91Ó°ÊÓ

The entropy and free energy of gas expansion. Two moles of an ideal gas undergo an irreversible isothermal expansion from \(V_{A}=100\) liters to \(V_{B}=300\) liters at \(T=300 \mathrm{~K}\). (a) What is the entropy change for this process? (b) What is the Gibbs free energy change?

Computing enthalpy and entropy with a temperaturedependent heat capacity. The heat capacity for liquid \(n\)-butane depends on temperature: $$ C_{p}(T)=a+b T, $$ where \(a=100 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\) and \(b=0.1067 \mathrm{JK}^{-2} \mathrm{~mol}^{-1}\), from its freezing temperature \(T_{f} \approx 140 \mathrm{~K}\) to \(T_{b} \approx 270 \mathrm{~K}\), its boiling temperature. (a) Compute \(\Delta H\) for heating liquid butane from \(T_{A}=\) \(170 \mathrm{~K}\) to \(T_{B}=270 \mathrm{~K}\). (b) Compute \(\Delta S\) for the same process.

What powers a hurricane? Example \(7.11\) of Chapter 7 describes the thermodynamics of a hurricane, called a Hadley cycle: air takes up heat and water vapor from a warm ocean, then rises, then releases the heat high in the atmosphere. The reason a hurricane is so strong is that its Hadley cycle is augmented by an additional ingredient, the vaporization of warm water. You can saturate \(1 \mathrm{~kg}\) of air with \(20 \mathrm{~g}\) of water vapor. (Consider air as consisting of \(\mathrm{N}_{2}\) gas.) (a) If the density of saturated air is \(1.25 \mathrm{~kg} \mathrm{~m}^{-3}\), how many grams of water vapor will occupy \(1 \mathrm{~m}^{3}\) of air? (b) If the enthalpy of vaporization of water is \(2.3 \times 10^{6} \mathrm{~J}(\mathrm{~kg} \text { of water })^{-1}\), how much heat \(q\) is given off by the water when its vapor condenses into rain, per \(\mathrm{m}^{3}\) of air? (c) If the heat \(q\) that is given off by water's condensation all goes into raising the temperature of the surrounding air (at \(p=1\) atm), what is the temperature increase \(\Delta T\) ?