Question

The specific heat at constant pressure of a real gas obeying \( PV^2 = RT \) equation is:

• A. \( C_V + R \)
• B. \( \frac{R}{3} + C_V \)
• C. \( R \)
• D. \( C_V + \frac{R}{2V} \)

Answer 1

The Correct Option is D

The first law of thermodynamics gives: \[ dQ = du + dW \]

At constant pressure, this becomes: \[ C dT = C_v dT + P dV \tag{1} \]

Given \( PV^2 = RT \), differentiating both sides with respect to \( T \) at constant \( P \): \[ P(2V dV) = R dT \] \[ P dV = \frac{R}{2V} dT \]

Substitute \( P dV \) into equation (1): \[ C dT = C_v dT + \frac{R}{2V} dT \] \[ C = C_v + \frac{R}{2V} \]

Thus, the specific heat at constant pressure is: \[ C = C_v + \frac{R}{2V}. \]

Answer 2

To determine the specific heat at constant pressure of a real gas obeying the equation \( PV^2 = RT \), we start by analyzing the properties and behavior of gases in relation to specific heats.

For an ideal gas, the specific heat at constant pressure \( C_P \) is given by:

\(C_P = C_V + R\)

where \( C_V \) is the specific heat at constant volume, and \( R \) is the gas constant.

In this problem, the given equation for the gas is:

\(PV^2 = RT\)

Rearranging this, we find the value of pressure \( P \) as:

\(P = \frac{RT}{V^2}\)

To find \( \frac{\partial V}{\partial T} \) (at constant pressure), use the chain rule and the properties of derivatives applied to the equation:

\(\frac{d(PV^2)}{dT} = \frac{d(RT)}{dT}\)

Since \( P \) is constant and differentiating both sides gives:

\(0 = R - 2PV \frac{\partial V}{\partial T}\)

Solving for \( \frac{\partial V}{\partial T} \):

\(\frac{\partial V}{\partial T} = \frac{R}{2PV}\)

Substituting \( P = \frac{RT}{V^2} \), we integrate these results to find \( C_P \):

\(C_P = C_V + V \left( \frac{\partial P}{\partial T} \right)_V = C_V + \frac{R}{2V}\)

This suggests that the specific heat at constant pressure for this gas is:

\(C_P = C_V + \frac{R}{2V}\)

Therefore, the correct answer is:

\( C_V + \frac{R}{2V} \)