Ideal gas partial differential calculus

[CyberShot]

Homework Statement

Use partial differential calculus to show that if 3 quantities p, V, T are related to each other by some
unknown but smooth (which means all derivatives are well defined) equation of state f (P, V, T ) = 0. Then the
partial derivatives must satisfy the relation∂p/∂T = - (∂V /∂T ) / ( (∂V /∂p) )

Homework Equations

Not sure any would help in this case.

The Attempt at a Solution

I'm not even sure where to start since I haven't a proper understanding of the problem.

If p, V, and T are related then

how does f (P, V, T ) = 0 help?

 

[George Jones]

This requires some playing around with the multivariable chain rule and the implicit function theorem.

 

[George Jones]

I'll try and provide more of a start.

The equation of state $f \left( P , V , T \right) = 0$ picks out a surface in $\left( P , V , T \right)$ space, and the implicit function theorem says that (locally) on this surface, any of the three variables can be written as a function of the other two, e.g., $P = P \left( V , T \right)$. Define a new (related) function
$$\tilde{f} \left( V , T \right) = f \left( P \left( V , T \right) , V , T \right)$$
Use this and the chain rule to find $\partial \tilde{f} / \partial T$.

This is just a start.