Question

One mole of a monatomic ideal gas is expanded by a process described by \( PV^3 = C \), where \( C \) is a constant. The heat capacity of the gas during the process is given by (R is the gas constant)

• A. 2R
• B. 2.5R
• C. 1.5R
• D. R

Hint:

For a monatomic ideal gas, the heat capacity during a polytropic process can be derived using the equation \( C = \frac{3R}{2} \), considering the value of \( \gamma \).

Answer 1

The Correct Option is D

Step 1: Understand the process.
The process is described by the equation \( PV^3 = C \), where \( P \) is the pressure, \( V \) is the volume, and \( C \) is a constant. This equation represents a polytropic process, where the exponent of \( V \) is 3.

Step 2: Derive the heat capacity.
The heat capacity \( C \) is given by: \[ C = \frac{dQ}{dT} \] For a polytropic process, the relationship between pressure, volume, and temperature can be used to derive the heat capacity. The relationship for heat capacity during an expansion or compression is: \[ C = \frac{R(1 + \gamma)}{\gamma - 1} \] Where \( \gamma \) is the adiabatic index. For a monatomic ideal gas, \( \gamma = \frac{5}{3} \), which gives: \[ C = \frac{3R}{2} \]
Step 3: Conclusion.
Thus, the heat capacity of the gas during the process is \( 1.5R \). Therefore, the correct answer is (3) 1.5R.

Answer 2

For any polytropic process of the form \( PV^{n} = \text{constant} \), the molar heat capacity is given by \( C = C_v + \dfrac{R}{1-n} \), where \( C_v \) is the heat capacity at constant volume. Here \( n = 3 \) and, for a monatomic ideal gas, \( C_v = \dfrac{3}{2}R \). Substituting,

\[ C = \frac{3}{2}R + \frac{R}{1-3} = \frac{3}{2}R - \frac{1}{2}R = R \]

Let's check each option against this value.

1. 2R: This is larger than the value we derived, \( R \). It would correspond to a process index different from \( n=3 \), so this option is incorrect.
2. 2.5R: This equals \( C_p \) for a diatomic gas, which is unrelated to this monatomic, \( PV^{3} \) process. It does not match our result.
3. 1.5R: This is simply \( C_v \) for a monatomic gas, that is, the heat capacity at constant volume alone. It ignores the extra work term coming from the \( PV^{3} \) process, so it is not the process heat capacity asked for here.
4. R: This matches the value obtained from \( C = C_v + \dfrac{R}{1-n} \) with \( n = 3 \), so this is the correct heat capacity for the given process.

Therefore, the correct answer is R.