The specific heat at constant pressure of a real gas obeying \( PV^2 = RT \) equation is:
- \( C_V + R \)
- \( \frac{R}{3} + C_V \)
- \( R \)
- \( C_V + \frac{R}{2V} \)
The Correct Option is D
Approach Solution - 1
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: \[ C = C_v + \frac{R}{2V}. \]
Approach Solution -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:
\( C_V + \frac{R}{2V} \)
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