Partial Derivatives for P, S, T in Differential Calculus

[Bill Foster]

Homework Statement

I'm looking for \frac{\partial{P}}{\partial{V}} at fixed T and fixed S.

Homework Equations

P=\frac{TS}{4V}

The Attempt at a Solution

\frac{dP}{dV}=\frac{\partial{P}}{\partial{V}}+\frac{\partial{P}}{\partial{T}}\frac{dT}{dV}+\frac{\partial{P}}{\partial{S}}\frac{dS}{dV}

\frac{\partial{P}}{\partial{V}}=-\frac{TS}{4V^2}

\frac{\partial{P}}{\partial{T}}=\frac{S}{4V}

\frac{\partial{P}}{\partial{S}}=\frac{T}{4V}

\frac{dP}{dV}=\frac{\partial{P}}{\partial{V}}+\frac{\partial{P}}{\partial{T}}\frac{dT}{dV}+\frac{\partial{P}}{\partial{S}}\frac{dS}{dV}=-\frac{TS}{4V^2}+\frac{S}{4V}\frac{dT}{dV}+\frac{T}{4V}\frac{dS}{dV}

At constant T, I get this: \frac{dP}{dV}=-\frac{TS}{4V^2}+\frac{T}{4V}\frac{dS}{dV}

At constant S, I get this: \frac{dP}{dV}=-\frac{TS}{4V^2}+\frac{S}{4V}\frac{dT}{dV}

What do I do about the other differentials: \frac{dS}{dV} and \frac{dT}{dV}?

Wouldn't this also be true?

\frac{dS}{dV}=\frac{\partial{S}}{\partial{V}}+\frac{\partial{S}}{\partial{T}}\frac{dT}{dV}+\frac{\partial{S}}{\partial{P}}\frac{dP}{dV}

\frac{dT}{dV}=\frac{\partial{T}}{\partial{V}}+\frac{\partial{T}}{\partial{S}}\frac{dS}{dV}+\frac{\partial{T}}{\partial{P}}\frac{dP}{dV}

 

[Office_Shredder]

You want to find <br /> \frac{\partial{P}}{\partial{V}}<br />

and your second line is

<br /> \frac{\partial{P}}{\partial{V}}=-\frac{TS}{4V^2}<br />

So... you're done?

 

[Bill Foster]

You want to find <br /> \frac{\partial{P}}{\partial{V}}<br />

and your second line is

<br /> \frac{\partial{P}}{\partial{V}}=-\frac{TS}{4V^2}<br />

So... you're done?

Yes. That's at constant T and constant S.

Is it the same if T is constant and S is not constant? Or vice-versa?