Question

Using the Maxwell relations, determine a relation for \((\partial s / \partial P)_{T}\) for a gas whose equation of state is \(P(v-b)=\) $R T .

Short answer

Question: Determine an expression for the partial derivative of entropy (s) with respect to pressure (P) for a gas at constant temperature (T), given the equation of state \(P(v-b) = R T\). Answer: \((\frac{\partial s}{\partial P})_T = - (\frac{R}{P})\)

Step-by-step solution

Identify the specific Maxwell relation needed

In this problem, we need the Maxwell's relation that involves \((\partial s / \partial P)_T\). The specific relation that we need is: \( (\frac{\partial s}{\partial P})_T = - (\frac{\partial v}{\partial T})_P \) With this relation, we need to find an expression for \((\partial v / \partial T)_P\) using the given equation of state \(P(v-b) = R T\).

Rearrange the given equation of state

To find the expression for \((\partial v / \partial T)_P\), we need to rearrange the given equation of state to make v the subject: \( P(v-b) = RT \) Divide through by P: \(v-b = \frac{RT}{P}\) Add b to both sides: \(v = \frac{RT}{P} + b\)

Differentiate v with respect to T at constant P

Now differentiate v with respect to T, while keeping P constant: \((\frac{\partial v}{\partial T})_P = \frac{R \partial (T)}{\partial (T)P} = \frac{R}{P}\)

Substitute back into the Maxwell relation

Substitute the expression we found for \((\partial v / \partial T)_P\) back into the Maxwell relation: \( (\frac{\partial s}{\partial P})_T = - (\frac{\partial v}{\partial T})_P \) \( (\frac{\partial s}{\partial P})_T = - (\frac{R}{P}) \) So, the relation we were looking for is: \((\frac{\partial s}{\partial P})_T = - (\frac{R}{P})\)