Chemistry 2000 Slide Set 12: Temperature dependence of the - - PowerPoint PPT Presentation

Chemistry 2000 Slide Set 12: Temperature dependence of the equilibrium constant

Marc R. Roussel February 6, 2020

Marc R. Roussel Temperature dependence of equilibrium February 6, 2020 1 / 15

Temperature dependence of the equilibrium constant

So far, all the problems we have solved have been at 25◦C because

• ur 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, ∆rG ◦

m(T) = −RT ln K(T).

Suppose we have two temperatures, T1 and T2. Let K1 = K(T1) and K2 = K(T2). Then ∆rG ◦

m(T1) = ∆rH◦ m − T1∆rS◦ m = −RT1 ln K1

∆rG ◦

m(T2) = ∆rH◦ m − T2∆rS◦ m = −RT2 ln K2

Marc R. Roussel Temperature dependence of equilibrium February 6, 2020 2 / 15

∆rH◦

m − T1∆rS◦ m

= −RT1 ln K1 ∆rH◦

m − T2∆rS◦ m

= −RT2 ln K2 Divide both sides by RT: ∆rH◦

m

RT1 − ∆rS◦

m

R = − ln K1 ∆rH◦

m

RT2 − ∆rS◦

m

R = − ln K2 If we assume that ∆rH◦

m and ∆rS◦ m are independent of temperature

and subtract the two equations, we get ∆rH◦

m

R 1 T1 − 1 T2

• = ln K2 − ln K1
• r

ln K2 K1

• = ∆rH◦

m

R 1 T1 − 1 T2

• Marc R. Roussel

Temperature dependence of equilibrium February 6, 2020 3 / 15

Exponentials and logarithms

ln ex = x eln x = x = ⇒ the exponential function and the natural logarithm are inverse functions.

Marc R. Roussel Temperature dependence of equilibrium February 6, 2020 4 / 15

Example: Temperature dependence of Kw

Kw is the equilibrium constant for the autodissociation of water: H2O(l) ⇋ H+

(aq) + OH− (aq)

Kw = 1.01 × 10−14 at 25◦C and ∆rH◦ = 55.8 kJ mol−1. What is Kw at 37◦C? Answer: 2.4 × 10−14

Marc R. Roussel Temperature dependence of equilibrium February 6, 2020 5 / 15

For neutral water, aH+ = aOH− so Kw = (aH+)(aOH−) = (aH+)2 ∴ aH+ =

• Kw

At 37◦C, aH+ =

• 2.4 × 10−14 = 1.5 × 10−7

∴ pH = − log10(1.5 × 10−7) = 6.81 The pH of neutral water is 7 only at 25◦C! Similarly, the rule pOH = 14 − pH only applies at 25◦C.

Marc R. Roussel Temperature dependence of equilibrium February 6, 2020 6 / 15

Effect of temperature on reaction of CO2 with water

For the reaction CO2(g) + H2O(l) ⇋ HCO−

3(aq) + H+ (aq)

∆f H◦/kJ mol−1 CO2(g) −393.5 HCO−

3(aq)

−691.11 H2O(l) −285.840 We previously calculated K = 1.47 × 10−8 at 25 ◦C for this reaction. The average surface temperature of the sea is 16.1 ◦C. Calculate

∆rH◦ K at 16.1 ◦C

Marc R. Roussel Temperature dependence of equilibrium February 6, 2020 7 / 15

Effect of temperature on reaction of CO2 with water

(continued)

Answers:

∆rH◦ = −11.8 kJ mol−1 K = 1.70 × 10−8 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 6, 2020 8 / 15

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 = 1.013 25 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 6, 2020 9 / 15

Boiling-point elevation

What effect does a solute have on the boiling point of a solvent? Again consider H2O(l) → H2O(g), K = p/p◦ XH2O ln K2 K1

• = ∆vapH◦

m

R 1 T1 − 1 T2

• Take T1 = boiling point of pure solvent, K1 = p/p◦, and note that

∆vapH◦

m > 0.

Since XH2O < 1 for a solution, K2 > K1.

Marc R. Roussel Temperature dependence of equilibrium February 6, 2020 10 / 15

If K2 > K1, then in the formula ln K2 K1

• = ∆vapH◦

m

R 1 T1 − 1 T2

• the left-hand side is positive, which means that

1 T1 − 1 T2 > 0 (since ∆vapH◦

m > 0), or

T1 < T2 The boiling point of a solution is higher than the boiling point of the pure solvent.

Marc R. Roussel Temperature dependence of equilibrium February 6, 2020 11 / 15

Example: What is the boiling point of a solution made by dissolving 0.5032 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 40.66 kJ/mol. Note: This is roughly the salt concentration you would use to boil pasta. Answer: 100.51 ◦C

Marc R. Roussel Temperature dependence of equilibrium February 6, 2020 12 / 15

Freezing-point depression

What effect does a solute have on the freezing point of a solvent? The process is H2O(s) ⇋ H2O(l) with K = al as = XH2O 1 ln K2 K1

• = ∆fusH◦

m

R 1 T1 − 1 T2

• Take T1 = freezing point of pure solvent, K1 = 1, and note that

∆fusH◦

m > 0.

Since XH2O < 1 for a solution, K2 < K1 = 1.

Marc R. Roussel Temperature dependence of equilibrium February 6, 2020 13 / 15

If K2 < K1, then in the formula ln K2 K1

• = ∆fusH◦

m

R 1 T1 − 1 T2

• the left-hand side is negative, which means that

1 T1 − 1 T2 < 0 (since ∆fusH◦

m > 0), or

T1 > T2 The freezing point of a solution is lower than the freezing point of the pure solvent.

Marc R. Roussel Temperature dependence of equilibrium February 6, 2020 14 / 15

Example: What is the freezing point of a solution made by dissolving 100 g of sucrose (C12H22O11) 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 6, 2020 15 / 15