Question 6.14: At 298 K, the thermal expansion coefficient and the isotherm...
At 298 K, the thermal expansion coefficient and the isothermal compressibility of liquid water are \beta=2.4 × 10^{-4} K^{-1} and \kappa=45.9 × 10^{-6} bar^{-1}.
a. Calculate \left(\partial U/ \partial V \right)_{T} for water at 320. K and Ρ= 1.00 bar.
b. If an external pressure equal to \left(\partial U/ \partial V \right)_{T}were applied to 1.00 m³ of liquid water at 320. K, how much would its volume change? What is the relative change in volume?
c. As shown in Example Problem 3.5, \left(\partial U/ \partial V_{m} \right)_{T}=\alpha/\mathrm{V}_{m}^{2} for a van der Waals gas. Calculate \left(\partial U/ \partial V_{m} \right)_{T}=\alpha/\mathrm{V}_{m}^{2} for N_{2}\left( g \right) at 320 K and given that \alpha=1.35 atm L^{2} mol^{-2} . Compare the value that you obtain with that of part (a) and discuss the difference.
a. \left( \frac{\partial U}{\partial V} \right)_{T}=\frac{\beta T-\kappa P}{\kappa}
=\frac{2.04×10^{-4}K^{-1}×320.K -45.9×10^{-6}bar^{-1}×1.00bar}{45.9 ×10^{-6} bar^{-1}}
=1.4 × 10^{3} bar
b. \kappa=-\frac{1}{V}\left( \frac{\partial V}{\partial P} \right)_{T}
\frac{\Delta V_{T}}{V}=-\int_{P_{i}}^{P_{f}} V\left( T \right) \kappa dP\approx -V\left( T \right)\kappa\left( P_{f}-P_{i} \right)
=-1.00 m^{3} × 45.9 × 10^{-6} bar ×\left( 1.4 × 10^{3} bar -1 bar \right)
=-6.4 × 10^{-2} m^{3}
\frac{\Delta V_{T}}{V}=-\frac{6.4 × 10^{-2}m^{3}}{ 1 m^{3}} =-6.4 \%
c. At atmospheric pressure, we can calculate V_{m} using the ideal gas law and
\left( \frac{\partial U}{\partial V_{m}} \right)_{T}=\frac{\alpha}{\mathrm{V}_{m}^{2}}=\frac{\alpha P^{2}}{R^{2}T^{2}}= \frac{1.35 atm L^{2} mol^{-2} × \left( 1.00 atm \right)^{2}}{\left( 8.206 ×10^{-2 } atm L mol^{-1} K^{-1}× 320. K\right)^{2}} =1.96 × 10^{-3} atmThe internal pressure is much smaller for N_{2} gas than for H_{2}O liquid. This is the case because H_{2}O molecules in a liquid are relatively strongly attracted to one another, whereas the interaction between gas phase N_{2} molecules is very weak. Gases, which have a small internal pressure, are more easily compressed by an external pressure than liquids or solids, which have a high internal pressure.