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 C

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

The heat capacity of a monatomic ideal gas is governed by how many independent ways its molecules can store kinetic energy. Let's work through each option using this idea.

A monatomic gas molecule has only 3 translational degrees of freedom (no rotational or vibrational modes contribute at ordinary temperatures), so by the equipartition theorem its molar internal energy is \[ U = \frac{3}{2}RT \] which gives a molar heat capacity of \[ C = \frac{3}{2}R = 1.5R \]

1. 2R: This value does not correspond to the 3 translational degrees of freedom of a monatomic gas; it is closer to values seen for gases with additional active modes.
2. 2.5R: This equals \( C_p = C_v + R = 1.5R + R = 2.5R \), the constant-pressure molar heat capacity of a monatomic gas, not the value being sought here.
3. 1.5R: This matches \( \tfrac{3}{2}R \), the molar heat capacity arising purely from the 3 translational degrees of freedom of a monatomic gas.
4. R: This value does not account for the full contribution of all 3 translational degrees of freedom of a monatomic gas molecule.

Since a monatomic gas stores energy only through translational motion, its heat capacity works out to \( \tfrac{3}{2}R \).

Therefore, the correct answer is 1.5R.