Chemistry 2000 Lecture 12: Temperature dependence of the equilibrium constant

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1 Chemistry 2000 Lecture 12: Temperature dependence of the equilibrium constant Marc R. Roussel February 12, 2019 Marc R. Roussel Temperature dependence of equilibrium February 12, / 15

2 Temperature dependence of the equilibrium constant So far, all the problems we have solved have been at 25 C because our tables of free energies of formation are at this temperature. H and S vary relatively slowly with temperature. However, G = H T S has a strong dependence on T. At any temperature, r G m(t ) = RT ln K(T ). Suppose we have two temperatures, T 1 and T 2. Let K 1 = K(T 1 ) and K 2 = K(T 2 ). Then r G m(t 1 ) = r H m T 1 r S m = RT 1 ln K 1 r G m(t 2 ) = r H m T 2 r S m = RT 2 ln K 2 Marc R. Roussel Temperature dependence of equilibrium February 12, / 15

3 r H m T 1 r S m = RT 1 ln K 1 r H m T 2 r S m = RT 2 ln K 2 Divide both sides by RT : r Hm RT 1 r S m R = ln K 1 r Hm r Sm = ln K 2 RT 2 R If we assume that r Hm and r Sm are independent of temperature and subtract the two equations, we get r Hm ( 1 1 ) = ln K 2 ln K 1 R T 1 T 2 or ( ) K2 ln = r H ( m 1 1 ) K 1 R T 1 T 2 Marc R. Roussel Temperature dependence of equilibrium February 12, / 15

4 Exponentials and logarithms ln e x = x e ln x = x = the exponential function and the natural logarithm are inverse functions. Marc R. Roussel Temperature dependence of equilibrium February 12, / 15

5 Example: Temperature dependence of K w K w is the equilibrium constant for the autodissociation of water: H 2 O (l) H + (aq) + OH (aq) K w = at 25 C and r H = 55.8 kj mol 1. What is K w at 37 C? Answer: Marc R. Roussel Temperature dependence of equilibrium February 12, / 15

6 For neutral water, a H + = a OH so K w = (a H +)(a OH ) = (a H +) 2 a H + = K w At 37 C, a H + = = ph = log 10 ( ) = 6.81 The ph of neutral water is only 7 at 25 C! Similarly, the rule poh = 14 ph only applies at 25 C. Marc R. Roussel Temperature dependence of equilibrium February 12, / 15

7 Effect of temperature on reaction of CO 2 with water For the reaction CO 2(g) + H 2 O (l) HCO 3(aq) + H+ (aq) f H /kj mol 1 CO 2(g) HCO 3(aq) H 2 O (l) We previously calculated K = at 25 C for this reaction. The average surface temperature of the sea is 16.1 C. Calculate r H K at 16.1 C Marc R. Roussel Temperature dependence of equilibrium February 12, / 15

8 Effect of temperature on reaction of CO 2 with water (continued) Answers: r H = 11.8 kj mol 1 K = at 16.1 C Note that the increase in K with decreasing temperature is consistent with Le Chatelier s principle. Marc R. Roussel Temperature dependence of equilibrium February 12, / 15

9 Boiling point as a function of pressure A liquid boils when its vapor pressure equals the atmospheric pressure. The normal boiling point is the boiling point at 1 atm pressure. 1 atm = bar Lethbridge is about 940 m above sea level. The atmospheric pressure here is about 90 kpa on a typical day. What is the boiling point of water in Lethbridge? Answer: 96.7 C Marc R. Roussel Temperature dependence of equilibrium February 12, / 15

10 Boiling-point elevation What effect does a solute have on the boiling point of a solvent? Again consider H 2 O (l) H 2 O (g), ln ( K2 K 1 ) = vaph m R K = p/p X H2 O ( 1 T 1 1 T 2 Take T 1 = boiling point of pure solvent, K 1 = p/p, and note that vap H m > 0. Since X H2 O < 1 for a solution, K 2 > K 1. ) Marc R. Roussel Temperature dependence of equilibrium February 12, / 15

11 If K 2 > K 1, then in the formula ( ) K2 ln = vaph ( m 1 1 ) K 1 R T 1 T 2 the left-hand side is positive, which means that (since vap H m > 0), or 1 T 1 1 T 2 > 0 T 1 < T 2 The boiling point of a solution is higher than the boiling point of the pure solvent. Marc R. Roussel Temperature dependence of equilibrium February 12, / 15

12 Example: What is the boiling point of a solution made by dissolving mol NaCl in 1.01 kg of water at (exactly) 1 atm pressure? Data: The enthalpy of vaporization of water at the normal boiling point is kj/mol. Note: This is roughly the salt concentration you would use to boil pasta. Answer: C Marc R. Roussel Temperature dependence of equilibrium February 12, / 15

13 Freezing-point depression What effect does a solute have on the freezing point of a solvent? The process is with ln ( K2 K 1 H 2 O (s) H 2 O (l) K = a l = X H 2 O a s 1 ) = fushm ( 1 1 ) R T 1 T 2 Take T 1 = freezing point of pure solvent, K 1 = 1, and note that fus H m > 0. Since X H2 O < 1 for a solution, K 2 < K 1 = 1. Marc R. Roussel Temperature dependence of equilibrium February 12, / 15

14 If K 2 < K 1, then in the formula ( ) K2 ln = fush ( m 1 1 ) K 1 R T 1 T 2 the left-hand side is negative, which means that (since fus H m > 0), or 1 T 1 1 T 2 < 0 T 1 > T 2 The freezing point of a solution is lower than the freezing point of the pure solvent. Marc R. Roussel Temperature dependence of equilibrium February 12, / 15

15 Example: What is the freezing point of a solution made by dissolving 100 g of sucrose (C 12 H 22 O 11 ) in 500 g of water? Data: The enthalpy of fusion of ice at the normal freezing point is 6007 J/mol. Answer: 1.08 C Marc R. Roussel Temperature dependence of equilibrium February 12, / 15

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