Question

N moles of an ideal gas undergo a two-step process as shown in the figure. Let \( P, V, T \) denote the pressure, volume and temperature of the gas. The gas, initially at state-1 \( (P_1, V_1, T_1) \), undergoes an isochoric (constant volume) process to reach state-A, and then undergoes an isobaric (constant pressure) expansion to reach state-2 \( (P_2, V_2, T_2) \). For an ideal gas, \( C_P - C_V = NR \), where \( C_P \) and \( C_V \) are heat capacities at constant pressure and volume. The heat gained by the gas in the two-step process is given by 

 

• A. \( P_2 (V_2 - V_1) + C_V (T_2 - T_1) \)
• B. \( P_2 (V_2 - V_1) + C_P (T_2 - T_1) \)
• C. \( C_P (T_2 - T_1) + C_V (T_2 - T_1) \)
• D. \( P_2 V_2 - P_1 V_1 \)

Hint:

For multistep thermodynamic processes, compute heat for each sub-process separately using the correct specific heat, then sum the contributions.

Answer 1

The Correct Option is A

The process occurs in two steps: an isochoric (constant volume) heating from state-1 to state-A, followed by an isobaric (constant pressure) expansion from state-A to state-2. The total heat added is the sum of heat added in both steps.

Step 1: Heat added during isochoric process.
At constant volume, the heat gained is \[ Q_{\text{isochoric}} = C_V (T_A - T_1). \] Since state-A lies vertically below state-2 in the figure and the final temperature is \( T_2 \), we have \( T_A = T_2 \). Therefore, \[ Q_{\text{isochoric}} = C_V (T_2 - T_1). \]

Step 2: Heat added during isobaric process.
At constant pressure \( P_2 \), heat added is \[ Q_{\text{isobaric}} = P_2 (V_2 - V_1). \]

Step 3: Total heat added.
Adding both contributions gives \[ Q = C_V (T_2 - T_1) + P_2 (V_2 - V_1). \] This matches option (A).

Final Answer: \( P_2 (V_2 - V_1) + C_V (T_2 - T_1) \)