Equilibrium Calculations

Equilibrium Calculation

A quantitative calculation that uses an equilibrium-constant expression together with concentration or pressure information to determine an equilibrium quantity.

Calculating Kc from Equilibrium Concentrations

Substitute the known equilibrium molar concentrations into the Kc expression and evaluate the resulting ratio.

Known-Equilibrium-Concentration Problem

An equilibrium problem in which the equilibrium concentrations are already known and are used directly to calculate the equilibrium constant.

Missing Equilibrium Concentration Problem

An equilibrium problem in which K and some equilibrium concentrations are known and the equilibrium expression is solved for an unknown equilibrium concentration.

Initial-to-Equilibrium Concentration Problem

An equilibrium problem in which initial concentrations and K are known and the concentrations at equilibrium must be determined.

Why Can Initial Concentrations Usually Not Be Substituted Directly into K?

The equilibrium-constant expression requires equilibrium concentrations, so initial concentrations must first be related to their equilibrium values.

Equilibrium Expression as a Solving Equation

Once equilibrium concentrations or their algebraic expressions are substituted into the K expression, the equilibrium relationship becomes an equation that can be solved for the unknown.

Direct Substitution Equilibrium Problem

A problem in which all quantities required by the equilibrium expression are already known at equilibrium and can therefore be substituted directly.

Algebraic Equilibrium Problem

A problem in which an unknown equilibrium quantity must be isolated algebraically from the equilibrium-constant expression.

ICE-Based Equilibrium Problem

A problem in which equilibrium concentrations must first be expressed in terms of an unknown change before they can be substituted into K.

Equilibrium Concentration Expression

An algebraic expression such as C₀ − x, C₀ + x, or C₀ ± nx representing a species concentration at equilibrium.

Substitution of ICE Expressions into K

The Equilibrium-row expressions are inserted into the equilibrium-constant expression to create an equation in the unknown change variable.

Solving for x in an Equilibrium Problem

Determine the value of the concentration change that makes the calculated equilibrium ratio equal to the specified equilibrium constant.

Using a Solved x Value

Substitute the physically acceptable value of x back into the equilibrium-concentration expressions to calculate the final concentrations.

Quadratic Equilibrium Equation

An equilibrium equation that reduces to a quadratic equation because the equilibrium-concentration expressions contain powers or products involving x.

Quadratic Equation Standard Form

An equation written as ax² + bx + c = 0.

Quadratic Formula

A method for solving ax² + bx + c = 0 using x = [-b ± √(b² − 4ac)]/(2a).

Why Can an Equilibrium Calculation Produce Two Mathematical Roots?

A quadratic equation can have two solutions even though both may not represent physically possible chemical states.

Physical Root

A mathematical solution for x that produces physically possible equilibrium concentrations.

Nonphysical Root

A mathematical solution that must be rejected because it produces an impossible chemical result, such as a negative concentration.

Negative Equilibrium Concentration

A mathematically obtained concentration below zero; because concentration cannot physically be negative, the corresponding root must be rejected.

Root Validation

The process of testing a mathematical solution against physical constraints before accepting it as the equilibrium solution.

Why Must Quadratic Roots Be Checked?

Mathematical solutions are not automatically chemically valid; each root must produce physically possible concentrations.

Maximum Possible Forward Change

The reaction cannot consume more of a reactant than was initially available, which places an upper physical limit on x.

Maximum Possible Reverse Change

The reverse reaction cannot consume more product than is available, which places a physical limit on the magnitude of the reverse change.

Concentration Constraint on x

A valid x must keep every calculated equilibrium concentration at or above zero.

Stoichiometric Constraint on x

The allowable magnitude of x depends on both the initial concentrations and the stoichiometric coefficients relating the species.

Limiting Species in an ICE Calculation

A reactant or product whose available initial amount places the strictest physical limit on how far the reaction can shift.

Checking Equilibrium Concentrations After Solving

After finding x, calculate every equilibrium concentration and verify that the results are physically meaningful.

Back-Substitution Check

Substitute the calculated equilibrium concentrations back into the equilibrium expression to verify that they reproduce the specified K.

Why Is Back-Substitution Useful?

It checks whether the calculated concentrations actually satisfy the equilibrium condition and can reveal algebraic or arithmetic errors.

Equilibrium Calculation Workflow from Known Equilibrium Concentrations

Write the balanced reaction → write the correct equilibrium expression → substitute equilibrium concentrations → calculate K.

Equilibrium Calculation Workflow for a Missing Equilibrium Concentration

Write the equilibrium expression → substitute K and the known equilibrium quantities → solve algebraically for the missing quantity → verify physical validity.

Equilibrium Calculation Workflow from Initial Concentrations

Write the balanced reaction → determine the direction if necessary → construct equilibrium concentrations using stoichiometric changes → substitute into K → solve for x → calculate and verify equilibrium concentrations.

Reaction Direction Before Solving

If a mixture initially contains both reactants and products and the direction is unknown, compare Q with K before assigning the direction of concentration changes.

Why Determine Reaction Direction Before Assigning x Signs?

The direction determines which species are consumed and which are formed, so it determines the signs of the concentration changes.

Equilibrium Calculation When Q < K

The reaction proceeds forward while approaching equilibrium, so reactants decrease and products increase.

Equilibrium Calculation When Q > K

The reaction proceeds in reverse while approaching equilibrium, so products decrease and reactants increase.

Equilibrium Calculation When Q = K

The initial mixture is already at equilibrium, so no net concentration change is required.

Equilibrium Constant as the Final Constraint

Stoichiometry relates the concentration changes, while K determines the particular equilibrium composition at which those changes must stop.

Initial Conditions and Final Equilibrium

The initial composition helps determine the actual equilibrium concentrations reached even though K has a fixed value for the reaction at a specified temperature.

Same K, Different Equilibrium Concentrations

Different initial compositions can lead to different equilibrium concentrations while still satisfying the same equilibrium constant at the same temperature.

Why Can Different Equilibrium Mixtures Have the Same K?

K fixes the required equilibrium ratio of the included species rather than requiring one unique set of individual concentrations.

Equilibrium Calculation Sanity Check

A final assessment that concentrations are nonnegative, changes do not exceed available amounts, stoichiometric relationships are satisfied, and the calculated equilibrium composition reproduces K.

Physically Meaningful Equilibrium Solution

The solution that satisfies both the equilibrium equation and the physical constraints of the chemical system.