Question

Identify the incorrect statement:

• A. \( dH = V dP - T dS \)
• B. \( \displaystyle \frac{d\left(\frac{G}{T}\right)}{d\left(\frac{1}{T}\right)} = H \)
• C. \( \displaystyle \frac{d\left(\frac{\Delta G}{T}\right)}{d\left(\frac{1}{T}\right)} = \Delta H \)
• D. \( T dG - G dT = -H dT \)

Hint:

The correct differential for enthalpy is: \( dH = T dS + V dP \). Always double-check signs when working with thermodynamic identities.

Answer 1

The Correct Option is A

Step 1: Understanding Enthalpy Differential
The total differential of enthalpy \( H \), using its definition \( H = U + PV \), is given by: \begin{equation} dH = dU + P dV + V dP \end{equation} Using the First Law of Thermodynamics and assuming reversible processes: \begin{equation} dU = T dS - P dV \end{equation} Substituting into \( dH \), we get: \begin{equation} dH = T dS + V dP \end{equation} So the correct expression is: \[ {dH = T dS + V dP} \] Therefore, option (A): \( dH = V dP - T dS \) is incorrect. Step 2: Verifying Other Options
- (B) and (C) are known thermodynamic identities derived from the Gibbs-Helmholtz equation.
- (D) is a correctly rearranged form of the temperature differential of Gibbs free energy. Conclusion: Among the options, only option (A) presents an incorrect expression for \( dH \).