Consider a gas obeying the relation
\[
P(v - b) = RT
\]
where \(b\) and \(R\) are constants. Which of the following statement(s) is/are CORRECT about the specific heat capacity at constant pressure?
Show Hint
- It is independent of temperature
- It is a function of pressure
- It is a function of temperature
- It is independent of both specific volume and pressure
The Correct Option is C, D
Solution and Explanation
To analyze the specific heat capacity at constant pressure for a gas obeying the relation \( P(v - b) = RT \), where \( b \) and \( R \) are constants, let's first understand the expression and its implications:
The given equation can be rearranged to find volume:
\( v = \frac{RT}{P} + b \)
For specific heat capacity at constant pressure, denoted by \( C_P \), we typically express it using the relation of enthalpy:
\( C_P = \left( \frac{\partial H}{\partial T} \right)_P \)
Since enthalpy \( H = U + Pv \), and for an ideal-like gas under this modified equation:
\( H = U + P\left(\frac{RT}{P} + b\right) = U + RT + Pb \)
Taking the partial derivative with respect to temperature \( T \) at constant pressure \( P \), we have:
\( C_P = \left( \frac{\partial U}{\partial T} \right)_P + R \)
The term \(\left( \frac{\partial U}{\partial T} \right)_P\) typically involves internal specific volumetric contributions, but with constant \( b \), \( U \) itself may not depend on \( P \), following assumptions for ideal gas-like behavior specific to this modified state equation.
Thus, \( C_P \) is directly influenced by the term dependent on temperature through internal energy and indirectly through interactions described by \( R \).
Conclusion: The specific heat capacity at constant pressure depends on temperature due to the relation with internal energy and is independent of specific volume and pressure based on the structural dependencies within the modified state equation provided in the problem.
Correct Statements:
- It is a function of temperature.
- It is independent of both specific volume and pressure.
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