Question

One mole of an ideal monatomic gas expands along the polytrope $P V^3 = \text{constant}$ from $V_1$ to $V_2$ at a constant pressure $P_1$. The temperature during the process is such that molar specific heat $C_v = \frac{3R}{2}$. The total heat absorbed during the process can be expressed as

• A. P1V1 (V1 2 / V2 2 + 1)
• B. P1V1 (V2 1 / V2 2 -1)
• C. P1V1 ( V3 1/V2 2 -1)
• D. P1V1 ( V1 / V2 2 -1)

Answer 1

The Correct Option is B

We are given the polytropic equation for an ideal monatomic gas: \[ P V^3 = \text{constant} \] and that the gas expands from volume \( V_1 \) to volume \( V_2 \) at a constant pressure \( P_1 \), with molar specific heat \( C_v = \frac{3R}{2} \).

Step 1: Understanding the Process
The heat absorbed \( Q \) during a thermodynamic process can be expressed as: \[ Q = n C_v \Delta T \] where \( n \) is the number of moles of the gas, \( C_v \) is the molar specific heat at constant volume, and \( \Delta T \) is the change in temperature. The polytropic process relationship gives us a way to relate the temperature change to the volume change. Since \( P V^3 = \text{constant} \), we can write: \[ P = \frac{\text{constant}}{V^3} \] This equation can be used to determine the temperature change, as \( P = \frac{nRT}{V} \), where \( R \) is the gas constant and \( T \) is the temperature. 

Step 2: Deriving the Heat Absorbed
We know that the total work done in an ideal gas process is given by: \[ W = P \Delta V \] The total heat absorbed is related to the work done and the change in internal energy \( \Delta U \), which for a monatomic ideal gas is \( \Delta U = n C_v \Delta T \). Using these relations, we can express the total heat absorbed \( Q \) during the polytropic expansion from \( V_1 \) to \( V_2 \). The expression for the total heat absorbed turns out to be: \[ Q = P_1 V_1 \left( \frac{V_1^2}{V_2^2} + 1 \right) \]

Answer:

\[ \boxed{P_1 V_1 \left( \frac{V_1^2}{V_2^2} + 1 \right)} \]