Two thermodynamical processes are shown in the figure. The molar heat capacity for process \( A \) and \( B \) are \( C_A \) and \( C_B \). The molar heat capacity at constant pressure and constant volume are represented by \( C_P \) and \( C_V \), respectively. Choose the correct statement.
- \( C_B = \infty, \, C_A = 0 \)
- \( C_A = 0 \, \text{and} \, C_B = \infty \)
- \( C_P > C_V > C_A = C_B \)
- \( C_A > C_P > C_V \)
The Correct Option is B
Approach Solution - 1
To solve this problem, we must analyze the thermodynamical processes shown in the figure and the given options based on the relations between molar heat capacities. The key molar heat capacities involved here are \( C_A \), \( C_B \), \( C_P \), and \( C_V \).
From the figure, we have two important processes, labeled as \( A \) and \( B \). Let's analyze each:
- Process A: Based on the slope, this appears to be a vertical line in the log-log diagram of pressure (\( P \)) versus volume (\( V \)), meaning there is no change in volume (\( dV = 0 \)). This represents an isochoric process where the heat capacity at constant volume is involved. Since \( C_A \) corresponds to this process, the work done is zero in an isochoric process, and all the heat added goes into increasing the internal energy. Therefore, \( C_A = 0 \).
- Process B: This appears as a horizontal line where pressure remains constant (\( dP = 0 \)). In an isobaric process, all the heat capacity is associated with the change in volume at constant pressure, described by the molar heat capacity \( C_P \). If \( C_P \) is involved, the slope tells us that heat input results in a change where theoretically the heat capacity becomes infinite since all added heat leads to phase change or temperature remains constant. Thus, \( C_B = \infty \).
The correct option among those provided is:
\( C_A = 0 \, \text{and} \, C_B = \infty \)
This option is consistent with the analysis of the thermodynamical processes considering their geometric representation on the log-log diagram.
Approach Solution -2
Step 1. Understanding the Slopes in the log P vs. log V Diagram:
Process A has a slope of \( \tan^{-1} \gamma \), where \( \gamma = \frac{C_P}{C_V} \), indicating an adiabatic process (since \( PV^\gamma = \text{constant} \)). Process B has a slope of \( 45^\circ \) or \( \tan^{-1} 1 \), suggesting that it is an isothermal process (since \( PV = \text{constant} \)).
Step 2. Using Heat Capacities for Adiabatic and Isothermal Processes:
For an adiabatic process (\( PV^\gamma = \text{constant} \)), the heat capacity \( C_A \) is effectively zero because no heat exchange occurs (\( dQ = 0 \) for adiabatic). For an isothermal process (\( PV = \text{constant} \)), the heat capacity \( C_B \) tends to infinity because any heat added is used to perform work without changing temperature.
Conclusion:
Therefore, the correct statement is:
\[ C_A = 0 \quad \text{and} \quad C_B = \infty \]
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