Chapter 5: Q 5.26 (page 171)
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.
As the pressure increases entropy increases.
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Suppose you have a box of atomic hydrogen, initially at room temperature and atmospheric pressure. You then raise the temperature, keeping the volume fixed.
(a) Find an expression for the fraction of the hydrogen that is ionised as a function of temperature. (You'll have to solve a quadratic equation.) Check that your expression has the expected behaviour at very low and very high temperatures.
(b) At what temperature is exactly half of the hydrogen ionised?
(c) Would raising the initial pressure cause the temperature you found in part (b) to increase or decrease? Explain.
(d) Plot the expression you found in part (a) as a function of the dimension- less variable t = kT/I. Choose the range of t values to clearly show the interesting part of the graph.
Suppose you have a liquid (say, water) in equilibrium with its gas phase, inside some closed container. You then pump in an inert gas (say, air), thus raising the pressure exerted on the liquid. What happens?
(a) For the liquid to remain in diffusive equilibrium with its gas phase, the chemical potentials of each must change by the same amount: Use this fact and equation 5.40 to derive a differential equation for the equilibrium vapour pressure, Pv as a function of the total pressure P. (Treat the gases as ideal, and assume that none of the inert gas dissolves in the liquid.)
(b) Solve the differential equation to obtain
where the ratio V/N in the exponent is that of the liquid. (The term Pv(Pv) is just the vapour pressure in the absence of the inert gas.) Thus, the presence of the inert gas leads to a slight increase in the vapour pressure: It causes more of the liquid to evaporate.
(c) Calculate the percent increase in vapour pressure when air at atmospheric pressure is added to a system of water and water vapour in equilibrium at 25°C. Argue more generally that the increase in vapour pressure due to the presence of an inert gas will be negligible except under extreme conditions.
Let the system be one mole of argon gas at room temperature and atmospheric pressure. Compute the total energy (kinetic only, neglecting atomic rest energies), entropy, enthalpy, Helmholtz free energy, and Gibbs free energy. Express all answers in SI units.
In a hydrogen fuel cell, the steps of the chemical reaction are
Calculate the voltage of the cell. What is the minimum voltage required for electrolysis of water? Explain briefly.
A mixture of one part nitrogen and three parts hydrogen is heated, in the presence of a suitable catalyst, to a temperature of 500° 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° C is 6.9 x 10-5. (Hint: You'l have to solve a quadratic equation.)
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