_T]dV + [(∂P/∂T)_V]dT

Similarly:

dV = [(∂V/∂P)_T]dP + [(∂V/∂T)_P]dT
dT = [(∂T/∂V)_P]dV + [(∂T/∂P)_V]dP

But I cannot combine these to make the required results, and can't think of anything else I can write down.

I did write that since P is a function of V and T and K is a function of V and T then K is a function of P -- but I've just thought this isn't actually (necessarily) correct?

Any help/hints on this would be appreciated.

 

[mighty2000]

Thank you for your question. The first step in proving these results is to use the chain rule for partial derivatives. Recall that for a function f(x,y), the chain rule states that:

(∂f/∂x)_y = (∂f/∂u)(∂u/∂x) + (∂f/∂v)(∂v/∂x)

Where u and v are intermediate variables that depend on x and y. In our case, u could be P and v could be T, or vice versa.

Now, let's apply this to our problem. We start with the first result:

(∂K/∂T)_P = (∂K/∂T)_V + [(∂K/∂V)_T]*[(∂V/∂T)_P]

Using the chain rule, we can rewrite the second term on the right-hand side as:

[(∂K/∂V)_T]*[(∂V/∂T)_P] = [(∂K/∂V)_T]*(∂P/∂T)_V

Notice that we have replaced (∂V/∂T)_P with (∂P/∂T)_V using the fact that P is a function of V and T. Now, using the definition of (∂K/∂T)_V, we can write:

(∂K/∂T)_V = (∂K/∂P)_V*(∂P/∂T)_V

Substituting this into our original expression, we get:

(∂K/∂T)_P = (∂K/∂P)_V*(∂P/∂T)_V + [(∂K/∂V)_T]*(∂P/∂T)_V

Factorizing (∂P/∂T)_V, we get the desired result:

(∂K/∂T)_P = [(∂K/∂P)_V + (∂K/∂V)_T]*(∂P/∂T)_V

Similarly, we can prove the second result:

(∂K/∂P)_T = [(∂K/∂V)_T]*(∂V/∂P)_T

Using the chain rule,