Question

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?

• A. It is independent of temperature
• B. It is a function of pressure
• C. It is a function of temperature
• D. It is independent of both specific volume and pressure

Hint:

For gases with non-ideal relationships like \( P(v - b) = RT \), specific heat can vary with temperature but is independent of pressure and specific volume under certain conditions.

Answer 1

The Correct Option is C, D

The given equation for the gas is: \[ P(v - b) = RT \] This equation indicates that the pressure \( P \) is related to the volume \( v \) and temperature \( T \), where \( b \) is a constant. Now, the specific heat capacity at constant pressure \( c_p \) is defined as: \[ c_p = \left( \frac{\partial h}{\partial T} \right)_P \] where \( h \) is the enthalpy, and the enthalpy \( h = u + Pv \), with \( u \) being the internal energy. From the given relation, we can see that the specific heat capacity at constant pressure will depend on the temperature because it influences the relationship between pressure and volume. Hence, the specific heat capacity is a function of temperature.
Now, let's evaluate the options:
- Option (A): Incorrect. The specific heat at constant pressure is not independent of temperature; it varies with temperature.
- Option (B): Incorrect. The specific heat at constant pressure in this case is not directly influenced by pressure, as the relationship \( P(v - b) = RT \) shows that pressure is related to volume and temperature.
- Option (C): Correct. The specific heat at constant pressure is a function of temperature. 
- Option (D): Correct. The specific heat capacity at constant pressure does not depend on the specific volume or pressure for this equation. 
Thus, the correct answer is (C) and (D).