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Use the vapor pressures for tetrachloromethane given in the following table to estimate the temperature and pressure of the triple point and also the enthalpies of fusion, vaporization, and sublimation. $$\begin{array}{lll} \text { Phase } & T(\mathbf{K}) & P(\mathbf{P a}) \\ \hline \text { Solid } & 230 . & 225.7 \\ \text { Solid } & 250 . & 905 \\ \text { Liquid } & 280 . & 6440 . \\ \text { Liquid } & 340 . & 62501 \end{array}$$

A cell is roughly spherical with a radius of \(20.0 \times 10^{-6} \mathrm{m} .\) Calculate the work required to expand the cell surface against the surface tension of the surroundings if the radius increases by a factor of three. Assume the cell is surrounded by pure water and that \(T=298.15 \mathrm{K}\).

It has been suggested that the surface melting of ice plays a role in enabling speed skaters to achieve peak performance. Carry out the following calculation to test this hypothesis. At 1 atm pressure, ice melts at \(273.15 \mathrm{K}\) \(\Delta H_{f u s i o n}=6010 . \mathrm{Jmol}^{-1},\) the density of ice is \(920 . \mathrm{kg} \mathrm{m}^{-3}\), and the density of liquid water is \(997 . \mathrm{kg} \mathrm{m}^{-3}\). a. What pressure is required to lower the melting temperature by \(4.0^{\circ} \mathrm{C} ?\) b. Assume that the width of the skate in contact with the ice has been reduced by sharpening to \(19 \times 10^{-3} \mathrm{cm}\) and that the length of the contact area is \(18 \mathrm{cm} .\) If a skater of mass \(78 \mathrm{kg}\) is balanced on one skate, what pressure is exerted at the interface of the skate and the ice? c. What is the melting point of ice under this pressure? d. If the temperature of the ice is \(-4.0^{\circ} \mathrm{C},\) do you expect melting of the ice at the ice-skate interface to occur?

The vapor pressure of ethanol( \(l\) ) is given by \\[\ln \left(\frac{P}{\mathrm{Pa}}\right)=23.58-\frac{3.6745 \times 10^{3}}{\frac{T}{\mathrm{K}}-46.702}\\] a. Calculate the standard boiling temperature. b. Calculate \(\Delta H_{\text {vaporization}}\) at \(298 \mathrm{K}\) and at the standard boiling temperature.

Benzene ( \(l\) ) has a vapor pressure of 0.1269 bar at \(298.15 \mathrm{K}\) and an enthalpy of vaporization of \(30.72 \mathrm{kJ} \mathrm{mol}^{-1}\). The \(C_{P, m}\) of the vapor and liquid phases at that temperature are 82.4 and \(136.0 \mathrm{JK}^{-1} \mathrm{mol}^{-1}\), respectively. Calculate the vapor pressure of \(\mathrm{C}_{6} \mathrm{H}_{6}(l)\) at \(340.0 \mathrm{K}\) assuming a. that the enthalpy of vaporization does not change with temperature. b. that the enthalpy of vaporization at temperature \(T\) can be calculated from the equation \(\Delta H_{\text {vaporization}}(T)=\) \(\Delta H_{\text {vaporiation}}\left(T_{0}\right)+\Delta C_{P}\left(T-T_{0}\right)\) assuming that \(\Delta C_{P}\) does not change with temperature.

The vapor pressure of an unknown solid is given by \(\ln (P / \text { Torr })=22.413-2035(\mathrm{K} / T),\) and the vapor pressure of the liquid phase of the same substance is approximately given by \(\ln (P / \mathrm{Torr})=18.352-1736(\mathrm{K} / T)\). a. Calculate \(\Delta H_{\text {vaporization}}\) and \(\Delta H_{\text {sublimation }}\). b. Calculate \(\Delta H_{\text {fusion}}\). c. Calculate the triple point temperature and pressure.

The vapor pressure of methanol( \(l\) ) is \(16.94 \times 10^{3}\) Pa at \(298.15 \mathrm{K} .\) Use this value to calculate \(\Delta G_{f}^{\circ}\left(\mathrm{CH}_{3} \mathrm{OH}, g\right)-\) \(\Delta G_{f}^{\circ}\left(\mathrm{CH}_{3} \mathrm{OH}, l\right)\). Compare your result with those in Table 4.1.

The normal melting point of \(\mathrm{H}_{2} \mathrm{O}\) is \(273.15 \mathrm{K}\), and \(\Delta H_{f u s i o n}=6010 . \mathrm{Jmol}^{-1} .\) Calculate the change in the normal freezing point at \(100 .\) and \(500 .\) bar compared to that at 1 bar assuming that the density of the liquid and solid phases remains constant at 997 and \(917 \mathrm{kg} \mathrm{m}^{-3},\) respectively. Explain why your answer is positive (or negative).

Carbon tetrachloride melts at \(250 .\) K. The vapor pressure of the liquid is 10,539 Pa at \(290 .\) K and 74,518 Pa at 340. K. The vapor pressure of the solid is \(270 .\) Pa at \(232 \mathrm{K}\) and 1092 Pa at \(250 .\) K. a. Calculate \(\Delta H_{\text {vaporization}}\) and \(\Delta H_{\text {sublimation}}\). b. Calculate \(\Delta H_{\text {fusion}}\). c. Calculate the normal boiling point and \(\Delta S_{\text {vaporization}}\) at the boiling point. d. Calculate the triple point pressure and temperature.

In this problem, you will calculate the differences in the chemical potentials of ice and supercooled water and of steam and superheated water, all at 1 atm pressure shown schematically in Figure \(8.1 .\) For this problem, \(S_{H_{2} O, s}^{\circ}=48.0 \mathrm{J} \mathrm{mol}^{-1} \mathrm{K}^{-1}, S_{H_{2}}^{\circ} 0, l=70.0 \mathrm{J} \mathrm{mol}^{-1} \mathrm{K}^{-1},\) and \(S_{H_{2}, O, g}^{\circ}=188.8 \mathrm{J} \mathrm{mol}^{-1} \mathrm{K}^{-1}\). a. By what amount does the chemical potential of water exceed that of ice at \(-2.25^{\circ} \mathrm{C} ?\) b. By what amount does the chemical potential of water exceed that of steam at \(102.25^{\circ} \mathrm{C} ?\)