Derive the expression, H = U + P V. - Vidyadip

Question

Derive the expression, $\Delta H =\Delta U + P \Delta V$.

✓

Answer

Enthalpy $( H )$ of a system is defined as
$H=U+P V$
where $U$ is internal energy
$P$ is pressure and $V$ is volume.
Consider a process in which a state of a system changes from an initial state $A$ to a final state $B$. Let $H_1, U_1, P_1, V_1$ and $H_2, U_2, P_2, V_2$ be the state functions of the system in initial and final states.

Then,
$H _1= U _1+ P _2 V _2 \text { and } H _2= U _2+ P _2 V _2$
The enthalpy change $\Delta H$ is given by,
$\begin{aligned}
& \Delta H=H_2-H_1 \\
& =\left(U_2+P_2 V_2\right)-\left(U_1+P_1 V_1\right) \\
& =\left(U_2-U 1\right)+\left(P_2 V_2-P_1 V_1\right) \\
& =\Delta U+\Delta P V
\end{aligned}$
where $\Delta U=U_2-U_1$
At constant pressure, $P _1= P _2= P$
$\begin{aligned}
& \therefore P_2 V_2-P_1 V_1=P V_2-P V_1 \\
& =P\left(V_2-V_1\right) \\
& =P \times \Delta V
\end{aligned}$
Hence, $\Delta H=\Delta U+P \Delta V$
This is a relation for enthalpy change.

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