# Thermodynamics problem

## Homework Statement

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Use the equation $$C_p - C_v = \left[ P + \left( \frac {∂U}{∂V} \right)_T\right] \left[ \left( \frac {∂V}{∂T} \right)_P \right]$$ to find ##C_p - C_v## for an ideal monatomic gas.

## Homework Equations

##U = \frac {3}{2} RT##
##PV = RT##

## The Attempt at a Solution

I substitute ##PV = RT## into the expression for ##U## to get ##U = \frac {3}{2} PV##, therefore

$$\left( \frac {∂U}{∂V} \right)_T = \frac {3}{2} P$$

Since ##PV=RT ⇒ V = \frac {RT}{P}##,

$$\left( \frac {∂V}{∂T} \right)_P = \frac {R}{P}$$

Therefore ##C_p - C_v = [P + \frac {3}{2} P][\frac {R}{P}] = \frac {5}{2} R## but the real answer is ##C_p - C_v = R##. What have I done wrong? I can't seem to find the mistake.

haruspex
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I substitute ##PV = RT## into the expression for ##U## to get ##U = \frac {3}{2} PV##, therefore

$$\left( \frac {∂U}{∂V} \right)_T = \frac {3}{2} P$$
The T suffix in
$$\left( \frac {∂U}{∂V} \right)_T$$
means keeping T constant. In performing the differentiation you appear to have kept P constant instead.

The T suffix in
$$\left( \frac {∂U}{∂V} \right)_T$$
means keeping T constant. In performing the differentiation you appear to have kept P constant instead.
Oh I see. So the partial derivative would be 0?

haruspex