Problem Set #3. Assigned September 6, 2013 Due Friday, September 13, Please show all work for credit

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1 Problem Set #3 Assigned September 6, 013 Due Frida, September 13, 013 Please show all work for credit To warm up or practice tr the Atkins Eercises, which are generall simple one step problems Partial Derivatives 1. Engle P. 3. (First and second partial derivatives) f 5 Cos5 Sin51 e Cos ln. f Sin 5 3 e Sin ln f 10 Cos 5 5 Sin51 e Cos 48e Cos ln f 3 e Cos ln f f ln Sin 5 5 Cos5 1 e Sin f f a) 5 Cos5 1 e Sin Sin5 ln f f b) f df Sin f d 5 3 e Sin d d ln 5 Cos 5 Sin51 e Cos ln d

2 . Atkins P.. (Eact differentials) Show that the following functions have eact diffreentials: (a) +3, (b) cos(), (c) 3, (d) t(t+e s )+s Real Gases 3. Atkins P. 1.8 (From last week, virial gas coefficients and compressibilit) At 73 K measurements on argon gave B= 1.7 cm 3 mol -1 and C=100cm 6 mol -, where B and C are the second and third virial coefficients in the epansion of Z in powers of 1/Vm. Assuming that the perfect gas law holds sufficientl well for the estimation of the second and third terms of the epansion, calculate the compression factor of argon at 100 atm and 73 K. From our result, estimate the molar volume of argon under these conditions.

3 Heat Capacit 4. Atkins E..4(b) (heat epansion) A sample consisting of.00 mol of perfect gas molecules, for which CV,m = 5/R, initiall at P1 = 111 kpa and T1 = 77 K, is heated reversibl to 356 K at constant volume. Calculate the final pressure, U, q, and w. 5. Atkins E..8(b) (heat capacit) The constant-pressure heat capacit of a sample of a perfect gas was found to var with temperature according to the epression Cp/(J K -1 ) = (T/K). Calculate q, w, U, and H when the temperature is raised from 0 C to 100 C (a) at constant pressure, (b) at constant volume.

4 6. Atkins Life Science P (Heat capacit derivation and calculation) (a) Show that for a perfect gas, Cp,m- Cv,m=R. (b) When 9 J of energ is supplied as heat at constant pressure to 3.00 mol CO (g), the temperature of the sample increases b.06k. Calculate the molar heat capacities at constant volume and constant pressure of the gas. Part1: 7. Atkins P..11 (Heat capacit simple use) An average human produces about 10 MJ of heat each da through metabolic activit. If a human bod were an isolated sstem of mass 65 kg with the heat capacit of water, what temperature rise would the bod eperience? Human bodies are actuall open sstems, and the main mechanism of heat loss is through the evaporation of water. What mass of water should be evaporated each da to maintain constant temperature? Part: First of all, everthing happens in an isobaric process ( 1, we can also treat the water vapor as a perfect gas, so. From the eample.3 on the book, the enthalp change of vaporization per mole of is

5 Work, Energ, and Enthalp 8. Atkins E..3(b) (epansion work) A sample consisting of.00 mol He is epanded isothermall at C from.8 dm 3 to 31.7 dm 3 (a) reversibl, (b) against a constant eternal pressure equal to the final pressure of the gas, and (c) freel (against zero eternal pressure). For the three processes calculate q, w, U, and H.

6 9. Atkins P.. (Work of Gas Epansion) A sample consisting of 1.0 mol CaCO3(s) was heated to 800 C, when it decomposed. The heating was carried out in a container fitted with a piston that was initiall resting on the solid. Calculate the work done during complete decomposition at 1.0 atm. What work would be done if instead of having a piston the container was open to the atmosphere? 10. Engel - P.9 (heat. metabolism) A hiker caught in a thunderstorm loses heat when her clothing becomes wet. She is packing emergenc rations that, if completel metabolized, will release 30. kj of heat per gram of rations consumed. How much rations must the hiker consume to avoid a reduction in bod temperature of 4.0 K as a result of heat loss? Assume the heat capacit of the bod equals that of water. Assume the hiker weighs 55 kg. State an additional assumptions. We start b calculating the heat that corresponds to a temperature decrease of 4 K. Using q Cp T, we obtain: q 4 K Cp T 75.3 J K mol 4.0 K J kg kg mol We then determine the heat lost for a 55 kg person as: q person q 4 K m J kg 55 kg J And finall the mass of rations that needs to be consumed is given b: 5 m q person J rations 31.0 g 4 - q J g 1 rations

7 11. Engel - P.14 (heat. work) According to a stor told b Lord Kelvin, one da when walking down from Chamoni to commence a tour of Mt. Blanc, "whom should I meet walking up (the trail) but (James) Joule, with a long thermometer in his hand, and a carriage with a lad in it not far off. He told me he had been married since we parted from Oford, and he was going to tr for (the measurement of the) elevation of temperature in waterfalls." Suppose Joule encountered a waterfall 30. meters height. Calculate the temperature difference between the top and bottom of this waterfall. The decrease in potential energ (mgh) must equal the heat evolved as the water falls. (mcpt). Note that the mass cancels, so we need to use the heat capacit for mass, 4.18 kj kg -1 K -1, and also note that J=. Heat is evolved so T increases, so the bottom temp is higher than the top, and the difference is : m g h m C p, m ΔT m s 30.0 m J m K g h ΔT 1 C p,m K

8 1. Atkins P..8 (D-ribose energ of combustion) A sample of the sugar D-ribose (C5H10O5) of mass 0.77 g was placed in a constantvolume calorimeter and then ignited in the presence of ecess ogen. The temperature rose b K. In a separate eperiment in the same calorimeter, the combustion of 0.85 g of benzoic acid, for which the internal energ of combustion is -351 kj mol -1, gave a temperature rise of K. Calculate the internal energ of combustion of D- ribose and its enthalp of formation.

9 Enthalp 13. Atkins Life Science P. 1.3 and Fig. 1.7 (Enthalp of protein folding) Figure 1.7 shows the eperimental DSC scan of hen white lsozme (G. Privalovet al., Anal. Biochem. 79, 3 (1995)) converted to kilojoules (from calories). Determine the enthalp of unfolding of this protein b integration of the curve and the change in heat capacit accompaning the transition.

10 14. Atkins Life Science P. 1.4 (Enthalp of water vaporization) Estimate the enthalp of vaporization of water at 100 C from its value at 5 C ( kj mol -1 ) given the constant-pressure heat capacities of 75.9 J K -l mol -l and J K -l mol -l for liquid and gas, respectivel.

11 Etra problems - Practice for eams, do not hand in: 1. Atkins P..6 (Heat capacit derivation) Starting from the epression Cp-CV=T partial derivatives to show that: Cp-CV= Evaluate Cp-CV for a perfect gas. )T ( )p, use the appropriate relations between

12 . Atkins P..6 (isothermal epansion) Calculate the work done during the isothermal reversible epansion of a van der Waals gas. Account phsicall for the wa in which the coefficients a and b appear in the final epression. Plot on the same graph the indicator diagrams for the isothermal reversible epansion of (a) a perfect gas, (b) a van der Waals gas in which a=0 and b= dm 3 mol -1, and (c) a=4.dm 6 atm mol - and b=0. The values selected eaggerate the imperfections but give rise to significant effects on the indicator diagrams. Take Vi=1.0 dm 3, n=1.0 mol, and T=98 K.

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14 3. Atkins P..1 (Glucose energ) Glucose and fructose are simple sugars with the molecular formula C6H1O6. Sucrose, or table sugar, is a comple sugar with molecular formula C1HO11 that consists of a glucose unit covalentl bound to a fructose unit (a water molecule is given off as a result of the reaction between glucose and fructose to form sucrose). (a) Calculate the energ released as heat when a tpical table sugar cube of mass 1.5 g is burned in air. (b) To what height could ou climb on the energ a table sugar cube provides assuming 5 per cent of the energ is available for work? (c) The mass of a tpical glucose tablet is.5 g. Calculate the energ released as heat when a glucose tablet is burned in air. (d) To what height could YOU climb on the energ a cube provides assuming 5 per cent of the energ is available for work?

15 4. Atkins P..47 (Differential scanning calorimetr) Differential scanning calorimetr is used to eamine the role of solvent-protein interactions in the denaturation process. Figure.34 shows the thermogram for ubiquitin in water with the signal observed for ubiquitin in methanol/water mitures. Suggest an interpretation of the thermograms. The simplest interpretation is that the structure of the protein in MeOH is more stable, i.e. it undergoes a denaturation (sometimes termed melting) transition at a higher temperature. Beond that the Cp suggests that parts of the protein that were not in solvent are now eposed to solvent and have different motions/conformations available and that this eposure results in a different distribution of energ states in MeOH than in HO. Methanol interacts with the protein to desolvate it, displacing water, with weaker H- bonds to the amides, which will denature the protein to some etent, but at the same time can stabilize a non-native state. 5. Engel question Q.4 (heat,work) Wh is it incorrect to speak of the heat or work associated with a sstem? Heat and work eist onl during the transitions of a given sstem between states. A sstem in a particular state does not have heat or work, onl energ.

16 6. Engel question Q.6 (microwave) A cup of water at 78 K (the sstem) is placed in a microwave oven and the oven is turned on for 1 minute during which it begins to boil. Which of q, w, and au are positive, negative, or zero? The heat, q, is positive since heat flows across the sstem-surrounding boundar into the sstem. The work, w, is negative because the vaporized water does work on the surroundings. U is positive because the temperature increases and some of the liquid is vaporized. 7. Atkins Life Science P. 1.4 (Endothermic vs. eothermic) Classif as endothermic or eothermic (a) a combustion reaction for which H = -00 kj mol -1, (b) a dissolution for which H = +4.0 kj mol -1, (c) vaporization, (d) fusion, and (e)sublimation.