Chemical Equilibrium - Chemistry...

Prof. Mueller Chemistry 451 - Fall 2003 Lecture 20 - 1

Chemical Equilibrium

Chapter 9 of Atkins: Sections 9.1 - 9.2

Spontaneous Chemical Reactions

The Gibbs Energy MinimumThe reaction Gibbs energyExergonic and endergonic reactions

The Description of EquilibriumPerfect gas equilibriaThe general case of a reactionThe relation between equilibrium constants

molality and mole fractions

The Boltzmann Distribution


Introduction

The concept of the chemical potential is used to account for theequilibrium composition of chemical reactions.

Some forms in which we have seen the chemical potential:

The composition at equilibrium corresponds to a minimum in the Gibbs energy plotted against the extent of reaction. This provides a powerful and direct relationship between the equilibrium constant and the standard Gibbs energy of reaction.

µA = µA⊕ + RT ln f

p⊕

⎛

⎝ ⎜

⎞

⎠ ⎟ µA = µA

⊕ + RT ln aA( )


Dynamic Equilibrium

Reactants Products

Static equilibrium - forward and reverse processes have both ceased.Dynamic equilibrium - both forward and reverse processes occur at the same rate.

P

R

time

conc

.


Dynamic Equilibrium

Reactants Products

time

forward

reverserate

of

reac

tion


Spontaneous Chemical Reactions

We have learned the general principle that at constant T and p, the direction of spontaneous change is to lower values of Gibbs energy (G). This idea can be generalized to apply to discussions of chemical reactions as well.

Idea: Calculate the Gibbs energy of a reaction mixture and thenidentify the composition that corresponds to minimum G.


The Extent of Reaction

Consider: A B

Suppose an infinitesimal amount dξ of A turns into B. Then:

The quantity ξ is called the extent of reaction. The units are the same as n, and therefore ξ is usually reported in moles. For a finite amount, write ∆ξ, and can then calculate:

dnA = −dξdnB = +dξ

nA = n0,A − ∆ξnB = n0,B + ∆ξ


The Reaction Gibbs Energy, I

Remember that for this two component system:

So:

This is the difference between the chemical potentials of the reactants and products at the composition of the reaction mixture. Therefore it is natural to define the reaction Gibbs energy as the slope of the graph of the Gibbs energy vs. the extent of reaction:

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

∂G∂ξ

⎛

⎝ ⎜

⎞

⎠ ⎟

p,T

= µB − µA

∆ rG =∂G∂ξ

⎛

⎝ ⎜

⎞

⎠ ⎟

p,T


The Reaction Gibbs Energy, II

∆ rG =∂G∂ξ

⎛

⎝ ⎜

⎞

⎠ ⎟

p,T

= µB − µAThe chemical potential varies with composition, and therefore ∆rGchanges as the reaction proceeds.

A → B spontaneous when µA > µB

B → A spontaneous when µA < µB

∆rG = 0 when µA = µB

If we can find the composition of the reaction mixture where µA = µB, then we can identify the composition at equilibrium.


Types of Reactions: Ability to Do Work

If we know ∆rG at constant T and p, then we know whether forward or reverse reaction is spontaneous.

∆rG > 0, the reverse reaction is spontaneousEndergonic - work consuming

∆rG < 0, the forward reaction is spontaneousExergonic - work producing

∆rG = 0, the reaction is at equilibrium


Perfect Gas Equilibria, I

We will first examine the case of two perfect gases:

where:

The ratio of the partial pressures is denoted by Q:

∆ rG = µB − µA = µB⊕ + RT ln pB( )− µA

⊕ + RT ln pA( )

= ∆ rG⊕ + RT ln pB

pA

⎛

⎝ ⎜

⎞

⎠ ⎟

∆ rG = ∆ rG⊕ + RT lnQ Q =

pB

pAreaction quotient

∆ rG⊕ = µB

⊕ − µA⊕


Perfect Gas Equilibria, II

We have seen the standard molar Gibbs energy of reaction before,and in practice it can be calculated from:

So these are tabulated (huge array of thermodynamic tables like Table 2.6 in Atkins, 7th Ed.)

At equilibrium, ∆rG = 0:

0 = ∆ rG⊕ + RT lnK K =

pB

pA

⎛

⎝ ⎜

⎞

⎠ ⎟

equilibrium

RT lnK = −∆ rG⊕

equilibrium constant

∆ rG⊕ = ∆ f G

⊕ B( )− ∆ f G⊕ A( )


The Importance of Mixing

The position of the equilibrium composition depends upon the mixing of the products and reactants.

If no mixing, then a linear change in Gibbs energy from that of pure A to pure B (slope depends on overall difference, position then depends on amount of B). The exact slope is:

But we have seen that if the mixing is complete for ideal gases, then:

Now a minimum occurs and its position marks the equilibrium composition.

∆ rG⊕

∆mixG = nRT xA ln xA + xB ln xB( )


General Reactions

To generalize the idea of extent of reaction:Consider the reaction:

Which we can rewrite as

In a more general form,

And in the above reaction

Stoichiometric numbers are positive for products and negative or reactants. Then we modify the definition of ξ so that if it changes by ∆ξ then the change in the amount of any species J is νJ∆ξ .

2A + B → 3C + D

0 = −2A − B + 3C + D

0 = ν J JJ∑

ν A = −2 ν B = −1 νC = +3 ν D = +1


The General Reaction Gibbs Energy, I

For a reaction with known stoichiometric numbers, νJ, the reaction can advance by dξ and the amounts of reactants/products changes by:

Then:

Since:

and in general we write:

then we can obtain a general expression for ∆rG.

dnJ = ν J dξ

dG = µJJ

∑ dnJ = ν JµJJ∑

⎛

⎝ ⎜

⎞

⎠ ⎟ dξ

∆ rG =∂G∂ξ

⎛

⎝ ⎜

⎞

⎠ ⎟

p,T

= ν JµJJ

∑

µJ = µJ⊕ + RT lnaJ


The General Reaction Gibbs Energy, II

Remember that:

A general expression for ∆rG.

aln x = ln xa ln x + ln y +L = ln xyL

∆ rG = ν JµJJ

∑ = ν J µJ⊕ + RT lnaJ( )

J∑

= ν JµJ⊕

J∑ + ν J RT lnaJ

J∑

= ∆ rG⊕ + RT lnaJ

ν J

J∑

= ∆ rG⊕ + RT ln

JΠaJ

ν J

= ∆ rG⊕ + RT lnQ

Q =J

ΠaJν J

2A + B → 3C + D

Q =J

ΠaJν J

= aA−2aB

−1aC3 aD

=aC

3 aD

aA2 aB


The General Reaction Gibbs Energy, III

At equilibrium:

And now drop the “equilibrium” subscript.

This is a thermodynamic equilibrium constant, defined in terms of activities or fugacities (for gas phase species). Since aJ and fJ are dimensionless, then K (and Q) are also dimensionless.

Approximations made in earlier classes may have included substituting molar concentration for activities, and partial pressures for fugacities.

K =J

ΠaJν J

⎛ ⎝ ⎜

⎞ ⎠ ⎟

equilibrium


A Most Useful and Important Result

At equilibrium (∆rG = 0)

Some useful things to do:Given standard Gibbs energies of formation for

reactants and products, calculate K.Given K, calculate extent of reaction (such as degree

of dissociation)

RT lnK = −∆ rG⊕


Work Page I


Work Page II


Relation to Molalities/Mole Fractions

We use the definitions of activities in terms of mole fractions or molalities:

For the reaction:

we write:

This can be a complicated set of calculations, since activity coefficients can vary with composition, and must be evaluated at the equilibrium composition. Often, we start with the assumption:

aJ = γ J xJ aJ = γ JbJ

bJ⊕

A + B → C + D

K =aCaD

aAaB

=γCγD

γAγB

×xC xD

xA xB

=γCγD

γAγB

×bCbD

bAbB

= KγKb

Kγ =1



Molecules of either reactants or products exist in particular energy states. Placement of atoms, electrons, etc. lead to a distribution of possible energy states for the reactants and products, and the distribution of populations within these states (in a particularly useful picture) are given at equilibrium by the Boltzmann distribution. In the field of statistical mechanics (or here, statistical thermodynamics) it is found that the population in the state with energy E follows a well-defined form:

PE ∝exp −EkT

⎛ ⎝ ⎜

⎞ ⎠ ⎟



Pn ∝exp −En

kT⎛ ⎝ ⎜

⎞ ⎠ ⎟

Pn

Pm

=exp −

En

kT⎛ ⎝ ⎜

⎞ ⎠ ⎟

exp −Em

kT⎛ ⎝ ⎜

⎞ ⎠ ⎟

= exp −En − Em( )

kT⎛

⎝ ⎜

⎞

⎠ ⎟ = exp −

∆EkT

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Pn

Pm


kT⎛

⎝ ⎜

⎞

⎠ ⎟ <1 if En > Em and T > 0

Pn

Pm

≈1−∆EkT

if kT >> ∆E (high T approx.)



Pn

Pm


kT⎛

⎝ ⎜

⎞

⎠ ⎟

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