Question

Using Gibbs Equation, dU=TdS-pdV show that (dS/dV) at a constant U =P/T. The reciprocal of (dS/dU)v = 1/T.

Answer 1

Explanation:

dU=TdS-pdV (given)

To prove = 1) $(\frac{dS}{dV})_U=\frac{P}{T}$ (at constant U)

2) $(\frac{dS}{dU})_v=\frac{1}{T}$ (at constant V)

Solution: 1)

dU=TdS-PdV

$PdV=TdS-dU$

$P=\frac{(TdS)}{dV}-\frac{dU}{dV}$

Derivative of constant is zero.

Given that internal energy is ,U = constant

$P=T\frac{dS}{dV}-0$

$\frac{dS)}{dV}=\frac{P}{T}$ (hence proved)

Solution: 2)

dU=TdS-PdV

Differentiating with respect to dU, we get:

$(\frac{dU}{dU})_v=T(\frac{dS}{dU})_v-P(\frac{dV}{dU})_v$

Derivative of constant is zero.

Given that volume is constant , V= constant

$1=T(\frac{dS}{dU})_v$

$(\frac{dS}{dU})_v=\frac{1}{T}$ (hence proved)