Chapter 5

In a hydrogen fuel cell, the steps of the chemical reaction are at - electrode: \(\mathrm{H}_{2}+2 \mathrm{OH}^{-} \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}+2 \mathrm{e}^{-}\)at \(+\) electrode: \(\frac{1}{2} \mathrm{O}_{2}+\mathrm{H}_{2}\mathrm{O}+2 \mathrm{e}^{-} \longrightarrow 2 \mathrm{OH}^{-}\) Calculate the voltage of the cell. What is the minimum voltage required for electrolysis of water? Explain briefly.

Imagine that you drop a brick on the ground and it lands with a thud. Apparently the energy of this system tends to spontaneously decrease. Explain why.

Is heat capacity \((C)\) extensive or intensive? What about specific heat \((c) ?\) Explain briefly.

How can diamond ever be more stable than graphite, when it has less entropy? Explain how at high pressures the conversion of graphite to diamond can increase the total entropy of the carbon plus its environment.

Graphite is more compressible than diamond. (a) Taking compressibilities into account, would you expect the transition from graphite to diamond to occur at higher or lower pressure than that predicted in the text? (b) The isothermal compressibility of graphite is about \(3 \times 10^{-6} \mathrm{bar}^{-1},\) while that of diamond is more than ten times less and hence negligible in comparison. (Isothermal compressibility is the fractional reduction in volume per unit increase in pressure, as defined in Problem \(1.46 .\) ) Use this information to make a revised estimate of the pressure at which diamond becomes more stable than graphite (at room temperature).

The density of ice is \(917 \mathrm{kg} / \mathrm{m}^{3}.\) (a) Use the Clausius-Clapeyron relation to explain why the slope of the phase boundary between water and ice is negative. (b) How much pressure would you have to put on an ice cube to make it melt \(a t-1^{\circ} \mathrm{C} ?\) (c) Approximately how deep under a glacier would you have to be before the weight of the ice above gives the pressure you found in part (b)? (Note that the pressure can be greater at some locations, as where the glacier flows over a protruding rock.) (d) Make a rough estimate of the pressure under the blade of an ice skate, and calculate the melting temperature of ice at this pressure. Some authors have claimed that skaters glide with very little friction because the increased pressure under the blade melts the ice to create a thin layer of water. What do you think of this explanation?

An inventor proposes to make a heat engine using water/ice as the working substance, taking advantage of the fact that water expands as it freezes. A weight to be lifted is placed on top of a piston over a cylinder of water at \(1^{\circ} \mathrm{C}\). The system is then placed in thermal contact with a low-temperature reservoir at \(-1^{\circ} \mathrm{C}\) until the water freezes into ice, lifting the weight. The weight is then removed and the ice is melted by putting it in contact with a high-temperature reservoir at \(1^{\circ} \mathrm{C}\). The inventor is pleased with this device because it can seemingly perform an unlimited amount of work while absorbing only a finite amount of heat. Explain the flaw in the inventor's reasoning, and use the Clausius-Clapeyron relation to prove that the maximum efficiency of this engine is still given by the Carnot formula, \(1-T_{c} / T_{h}\)

When plotting graphs and performing numerical calculations, it is convenient to work in terms of reduced variables, $$t \equiv T / T_{c}, \quad p \equiv P / P_{c}, \quad v \equiv V / V_{c}$$ Rewrite the van der Waals equation in terms of these variables, and notice that the constants \(a\) and \(b\) disappear.

Plot the van der Waals isotherm for \(T / T_{c}=0.95,\) working in terms of reduced variables. Perform the Maxwell construction (either graphically or numerically) to obtain the vapor pressure. Then plot the Gibbs free energy (in units of \(N k T_{c}\) ) as a function of pressure for this same temperature and check that this graph predicts the same value for the vapor pressure.

In this problem you will investigate the behavior of a van der Waals fluid near the critical point. It is easiest to work in terms of reduced variables throughout. (a) Expand the van der Waals equation in a Taylor series in \(\left(V-V_{c}\right)\), keeping terms through order \(\left(V-V_{c}\right)^{3} .\) Argue that, for \(T\) sufficiently close to \(T_{c}\) the term quadratic in \(\left(V-V_{c}\right)\) becomes negligible compared to the others and may be dropped. (b) The resulting expression for \(P(V)\) is antisymmetric about the point \(V=V_{c}\) Use this fact to find an approximate formula for the vapor pressure as a function of temperature. (You may find it helpful to plot the isotherm.) Evaluate the slope of the phase boundary, \(d P / d T\), at the critical point. (c) Still working in the same limit, find an expression for the difference in volume between the gas and liquid phases at the vapor pressure. You should find \(\left(V_{g}-V_{l}\right) \propto\left(T_{c}-T\right)^{\beta},\) where \(\beta\) is known as a critical exponent. Experiments show that \(\beta\) has a universal value of about \(1 / 3,\) but the van der Waals model predicts a larger value. (d) Use the previous result to calculate the predicted latent heat of the transformation as a function of temperature, and sketch this function. (e) The shape of the \(T=T_{c}\) isotherm defines another critical exponent, called \(\delta\) : \(\left(P-P_{c}\right) \propto\left(V-V_{c}\right)^{\delta} .\) Calculate \(\delta\) in the van der Waals model. (Experimental values of \(\delta\) are typically around 4 or \(5 .\) ) (f) A third critical exponent describes the temperature dependence of the isothermal compressibility, $$\kappa \equiv-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T},$$ This quantity diverges at the critical point, in proportion to a power of \(\left(T-T_{c}\right)\) that in principle could differ depending on whether one approaches the critical point from above or below. Therefore the critical exponents \(\gamma\) and \(\gamma^{\prime}\) are defined by the relations $$\kappa \propto\left\\{\begin{array}{ll} \left(T-T_{c}\right)^{-\gamma} & \text { as } T \rightarrow T_{c} \text { from above } \\ \left(T_{c}-T\right)^{-\gamma^{\prime}} & \text { as } T \rightarrow T_{c} \text { from below } \end{array}\right.$$ Calculate \(\kappa\) on both sides of the critical point in the van der Waals model, and show that \(\gamma=\gamma^{\prime}\) in this model.

Consider an ideal mixture of just 100 molecules, varying in composition from pure \(A\) to pure \(B\). Use a computer to calculate the mixing entropy as a function of \(N_{A},\) and plot this function (in units of \(k\) ). Suppose you start with all \(A\) and then convert one molecule to type \(B ;\) by how much does the entropy increase? By how much does the entropy increase when you convert a second molecule, and then a third, from \(A\) to \(B ?\) Discuss.

Suppose you need a tank of oxygen that is \(95 \%\) pure. Describe a process by which you could obtain such a gas, starting with air.

Everything in this section assumes that the total pressure of the system is fixed. How would you expect the nitrogen-oxygen phase diagram to change if you increase or decrease the pressure? Justify your answer.

What happens when you spread salt crystals over an icy sidewalk? Why is this procedure rarely used in very cold climates?

What happens when you add salt to the ice bath in an ice cream maker? How is it possible for the temperature to spontaneously drop below \(0^{\circ} \mathrm{C} ?\) Explain in as much detail as you can.

Seawater has a salinity of 3.5\%, meaning that if you boil away a kilogram of seawater, when you're finished you'll have 35 g of solids (mostly \(\mathrm{NaCl}\) ) left in the pot. When dissolved, sodium chloride dissociates into separate \(\mathrm{Na}^{+}\) and \(\mathrm{Cl}^{-}\) ions. (a) Calculate the osmotic pressure difference between seawater and fresh water. Assume for simplicity that all the dissolved salts in seawater are \(\mathrm{NaCl}\). (b) If you apply a pressure difference greater than the osmotic pressure to a solution separated from pure solvent by a semipermeable membrane, you get reverse osmosis: a flow of solvent out of the solution. This process can be used to desalinate seawater. Calculate the minimum work required to desalinate one liter of seawater. Discuss some reasons why the actual work required would be greater than the minimum.

Most pasta recipes instruct you to add a teaspoon of salt to a pot of boiling water. Does this have a significant effect on the boiling temperature? Justify your answer with a rough numerical estimate.

Write down the equilibrium condition for each of the following reactions: (a) \(2 \mathrm{H} \leftrightarrow \mathrm{H}_{2}\) (b) \(2 \mathrm{CO}+\mathrm{O}_{2} \leftrightarrow 2 \mathrm{CO}_{2}\) (c) \(\mathrm{CH}_{4}+2 \mathrm{O}_{2} \leftrightarrow 2 \mathrm{H}_{2} \mathrm{O}+\mathrm{CO}_{2}\) (d) \(\mathrm{H}_{2} \mathrm{SO}_{4} \leftrightarrow 2 \mathrm{H}^{+}+\mathrm{SO}_{4}^{2-}\) (e) \(2 p+2 n \leftrightarrow^{4} \mathrm{He}\)

A mixture of one part nitrogen and three parts hydrogen is heated, in the presence of a suitable catalyst, to a temperature of \(500^{\circ} \mathrm{C}\). What fraction of the nitrogen (atom for atom) is converted to ammonia, if the final total pressure is 400 atm? Pretend for simplicity that the gases behave ideally despite the very high pressure. The equilibrium constant at \(500^{\circ} \mathrm{C}\) is \(6.9 \times 10^{-5}\). (Hint: You'll have to solve a quadratic equation.)

Derive the van't Hoff equation. $$\frac{d \ln K}{d T}=\frac{\Delta H^{\circ}}{R T^{2}}$$ which gives the dependence of the equilibrium constant on temperature." Here \(\Delta H^{\circ}\) is the enthalpy change of the reaction, for pure substances in their standard states (1 bar pressure for gases). Notice that if \(\Delta H^{\circ}\) is positive (loosely speaking, if the reaction requires the absorption of heat), then higher temperature makes the reaction tend more to the right, as you might expect. Often you can neglect the temperature dependence of \(\Delta H^{\circ}\); solve the equation in this case to obtain $$\ln K\left(T_{2}\right)-\ln K\left(T_{1}\right)=\frac{\Delta H^{\circ}}{R}\left(\frac{1}{T_{1}}-\frac{1}{T_{2}}\right)$$

Sulfuric acid, \(\mathrm{H}_{2} \mathrm{SO}_{4},\) readily dissociates into \(\mathrm{H}^{+}\) and \(\mathrm{HSO}_{4}^{-}\) ions: $$\mathrm{H}_{2} \mathrm{SO}_{4} \longrightarrow \mathrm{H}^{+}+\mathrm{HSO}_{4}^{-}$$ The hydrogen sulfate ion, in turn, can dissociate again: $$\mathrm{HSO}_{4}^{-} \longmapsto \mathrm{H}^{+}+\mathrm{SO}_{4}^{2-}$$ 'The equilibrium constants for these reactions, in aqueous solutions at \(298 \mathrm{K},\) are approximately \(10^{2}\) and \(10^{-1.9}\), respectively. (For dissociation of acids it is usually more convenient to look up \(K\) than \(\Delta G^{\circ} .\) By the way, the negative base- 10 logarithm of \(K\) for such a reaction is called \(\mathbf{p K},\) in analogy to pH. So for the first reaction \(\mathrm{pK}=-2,\) while for the second reaction \(\mathrm{pK}=1.9 .2\) (a) Argue that the first reaction tends so strongly to the right that we might as well consider it to have gone to completion, in any solution that could possibly be considered dilute. At what pH values would a significant fraction of the sulfuric acid not be dissociated? (b) In industrialized regions where lots of coal is burned, the concentration of sulfate in rainwater is typically \(5 \times 10^{-5} \mathrm{mol} / \mathrm{kg}\). The sulfate can take any of the chemical forms mentioned above. Show that, at this concentration, the second reaction will also have gone essentially to completion, so all the sulfate is in the form of \(\mathrm{SO}_{4}^{2-} .\) What is the pH of this rainwater? (c) Explain why you can neglect dissociation of water into \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\) in answering the previous question. (d) At what pH would dissolved sulfate be equally distributed between HSO \(_{4}^{-}\) and \(\mathrm{SO}_{4}^{2-} ?\)