(i) Make notes for yourself on the following and *learn by heart* the following derivations:

(a) 1st and 2nd laws of thermodynamics, and the derivation from them of the Gibbs-Helmholtz equation, Clapeyron equation, Clapeyron-Clausius equation, Hess' Law, Kirchoff's equation. These are derivations that you will need to know thoroughly and you should also be clear about any assumptions made in any of the derivations. Your knowledge of the 1st and 2nd laws should be in terms of equations rather than wordy discussions.

(ii) Write notes on what you understand entropy to be, including in your notes a discussion of the typical values of entropies for fusion and vaporization. You will have to look up some representative examples. (iii) Draw the phase diagrams for (a) water and (b) sulphur and indicate as much information as possible on each diagram. Make sure that you understand every detail of the two phase diagrams. You may have to search more widely than the general physical chemistry text-books to find these phase diagrams.

There are many books on thermodynamics, each one approaching the subject in quite different ways. You should find what suits you best. The main introductory books are: Smith: *Basic Chemical Thermodynamics*, OUP. Everett: *An Introduction to the Study of Chemical Thermodynamics*. Atkins: *The Second Law*. Atkins: *Physical Chemistry*. A more extended approach is Denbigh: *The Principles of Chemical Equilibrium*.

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(a) A bomb calorimeter was calibrated by igniting a 0.825 g sample of benzoic acid in the presence of excess oxygen. The heat given out made the temperature of the calorimeter rise by 1.940 K from 298 K. The internal energy of combustion of benzoic acid is -3251 kJ mol^{-1}. Calculate the heat capacity of the calorimeter.

(b) In two separate experiments in the same apparatus, 0.498 g of fumaric and 0.509 g of maleic acid (both of RMM 116) were ignited and gave temperature rises of 0.507 K and 0.528 K respectively. Use the heat capacity of the calorimeter, calculated in (a), to calculate (i) the molar internal energy of combustion, (ii) the molar enthalpy of combustion, and (iii) the molar enthalpy of formation of fumaric and maleic acids. Comment on the difference. The standard enthalpy of formation of water is -285.8 kJ mol^{-1} and of CO_{2} -393.5 kJ mol^{-1}.

Derive an equation relating the change in enthalpy of a chemical reaction with temperature and the heat capacities of the reactants and products.

At 298 K, the standard enthalpy of formation of NH_{3}(g) is -46.11 kJ mol^{-1}. Assuming that the molar heat capacities can be represented by expressions of the form: *C*_{P} = *a* + *bT* with the coefficients given below, calculate the enthalpy of formation at 1000 K.

- | N_{2} | H_{2} | NH_{3} |

a/J K^{-1} mole^{-1} | 28.58 | 27.28 | 29.75 |

10^{3}b/J K^{-2} mol^{-1} | 3.77 | 3.26 | 25.1 |

**3.** (a) At 298 K the enthalpy change of the graphite to diamond phase transition is 1.8961 kJ mol^{-1} and the entropy change is -3.2552 J K^{-1} mol^{-1}.

(i) Which is the spontaneous direction at 298 K?

(ii) Which direction is favoured by a rise in temperature?

(iii) Outline how the enthalpy and entropy changes of the transition could be determined.

(b) Starting from *dG* = *VdP* - *SdT* derive an expression showing how Δ*G* varies with pressure.

Aragonite and calcite are two forms of crystalline CaCO_{3}. For the transition aragonite to calcite Δ*G*^{†}(298) = -800 J mol^{-1} and Δ*V* = 2.75 cm^{3} mol^{-1}. Assuming Δ*V* to be independent of pressure, at what pressure would aragonite become the stable form at 298 K?

Calculate the difference in molar entropy (i) between water at -5^{o}C and ice at -5^{o}C and (ii) between water at 95^{o}C and steam at 95^{o}C and 1 atm pressure.

Calculate the entropy changes for the two cases. Discuss the spontaneity of transitions between phases at these temperatures. The difference in heat capacities on melting is 37.3 J K^{-1} mol^{-1} and on vaporisation is -41.9 J K^{-1} mol^{-1}. Further, Δ*H*_{fusion}(273) = 6.01 kJ mol^{-1} and Δ*H*_{vap}(373) = 40.7 kJ mol^{-1}.

Using the following data together with the Clapeyron and Clapeyron-Clausius equations, plot the phase diagram for benzene over the temperature range -10^{o}C to +15^{o}C. The triple point of benzene occurs at 36 mmHg and 5.50^{o}C.

Δ*H*_{fusion} = 10.6 kJ mol^{-1}, Δ*H*_{vap} = 30.8 kJ mol^{-1}, ρ(solid) = 0.910 g cm^{-3}, ρ(liquid) = 0.899 g cm^{-3}.

The enthalpy and entropy of vaporization of a liquid both vary with temperature and the entropy varies the more rapidly of the two. Explain.

The entropy of vaporization of most liquids *at their boiling points* has been found to be approximately the same at 90 J K^{-1} mol^{-1} (Trouton's Rule). This can be put to use in optimizing the reduced pressure distillation of organic compounds. An organic compound, A, boils at 407 K at a pressure of 9 mmHg. It can only be distilled at low pressures but the lowest pressure the pump can reach is 20 mmHg. Assuming the result of Trouton's Rule above, calculate the temperature at which A will boil at a pressure of 20 mmHg.