Thermodynamics: Identifying the Incorrect Equation

This section identifies the incorrect thermodynamic equation among the provided options, focusing on fundamental relationships.

Analyzing Thermodynamic Relations

We will verify each option against the standard definitions and differentials for thermodynamic potentials:

• Internal Energy ($U$): The fundamental equation derived from the first and second laws for a reversible process ($dU = TdS - pdV$) represents the change in internal energy.

  $ dU = TdS - pdV $

  This matches Option 4. Hence, Option 4 is a correct equation.
• Enthalpy ($H$): Defined as $H = U + pV$. Taking the differential yields:

  $ dH = dU + pdV + Vdp $

  Substituting the expression for $dU$:

  $ dH = (TdS - pdV) + pdV + Vdp = TdS + Vdp $

  This matches Option 3. Hence, Option 3 is a correct equation.
• Helmholtz Free Energy ($A$): Defined as $A = U - TS$. Its differential is:

  $ dA = dU - SdT - TdS $

  Substituting the expression for $dU$:

  $ dA = (TdS - pdV) - SdT - TdS = -pdV - SdT $

  Option 2 provides $dA = -pdV + SdT$. Comparing this with the derived correct equation ($dA = -pdV - SdT$), we see the sign of the $SdT$ term is incorrect. Thus, Option 2 is the incorrect equation.
• Gibbs Free Energy ($G$): Defined as $G = H - TS$. Its differential is:

  $ dG = dH - SdT - TdS $

  Substituting the expression for $dH$:

  $ dG = (TdS + Vdp) - SdT - TdS = Vdp - SdT $

  This matches Option 1. Hence, Option 1 is a correct equation.

Conclusion on Incorrect Equation

The analysis confirms that Option 2, presenting $dA = -pdV + SdT$, is the incorrect thermodynamic equation.