Chapter 6. Chemical Equilibrium

2011 Fall Semester Physical Chemistry 1

(CHM2201)

Contents Spontaneous chemical reactions

6.1 The Gibbs energy minimum 6.2 The description of equilibrium

The response of equilibria to the conditions 6.3 How equilibria respond to changes of pressure 6.4 The response of equilibria to changes of temperature

Equilibrium electrochemistry 6.5 Half-reaction and electrodes 6.6 Varieties of cells 6.7 The cell potential 6.8 Standard electrode potentials 6.9 Applications of standard potentials

6.1 The Gibbs energy minimum

Key points 1.  The reaction Gibbs energy is the slope of the

plot of Gibbs energy against extent of reaction 2.  Reactions either exergonic or endergonic

(a) The reaction Gibbs energy •  Consider the equilibrium A ⇌ B •  Suppose an infinitesimal

amount dξ of A turns into B •  The change in the amount of A

present is dnA = -dξ •  The change in the amount of B

present is dnB = dξ •  ξ is the extent of reaction •  The reaction Gibbs energy, ΔrG, is defined as the slope of the graph of the Gibbs energy plotted against ξ

6.1 The Gibbs energy minimum

ΔrG =∂G∂ξ

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(a) The reaction Gibbs energy

6.1 The Gibbs energy minimum

dG = µAdnA +µBdnB = −µAdξ +µBdξ = µB −µA( )dξ

∂G∂ξ

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= µB −µA

ΔrG = µB −µA

•  ΔrG is the difference between the chemical potentials at the composition of the reaction mixture

•  Because the chemical potential varies with composition, ΔrG changes as the reaction proceeds

•  When the slope is zero, the reaction is at equilibrium

ΔrG = 0

(b) Exergonic and endergonic reactions

6.1 The Gibbs energy minimum

•  ΔrG < 0 : the forward reaction is spontaneous •  ΔrG > 0 : the reverse reaction is spontaneous •  ΔrG = 0 : the reaction is at equilibrium •  A reaction for which ΔrG < 0 is called

“exergonic” (from the Greek word for work-producing)

•  A reaction for which ΔrG > 0 is called “endergonic” (from the Greek word for work-consuming)

6.2 The description of equilibrium

Key points 1.  The reaction Gibbs energy depends

logarithmically on the reaction quotient. When the reaction Gibbs energy is zero the reaction quotient has a value called the equilibrium constant

2.  Under ideal conditions, the thermodynamic equilibrium constant may be approximated by expressing it in terms of concentrations and partial pressures

3.  The presence of the enthalpy and entropy contributions to K are related to the role of the Boltzmann distribution

(a) Perfect gas equilibria •  When A and B are perfect gases

•  If we denote the ratio of partial pressures by Q,

•  Q is an example of reaction quotient •  The standard reaction Gibbs energy ΔrGΘ is the difference in the

standard molar Gibbs energy of reactants and products

6.2 The description of equilibrium

ΔrG = µB −µA = µBΘ + RT ln pB( )− µA

Θ + RT ln pA( ) = ΔrG

Θ + RT ln pB

pA

ΔrG = ΔrGΘ + RT lnQ Q= pB

pA

ΔrGΘ =Gm

Θ B( )−GmΘ A( ) = µΘ

B −µΘA

(a) Perfect gas equilibria •  The difference in standard molar Gibbs energies of the

products and reactants is equal to the difference in their standard Gibbs energy of formation (Section 3.6)

•  At equilibrium ΔrG = 0

•  Equilibrium constant, K

6.2 The description of equilibrium

ΔrGΘ = ΔfG

Θ B( )−ΔfGΘ A( )

0 = ΔrGΘ + RT lnK

RTlnK = −ΔrGΘ

K =pBpA

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(a) Perfect gas equilibria •  If only the enthalpy were important,

then H (therefore, G) would change linearly; there is no intermediate minimum in the graph

•  The entropic contribution is U-shaped

•  By combining both the enthalpy and entropy, there is an intermediate minimum in the total Gibbs energy; the minimum corresponds to the equilibrium composition

6.2 The description of equilibrium

(b) The general case of a reaction •  Let’s extend the previous argument to a general reaction

•  If the extent of reaction ξ changes by Δξ, the change in the amount of any species J is νJΔξ

•  Recall that for pure solids and liquids, the activity is 1. Such

substances make no contribution to Q

6.2 The description of equilibrium

0 = vJJJ∑ : symbolic form of a chemical reaction

ΔrG = ΔrGΘ + RT lnQ

ΔrGΘ = vΔfG

Θ

Products∑ − vΔfG

Θ

Reactants∑

ΔrGΘ = vJΔfG

Θ(J)J∑

Q= activities of productsactivities of reactants

Q= aJvJ

J∏

(b) The general case of a reaction •  At equilibrium the slope of G is zero: ΔrG = 0. The activities have

their equilibrium values

•  K is “a thermodynamic equilibrium constant” •  In elementary applications, the activities are often repl aced by

•  molalities, by replacing aJ by bJ/bΘ

•  molar concentrations, by replacing aJ by [J]/cΘ •  partial pressures, by replacing aJ by pJ/pΘ

6.2 The description of equilibrium

K= aJvJ

J∏"

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RT lnK = −ΔrGΘ

(c) The relation between equilibrium constants •  For an equilibrium of the form A + B ⇌ C + D,

•  The activity coefficients must be evaluated at the equilibrium composition of the mixture

•  In elementary applications, the assumption is often made that the activity coefficients are all so close to “unity” that Kϒ = 1.

•  A special case arises when we need to express K in terms of molar concentration.

6.2 The description of equilibrium

K= aCaDaAaB

=γCγDγAγB

×bCbDbAbB

=KγKb

K= aJ( )J∏

vJ=

pJpΘ

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J∏

vJ

= [J]vJ RTpΘ

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J∏

vJ

= [J]vJ × RTpΘ

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J∏

J∏

vJ

Kc=[J]cΘ"

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J∏

vJ

K=Kc ×cΘRTpΘ

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J∏

vJΔv= vJ

J∏ K=Kc ×

cΘRTpΘ

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Δv

(d) Molecular interpretation of the equilibrium constant

•  The Boltzmann distribution represents the distributions of occupied energy states in equilibrium

•  The Boltzmann distribution does not distinguish between reactants and products

6.2 The description of equilibrium

K=e−ΔrHΘ RTeΔrS

Θ R

More closely spaced B could dominate

Evenly distributed A would dominate

(e) Equilibria in biological systems

•  It is appropriate to adopt the biological standard state (Section 5.11)

•  In the biological standard state, both P and R are at unit activity

6.2 The description of equilibrium

R+ vH+ aq( )→ P

ΔrG = ΔrGΘ + RT ln aP

aRavH+

= ΔrGΘ + RT ln aP

aR− vRT lna

H+

ΔrG = ΔrGΘ − vRT ln10loga

H+ = ΔrGΘ + vRT ln10pH

ΔrG⊕ = ΔrG

Θ + 7vRT ln10