Question

Derive relationship between H and U.

Answer 1

Let $H_i$ be the enthalpy of a system in the initial state and $H_f$ be the enthalpy of a system in the final state. Let $U_i$ and $V_i$ be the internal energy and volume in the initial state and $U_f$ and $V_f$ be the internal energy and volume in a final state.

Now, as we know that,

$H=U+PV$

Therefore,

$H_i=U_i+P{V}_i.....\left(1\right)$

$H_f=U_f+P{V}_f.....\left(2\right)$

Subtracting equation $\left(1\right)$ from $\left(2\right)$, we have

$H_f-H_i=(U_f+P{V}_f)-(U_i+P{V}_f)$

$\Rightarrow H_f-H_i=(U_f-U_i)+P(V_f-V_i)$

$\Rightarrow \Delta H=\Delta U+P\Delta V.....\left(3\right)$

Here, $\Delta U$ and $P\Delta V$ are the change in internal energy and work energy respectively.

Hence equation $\left(3\right)$ is the relationship between $H$ and $U$.