Question

\(12-17 \quad\) Using the Maxwell relations and the ideal-gas equation of state, determine a relation for \((\partial s / \partial v)_{T}\) for an ideal gas. Answer: \(R / v\).

Short answer

Question: Determine \((\partial s / \partial v)_{T}\) for an ideal gas using Maxwell relations and the ideal gas equation of state. Answer: \(R / v\).

Step-by-step solution

Identify the Maxwell relation involving entropy and specific volume

The relevant Maxwell relation can be found from the Second Partial Derivative of Gibbs free energy with respect to Temperature (T) and specific volume (v): $$ \left(\frac{\partial T}{\partial s}\right)_v = \left( \frac{\partial P}{\partial v} \right)_T $$

Express the relevant partial derivatives in terms of entropy and specific volume

Since we are looking for a relation for \((\partial s / \partial v)_{T}\), we can rewrite the Maxwell relation in terms of \((\partial s / \partial v)_{T}\) as: $$ \left( \frac{\partial s}{\partial v} \right)_T = \frac{1}{\left(\frac{\partial T}{\partial s}\right)_v \left( \frac{\partial P}{\partial v} \right)_T} $$

Substitute the ideal gas equation of state

From the ideal gas equation \(PV = RT\), we have \(P = \frac{RT}{v}\). Now, the partial derivative of Pressure with respect to specific volume at constant temperature is: $$ \left(\frac{\partial P}{\partial v}\right)_T = \frac{-RT}{v^2} $$

Find the partial derivative of Temperature with respect to entropy at constant specific volume

To find the partial derivative of Temperature with respect to entropy at constant specific volume, we first need to recall the definition of specific heat capacity at constant volume \(c_v\): $$ c_v = T \left(\frac{\partial s}{\partial T}\right)_v $$ Now, we can find the inverse of this relation: $$ \left(\frac{\partial T}{\partial s}\right)_v = \frac{c_v}{T} $$

Substitute the partial derivatives into the Maxwell relation

Now, we will substitute the expressions for the partial derivatives of Temperature and Pressure into the Maxwell relation derived in Step 2: $$ \left( \frac{\partial s}{\partial v} \right)_T = \frac{1}{\left(\frac{c_v}{T}\right)\left(\frac{-RT}{v^2}\right)} $$

Simplify the expression

By simplifying the expression obtained in Step 5, we get the desired relation for \((\partial s / \partial v)_{T}\): $$ \left( \frac{\partial s}{\partial v} \right)_T = \frac{R}{v} $$ Answer: \(R / v\).