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Ch 13 Chemical Equilibrium. Partition Functions  K LeChatelier’s Principle K(T), K(p). I. Chemical Equilibrium; K = f(N). Consider gas phase rxn A(g) ↔ B(g) Equilibrium constant = K = N B /N A

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Ch 13 chemical equilibrium

Ch 13 Chemical Equilibrium

Partition Functions  K

LeChatelier’s Principle

K(T), K(p)


I chemical equilibrium k f n
I. Chemical Equilibrium; K = f(N)

  • Consider gas phase rxn A(g) ↔ B(g)

  • Equilibrium constant = K = NB /NA

  • At constant T and p, G is the thermodynamic function that characterizes equilibrium. dG = -SdT + Vdp + ΣμidNi = ΣμidNi = 0 at equilibrium or μA = μB

  • Eqn 11.47 defines μ = -kT ℓn (q/N)


K f q
K = f(q)

  • A and B have their own set (ladder) of energy levels beginning at ground state energies ε0A and ε0B, respectively. Fig 13.1

  • q’A = partition function = [exp(-ε0Aβ) + exp(ε1Aβ) + exp(ε2Aβ) + …]

  • μA = -kT ℓn (q’A/N)

  • qA = reduced partition fnt with ε0A factored out = exp(ε0Aβ) [exp(-ε0Aβ) + exp(-ε1Aβ) + exp(-ε2Aβ) + …] = exp(ε0Aβ) q’A


K f q1
K = f(q)

  • K = NB/NA= q’B/q’A =

    qB/qA exp(- [ε0B - ε0A]β) = qB/qA exp(- Δε0β) Eqn 13.10

  • More complex rxn: aA +bB ↔ cC

    • K = NCc/[NAaNBb] = Eqn 13.18 which shows q and ε0 terms

  • Ex. 13.2, prob 6


Vibrational energy reference level
Vibrational Energy Reference Level

  • If you use bottom of the well = 0  Eqn 11.26 for qvib = exp (-hνβ/2)/[1- exp (-hνβ)]. Then energy from dissociation limit to bottom of well = - ε0

  • If you use ZPE as 0  Eqn 13.21 for qvz = exp [1- exp (-hνβ)]-1. Then energy from ZPE to dissociation limit = D or D0 (T11.2)


K f p
K=f(p)

  • K  Kp using n = pV/RT or N = pV/kT

  • Kp = [pCc/[pAa pBc] Eqn 13.29

    = (kT)c-a-b [q0Cc/[q0Aa q0Bb] exp(ΔDβ)

  • q0 = q/V

  • μ = μ0 + kT ℓn p = std state chem pot and term that depends on p

  • Ex 13.3, prob 1


Ii lechatelier s principle
II. LeChatelier’s Principle

  • Given a system at equilibrium at constant T and p, ΔG = 0 and G is at a minimum. Any disturbance on the system will increase G.

  • The LeChatelier Prin says that the system will return to equilibrium in a way that opposes disturbance. Note that K does not change for all perturbations. Fig 13.4


Ii a k f t
II.A. K = f(T)

  • See Thermody. Relationships (p 2) for van’t Hoff Eqn and Gibbs-Helmholtz Eqn.

  • A(g) ↔ B(g) μA = μB at equilibrium or

  • μA = μA0 + kT ℓn pA = μB = μB0 + kT ℓn pB

  • ℓn (pB/pA) = ℓn Kp = -(μB0 - μA0)kT = -∆μ0/kT

  • ∆μ0 = ∆h0 - T∆s0 (partial molar quant)

  • ℓn Kp = -∆μ0/kT = [∆h0 - T∆s0]/kT


Van t hoff eqn
Van’t Hoff Eqn

  • ℓn Kp = -∆μ0/kT = - [∆h0 - T∆s0]/kT

  • Take partial w/respect to T and assume ∆h0 and ∆s0 are independent of T.

  • δ ℓn Kp /δT = - δ{[∆h0 - T∆s0]/kT}/δT = ∆h0/kT2 Eqn 13.37

  • Or δ ℓn Kp /δ(1/T) = - ∆h0/k Eqn 13.38

  • Plot ℓn Kp vs 1/T and slope = - ∆h0/k

  • Ex 13.4, Prob 5


Gibbs helmholtz eqn g t
Gibbs-Helmholtz Eqn; G(T)

  • δ (G/T)/δT = - H(T)/T2 Eqn 13.43; see also p 2 of Thermodynamic Relationships

  • Recall ∆G = -RT ℓn K


Ii b k f p
II.B. K = f(p)

  • (δ ℓn K/δp)T = - ∆v/kT PMV

  • Ex 13.6


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