Question

Which of the following equation is the basis for the construction of Mollier diagram? (P- Pressure, V- Volume, T- Temperature, U- Internal Energy, h - Enthalpy, s - Entropy)

• A. \( \left( \frac{\partial h}{\partial s} \right)_p = P \)
• B. \( \left( \frac{\partial h}{\partial s} \right)_p = V \)
• C. \( \left( \frac{\partial h}{\partial s} \right)_p = T \)
• D. \( \left( \frac{\partial h}{\partial s} \right)_p = U \)

Hint:

The relationship \( dh = T ds + V dP \) is a crucial thermodynamic identity. Understanding how enthalpy changes with entropy and pressure is key to comprehending the Mollier diagram.

Answer 1

The Correct Option is C

Step 1: Recall the definition of enthalpy.
Enthalpy \( h \) is a thermodynamic property defined as the sum of the internal energy \( U \) and the product of pressure \( P \) and volume \( V \): $$h = U + PV$$ Step 2: Write the differential form of enthalpy.
The differential of enthalpy \( dh \) can be expressed as:
$$dh = dU + P dV + V dP$$
Step 3: Recall the first law of thermodynamics for a reversible process.
For a reversible process, the change in internal energy \( dU \) is given by:
$$dU = T ds - P dV$$
where \( T \) is the temperature and \( s \) is the entropy.
Step 4: Substitute the expression for \( dU \) into the differential of enthalpy.
Substituting \( dU = T ds - P dV \) into \( dh = dU + P dV + V dP \), we get: $$dh = (T ds - P dV) + P dV + V dP$$
$$dh = T ds + V dP$$
Step 5: Rearrange the equation to find the partial derivative \( \left( \frac{\partial h}{\partial s} \right)_p \).
We want to find the partial derivative of enthalpy with respect to entropy at constant pressure. This means we consider \( dP = 0 \). When pressure is constant, the equation \( dh = T ds + V dP \) simplifies to: $$dh = T ds$$ Now, divide both sides by \( ds \) and apply the condition of constant pressure \( p \):
$$\left( \frac{\partial h}{\partial s} \right)_p = T$$
Step 6: Understand the significance for the Mollier diagram.
The Mollier diagram (also known as the \( h-s \) diagram) is a plot of enthalpy against entropy. The slope of a constant pressure line on the Mollier diagram is given by \( \left( \frac{\partial h}{\partial s} \right)_p \), which we have shown to be equal to the temperature \( T \). This relationship is fundamental to the construction and use of the Mollier diagram in thermodynamic analysis, particularly for steam and other fluids.