Question 2.22: Estimate the change in Û when this compound melts at atmosph...

For Solids (1) and Liquids (2):

\mathrm{H}=\mathrm{U}+\mathrm{PV}

\mathrm{U}=\mathrm{H}-\mathrm{PV}

\Delta \mathrm{U}=\mathrm{U}_{2}-\mathrm{U}_{1}=\left(\mathrm{H}_{2}-\mathrm{H}_{1}\right)-\mathrm{P}\left(\mathrm{V}_{2}-\mathrm{V}_{1}\right) \quad (Only valid for constant pressure)

Divide both sides by mass to get,

\Delta \widehat{\mathrm{U}}=\widehat{\mathrm{U}}_{2}-\widehat{\mathrm{U}}_{1}=\left(\widehat{\mathrm{H}}_{2}-\widehat{\mathrm{H}}_{1}\right)-\mathrm{P}\left(\widehat{\mathrm{V}}_{2}-\widehat{\mathrm{V}}_{1}\right)

Since the process is melting, \widehat{\mathrm{H}}_{2}-\widehat{\mathrm{H}}_{1} is \Delta \widehat{\mathrm{H}}^{\text {fus }}.

\Delta \widehat{\mathrm{U}}=\widehat{\mathrm{U}}_{2}-\widehat{\mathrm{U}}_{1}=\Delta \widehat{\mathrm{H}}^{\text {fus }}-\mathrm{P}\left(\widehat{\mathrm{V}}_{2, \text { liquid }}-\widehat{\mathrm{V}}_{1, \text { solid }}\right)

Plugging in given values, \widehat{\mathrm{V}}_{\text {solid } / \text { liquid }}=\frac{1}{\rho}

\Delta \widehat{\mathrm{U}}=\frac{75~ \mathrm{BTU}}{\mathrm{lb}_{\mathrm{m}}}-(1 \mathrm{~atm})\left(\frac{\mathrm{ft}^{3}}{30 \,\mathrm{lb}_{\mathrm{m}}}-\frac{\mathrm{ft}^{3}}{40 \,\mathrm{lb}_{\mathrm{m}}}\right)

Converting from \frac{\operatorname{atm} \mathrm{ft}^{3}}{\mathrm{lb}_{\mathrm{m}}} to \frac{\mathrm{BTU}}{\mathrm{lb}_{\mathrm{m}}},

Notice that the PV term is small and can often be neglected when we are dealing with liquids and solids. The PV term is not small for typical gases and vapors.