Understanding the Property Relation for Enthalpy Change, dh

The question asks for the correct property relation for an infinitesimal change in enthalpy, denoted as dh. This relation is a fundamental equation in thermodynamics derived from the definitions of thermodynamic properties and the laws of thermodynamics.

Enthalpy (H) is a thermodynamic property defined as the sum of internal energy (U) and the product of pressure (P) and volume (V):

\begin{equation*} H = U + PV \end{equation*}

To find the differential change in enthalpy (dH or dh), we differentiate this equation:

\begin{equation*} dH = dU + d(PV) \end{equation*}

Using the product rule for differentiation, $d(PV) = PdV + VdP$, so the equation becomes:

\begin{equation*} dH = dU + PdV + VdP \end{equation*}

Now, we need to express dU using the laws of thermodynamics. According to the First Law of Thermodynamics, the change in internal energy (dU) is equal to the heat added to the system (dQ) minus the work done by the system (dW):

\begin{equation*} dU = dQ - dW \end{equation*}

For a reversible process, the heat transfer dQ can be expressed using the definition of entropy (S) and temperature (T): $dQ = TdS$. The work done by the system for a reversible process involving only expansion or compression is $dW = PdV$. Substituting these into the First Law equation gives the fundamental property relation for internal energy:

\begin{equation*} dU = TdS - PdV \end{equation*}

Now, substitute this expression for dU back into the equation for dH:

\begin{equation*} dH = (TdS - PdV) + PdV + VdP \end{equation*}

Notice that the $PdV$ terms cancel out:

\begin{equation*} dH = TdS + VdP \end{equation*}

This is the fundamental property relation for enthalpy change, dh. It expresses the change in enthalpy in terms of changes in entropy (dS) and pressure (dP), and the intensive properties temperature (T) and specific volume (v, often represented by V in differential relations depending on context, assuming a unit mass or total quantity). Thus, dh = Tds + vdp (using lowercase to denote specific properties as is common).

Let's examine the given options:

• Option 1: Tds – pdv (This is the relation for dU)
• Option 2: Tds + vdp (This matches our derived relation)
• Option 3: Tds – vdp (Incorrect)
• Option 4: Tds + pdv (Incorrect)

Therefore, the correct property relation for enthalpy change, dh, is Tds + vdp.