Problem 28
The amino acid glycine dimerizes to form the dipeptide glycylglycine according to the reaction \\[ 2 \text { Glycine }(s) \rightarrow \text { Glycylglycine }(s)+\mathrm{H}_{2} \mathrm{O}(l) \\] Calculate \(\Delta S, \Delta S_{\text {surr}},\) and \(\Delta S_{\text {universe}}\) at \(T=298 \mathrm{K}\). Useful thermodynamic data follow: $$\begin{array}{lccc} & \text { Glycine } & \text { Glycylglycine } & \text { Water } \\ \hline \Delta H_{f}^{\circ}\left(\mathrm{kJ} \mathrm{mol}^{-1}\right) & -537.2 & -746.0 & -285.8 \\ S_{m}^{\circ}\left(\mathrm{JK}^{-1} \mathrm{mol}^{-1}\right) & 103.5 & 190.0 & 70.0 \end{array}$$
Problem 29
One mole of \(\mathrm{H}_{2} \mathrm{O}(l)\) is supercooled to \(-3.75^{\circ} \mathrm{C}\) at 1 bar pressure. The freezing temperature of water at this pressure is \(0.00^{\circ} \mathrm{C} .\) The transformation \(\mathrm{H}_{2} \mathrm{O}(l) \rightarrow \mathrm{H}_{2} \mathrm{O}(s)\) is suddenly observed to occur. By calculating \(\Delta S\) \(\Delta S_{\text {surroundings}},\) and \(\Delta S_{\text {total}},\) verify that this transformation is spontaneous at \(-3.75^{\circ} \mathrm{C}\). The heat capacities are given by \(C_{P, m}\left(\mathrm{H}_{2} \mathrm{O}(l)\right)=75.3 \mathrm{J} \mathrm{K}^{-1} \mathrm{mol}^{-1}\) and \(C_{P m}\left(\mathrm{H}_{2} \mathrm{O}(s)\right)=37.7 \mathrm{J} \mathrm{K}^{-1}\) \(\mathrm{mol}^{-1},\) and \(\Delta H_{f u s i o n}=6.008 \mathrm{kJ} \mathrm{mol}^{-1}\) at \(0.00^{\circ} \mathrm{C} .\) Assume that the surroundings are at \(-3.75^{\circ} \mathrm{C}\). [Hint: Consider the two pathways at 1 bar: (a) \(\mathrm{H}_{2} \mathrm{O}\left(l,-3.75^{\circ} \mathrm{C}\right) \rightarrow \mathrm{H}_{2} \mathrm{O}\left(s,-3.75^{\circ} \mathrm{C}\right)\) and (b) \(\mathrm{H}_{2} \mathrm{O}\left(l,-3.75^{\circ} \mathrm{C}\right) \rightarrow \mathrm{H}_{2} \mathrm{O}\left(l, 0.00^{\circ} \mathrm{C}\right) \rightarrow \mathrm{H}_{2} \mathrm{O}(s\) \(\left.0.00^{\circ} \mathrm{C}\right) \rightarrow \mathrm{H}_{2} \mathrm{O}\left(s,-3.75^{\circ} \mathrm{C}\right) .\) Because \(S\) is a state function, \(\Delta S\) must be the same for both pathways.
Problem 33
A refrigerator is operated by a 0.25-hp (1 hp = \(746 \text { watts })\) motor. If the interior is to be maintained at \(4.50^{\circ} \mathrm{C}\) and the room temperature on a hot day is \(38^{\circ} \mathrm{C}\), what is the maximum heat leak (in watts) that can be tolerated? Assume that the coefficient of performance is \(50 \%\) of the maximum theoretical value. What happens if the leak is greater than your calculated maximum value?
Problem 35
Between \(0^{\circ} \mathrm{C}\) and \(100^{\circ} \mathrm{C}\), the heat capacity of \(\mathrm{Hg}(l)\) is given by \\[ \frac{C_{P, m}(\mathrm{Hg}, l)}{\mathrm{JK}^{-1} \mathrm{mol}^{-1}}=30.093-4.944 \times 10^{-3} \frac{T}{\mathrm{K}} \\] Calculate \(\Delta H\) and \(\Delta S\) if 2.25 moles of \(\mathrm{Hg}(l)\) are raised in temperature from \(0.00^{\circ}\) to \(88.0^{\circ} \mathrm{C}\) at constant \(P\)
Problem 39
a. Calculate \(\Delta S\) if 1.00 mol of liquid water is heated from \(0.00^{\circ}\) to \(10.0^{\circ} \mathrm{C}\) under constant pressure and if \(C_{P, m}=\) \(75.3 \mathrm{JK}^{-1} \mathrm{mol}^{-1}\) b. The melting point of water at the pressure of interest is \(0.00^{\circ} \mathrm{C}\) and the enthalpy of fusion is \(6.010 \mathrm{kJ} \mathrm{mol}^{-1} .\) The boiling point is \(100 .^{\circ} \mathrm{C}\) and the enthalpy of vaporization is \(40.65 \mathrm{kJ} \mathrm{mol}^{-1} .\) Calculate \(\Delta S\) for the transformation \(\mathrm{H}_{2} \mathrm{O}\left(s, 0^{\circ} \mathrm{C}\right) \rightarrow \mathrm{H}_{2} \mathrm{O}\left(g, 100 \cdot^{\circ} \mathrm{C}\right)\)
Problem 42
The mean solar flux at Earth's surface is \(\sim 2.00 \mathrm{J}\) \(\mathrm{cm}^{-2} \min ^{-1} .\) In a nonfocusing solar collector, the temperature reaches a value of \(79.5^{\circ} \mathrm{C}\). A heat engine is operated using the collector as the hot reservoir and a cold reservoir at \(298 \mathrm{K}\). Calculate the area of the collector needed to produce \(1000 .\) W. Assume that the engine operates at the maximum Carnot efficiency.
Problem 43
An ideal gas sample containing 1.75 moles for which \(C_{V, m}=5 R / 2\) undergoes the following reversible cyclical process from an initial state characterized by \(T=275 \mathrm{K}\) and \(P=1.00\) bar: a. It is expanded reversibly and adiabatically until the volume triples. b. It is reversibly heated at constant volume until \(T\) increases to \(275 \mathrm{K}\) c. The pressure is increased in an isothermal reversible compression until \(P=1.00\) bar. Calculate \(q, w, \Delta U, \Delta H,\) and \(\Delta S\) for each step in the cycle, and for the total cycle.
Problem 45
The standard entropy of \(\mathrm{Pb}(s)\) at \(298.15 \mathrm{K}\) is \(64.80 \mathrm{J}\) \(\mathrm{K}^{-1} \mathrm{mol}^{-1}\). Assume that the heat capacity of \(\mathrm{Pb}(s)\) is given by \\[ \frac{C_{P, m}(\mathrm{Pb}, s)}{\mathrm{J} \mathrm{mol}^{-1} \mathrm{K}^{-1}}=22.13+0.01172 \frac{T}{\mathrm{K}}+1.00 \times 10^{-5} \frac{T^{2}}{\mathrm{K}^{2}} \\] The melting point is \(327.4^{\circ} \mathrm{C}\) and the heat of fusion under these conditions is \(4770 . \mathrm{J} \mathrm{mol}^{-1}\). Assume that the heat capacity of \(\mathrm{Pb}(l)\) is given by \\[ \frac{C_{P, m}(\mathrm{Pb}, l)}{\mathrm{J} \mathrm{K}^{-1} \mathrm{mol}^{-1}}=32.51-0.00301 \frac{T}{\mathrm{K}} \\] a. Calculate the standard entropy of \(\mathrm{Pb}(l)\) at \(725^{\circ} \mathrm{C}\). b. Calculate \(\Delta H\) for the transformation \(\mathrm{Pb}\left(s, 25.0^{\circ} \mathrm{C}\right) \rightarrow\) \(\mathrm{Pb}\left(L, 725^{\circ} \mathrm{C}\right)\)
