Question

If 
\(h\) is the mass specific enthalpy,  
\(s\) is the mass specific entropy, 
\(P\) is the pressure, 
\(T\) is the temperature, 
\(C_V\) is the mass specific heat at constant volume, 
\(C_P\) is the mass specific heat at constant pressure,  
\(\beta\) is the coefficient of thermal expansion,  
\(v\) is the mass specific volume,  
\(\kappa\) is the isothermal compressibility,
then the partial derivative  \(\left( \frac{\partial h}{\partial s} \right)_P\) is

• A. \( \left( T - \frac{1}{\beta} \right) \left( \frac{C_P}{C_V} \right) \)
• B. \( \left( T - \frac{1}{\beta} \right) \)
• C. \( T \left( 1 - \frac{v \beta}{\kappa C_V} \right) \)
• D. \( T \)

Hint:

The partial derivative \( \left( \frac{\partial h}{\partial s} \right)_P \) simplifies to the temperature \( T \) at constant pressure, as shown by the first law of thermodynamics.

Answer 1

The Correct Option is D

The partial derivative \( \left( \frac{\partial h}{\partial s} \right)_P \) can be derived from the first law of thermodynamics for a closed system. The first law is given by: \[ dh = T \, ds + v \, dP \] At constant pressure (\( dP = 0 \)), the equation simplifies to: \[ dh = T \, ds \] Thus, the partial derivative of \( h \) with respect to \( s \) at constant pressure is: \[ \left( \frac{\partial h}{\partial s} \right)_P = T \] Therefore, the correct answer is \( T \).
Final Answer: \( T \)