Equilibrium Kinetics
Supporting original reading. Follow the AP topic pathway for reviewed scope, worked reasoning and targeted practice.
Chemistry II · original Chapter 5 · combined Chapter 13
Section Breakdown
- Determining reaction rate.
- Finding the time to equilibrium.
Section 2
Equilibrium Kinetics
Knowing just which way a reaction is progressing is most of the time not very useful. Sure, you know what’s getting used up, but generally, you intended it to work that way.
Equilibrium constants determine equilibrium composition, rather than how long a reaction takes to approach it. To estimate an approach time, specify reversible rate laws and a closeness threshold. The integrated rate laws below describe the stated disappearance laws and must not be used as finite exact equilibrium-time formulas.
Zeroth-Order:
The rate equation:
The integrated rate equation:
The half-life equation:
First-Order:
The rate equation:
The integrated rate equation:
The half-life equation:
Second-Order:
For the specific disappearance law −d[A]/dt=k[A]², integration gives 1/[A]=1/[A]₀+kt and t½=1/(k[A]₀). The law k[A][B] does not have this form unless the circumstances actually make [A] and [B] equal throughout the process. Also distinguish a species disappearance rate from a stoichiometrically normalized reaction rate.
These equations allow us to determine the time it takes for the reaction to reach a certain state. Normally, you’ve only used these equations to determine the time until one reaction component reaches a certain concentration. So how can you use them to determine time to equilibrium, when that typically uses many reaction components?
Well, we simply have to relate every component of the reaction to one variable. That way, we know the concentration of that reactant at equilibrium.
To determine that, we use what is known as an ICE table. An ICE table is a table of the Initial Conditions, the Change, and the Equilibrium Conditions for a reaction’s components. The ICE table for a generic reaction:
| COndition | A | B | C | D |
|---|---|---|---|---|
| Initial | [A]0 | [B]0 | [C]0 | [D]0 |
| Change | -x | -x | +x | +x |
| Equilibrium | [A]0-x | [B]0-x | [C]0+x | [D]0+x |
Using this table, you can determine the concentrations of a reactions components at equilibrium, given its initial state and the reaction’s Equilibrium Constant.
Example Problem 5.2.1
Determine the concentrations of all components of the following reaction when the reaction reaches equilibrium.
Let x be the forward extent per litre. [A]=0.960−3x; [B]=0.560−2x; [C]=0.0760+x; [D]=0.0110+4x. All four must be nonnegative.
Qinitial=(0.0760)(0.0110)⁴/[(0.960)³(0.560)²]≈4.01×10⁻⁹, below Kc=43, so the net change is forward.
Solve (0.0760+x)(0.0110+4x)⁴/[(0.960−3x)³(0.560−2x)²]=43 on −0.00275<x<0.280. The physical numerical root is x=0.1881492235 M.
[A]=0.395552330 M; [B]=0.183701553 M; [C]=0.264149223 M; [D]=0.763596894 M. Substitution returns Q=43.0000. Round only at the end.
An ICE table supplies initial and equilibrium concentrations. It does not supply forward and reverse rate constants. Exact equilibrium is commonly approached asymptotically; a practical time requires a tolerance such as “within 1% of the equilibrium value.”
Reviewed irreversible first-order threshold task: −d[A]/dt=k[A], k=0.0112 day⁻¹, [A]₀=0.960 M. Find the time to [A]=0.396 M. This is a separate kinetic task; an equilibrium constant does not supply an exact approach time.
Reviewed irreversible first-order threshold task: −d[A]/dt=k[A], k=0.0112 day⁻¹, [A]₀=0.960 M. Find the time to [A]=0.396 M. This is a separate kinetic task; an equilibrium constant does not supply an exact approach time.
Integrate the stated disappearance law: [A]=[A]₀e⁻ᵏᵗ, or ln([A]₀/[A])=kt.
t=ln(0.960/0.396)/0.0112=79.0642 days≈79.1 days.
The specified threshold is reached at this finite time. Exact reversible equilibrium requires a reversible rate model and is generally approached asymptotically.
Reviewed ideal composition exercise: A+B⇌C, Kc=0.023, with initial concentrations 0.362 M, 0.691 M and 0.457 M. Define x as the forward extent per litre: [A]=0.362−x, [B]=0.691−x and [C]=0.457+x.
Kc=(0.457+x)/[(0.362−x)(0.691−x)]=0.023. The initial Q=0.457/(0.362×0.691)≈1.827>K, so x must be negative: the net reaction proceeds toward A and B.
The correctly expanded product is 0.250142−1.053x+x². Rearranging gives 0.023x²−1.024219x−0.451246734=0.
The physical root is x=−0.436301683 M. The other root is approximately 44.9676 M and gives negative reactant concentrations, so reject it.
The equilibrium concentrations are [A]=0.798301683 M, [B]=1.127301683 M and [C]=0.0206983171 M. Substitution gives Q=0.0230000, and each concentration is nonnegative.
This ICE calculation establishes equilibrium composition only. An irreversible rate constant cannot identify an exact finite time to reversible equilibrium. A separate reviewed kinetic threshold exercise must supply its own disappearance law and target concentration.
For part B only: t=ln(0.457/0.0360)/0.320=7.94114 min≈7.94 min. Neither the equilibrium constant nor the ICE table supplies a reversible kinetic time.
Reviewed ideal composition exercise: A+B⇌C, Kc=0.023, with initial concentrations 0.362 M, 0.691 M and 0.457 M. Define x as the forward extent per litre: [A]=0.362−x, [B]=0.691−x and [C]=0.457+x.
Kc=(0.457+x)/[(0.362−x)(0.691−x)]=0.023. The initial Q=0.457/(0.362×0.691)≈1.827>K, so x must be negative: the net reaction proceeds toward A and B.
The correctly expanded product is 0.250142−1.053x+x². Rearranging gives 0.023x²−1.024219x−0.451246734=0.
The physical root is x=−0.436301683 M. The other root is approximately 44.9676 M and gives negative reactant concentrations, so reject it.
The equilibrium concentrations are [A]=0.798301683 M, [B]=1.127301683 M and [C]=0.0206983171 M. Substitution gives Q=0.0230000, and each concentration is nonnegative.
This ICE calculation establishes equilibrium composition only. An irreversible rate constant cannot identify an exact finite time to reversible equilibrium. A separate reviewed kinetic threshold exercise must supply its own disappearance law and target concentration.
For part B only: t=ln(0.457/0.0360)/0.320=7.94114 min≈7.94 min. Neither the equilibrium constant nor the ICE table supplies a reversible kinetic time.
Glossary and deeper reading · 5 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Reaction rate
Reaction rate describes how quickly a reactant or product quantity changes with time. Rates expressed using different species must account for their stoichiometric coefficients. For example, a product's concentration–time slope can be used to determine its rate of formation.
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Equilibrium constant
An equilibrium constant is the value of the reaction quotient at equilibrium for a specified reaction and temperature. Its magnitude indicates equilibrium composition preference rather than reaction speed. For example, a large constant often indicates a product-favored equilibrium, but the starting composition still determines the immediate net direction.
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Chemical equilibrium
Chemical equilibrium is a dynamic state in which forward and reverse reaction rates are equal. Macroscopic composition remains constant in a closed system under fixed conditions, but reactant and product concentrations need not be equal. For example, a reversible reaction can reach equilibrium with much more reactant than product.
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Stoichiometry
Stoichiometry uses a balanced chemical equation to relate amounts of reactants and products. Equation coefficients provide mole ratios rather than direct mass ratios. For example, forming water from hydrogen and oxygen consumes two moles of hydrogen molecules per mole of oxygen molecules.
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ICE table
An ICE table organizes initial concentrations, stoichiometric changes, and equilibrium concentrations. Its change entries follow the balanced equation and do not imply equal changes for every species. For example, producing two units of a product for each reactant consumed requires twice the concentration-change entry for that product.