Chemical Equilibrium: Concepts, Calculations, and Applications

Chemical Equilibrium

Definition of Equilibrium State

Chemical equilibrium is the state in which the rates of the forward and reverse reactions are equal, resulting in no net change in the concentrations of reactants and products over time. This is known as dynamic equilibrium because the reactions continue to occur, but at equal rates.

• Dynamic Equilibrium: Both forward and reverse reactions occur simultaneously and continuously.

• Ratefwd = Raterev: At equilibrium, the rate of the forward reaction equals the rate of the reverse reaction.

The Equilibrium Constant (K)

The equilibrium constant, K, quantifies the ratio of product concentrations to reactant concentrations at equilibrium, each raised to the power of their stoichiometric coefficients.

• Large K (>>1): Products are favored at equilibrium; the reaction proceeds nearly to completion.

• Small K (<<1): Reactants are favored at equilibrium; little product is formed.

General form for a reaction:

$aA + bB \rightleftharpoons cC + dD$

Equilibrium constant expression (Kc):

$K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b}$

• Pure solids and liquids are not included in the Kc expression.

The Reaction Quotient (Q)

The reaction quotient, Q, has the same form as the equilibrium constant but uses initial or non-equilibrium concentrations.

• Q < K: The reaction proceeds forward (toward products) to reach equilibrium.

• Q > K: The reaction proceeds in reverse (toward reactants) to reach equilibrium.

• Q = K: The system is at equilibrium.

Expression for Q:

$Q_c = \frac{[C]^c [D]^d}{[A]^a [B]^b}$

Writing and Using the Reaction Quotient

• Write the balanced chemical equation.

• Write the Qc expression, omitting pure solids and liquids.

• Substitute the given concentrations to calculate Q.

Multiple-Step Reactions and Overall Equilibrium

For reactions occurring in multiple steps:

• Overall Q: $Q_{overall} = Q_1 \times Q_2 \times Q_3$

• Overall K: $K_{overall} = K_1 \times K_2 \times K_3$

• Each step's Q and K are multiplied to get the overall values.

Relationship Between Kc and Kp

For gaseous reactions, equilibrium can be expressed in terms of concentrations (Kc) or partial pressures (Kp).

• Relationship:

$K_p = K_c (RT)^{\Delta n}$

• R = gas constant (0.0821 L·atm·mol−1·K−1)

• T = temperature in Kelvin

• $\Delta n$ = (moles of gaseous products) − (moles of gaseous reactants)

Predicting Reaction Direction Using Q and K

• Calculate Q using initial concentrations.

• Compare Q to K to determine the direction the reaction will proceed to reach equilibrium.

• Summary Table:

Q vs. K	Direction of Reaction
Q < K	Forward (toward products)
Q > K	Reverse (toward reactants)
Q = K	At equilibrium

Solving Equilibrium Problems

To solve equilibrium problems, use a reaction table (ICE table) to track changes in concentrations:

• I: Initial concentrations

• C: Change in concentrations

• E: Equilibrium concentrations

Example ICE Table:

	A	B	C	D
Initial (I)	a0	b0	c0	d0
Change (C)	−ax	−bx	+cx	+dx
Equilibrium (E)	a0−ax	b0−bx	c0+cx	d0+dx

• Substitute equilibrium concentrations into the K expression and solve for x.

• If the equation is quadratic, use the quadratic formula:

$x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a}$

• If Kc is very small and initial concentration is large, the change in concentration may be negligible (the "small x" assumption). Check the validity of this assumption by comparing x to the initial value (good if x < 5% of initial concentration).

Le Chatelier's Principle

Le Chatelier's Principle states that if a system at equilibrium is disturbed, the system will shift in a direction that minimizes the disturbance and restores equilibrium.

• Change in Concentration: Adding/removing reactants or products shifts equilibrium to oppose the change.

• Change in Pressure (for gases): Increasing pressure (by decreasing volume) shifts equilibrium toward the side with fewer moles of gas.

• Change in Temperature: For endothermic reactions, increasing temperature shifts equilibrium toward products; for exothermic, toward reactants.

The Van't Hoff Equation

The Van't Hoff equation describes how the equilibrium constant changes with temperature, given the standard enthalpy change (ΔH°) for the reaction.

$\ln \left( \frac{K_2}{K_1} \right) = -\frac{\Delta H^\circ}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)$

• K1, K2: Equilibrium constants at temperatures T1 and T2

• ΔH°: Standard enthalpy change

• R: Gas constant

Types of Equilibrium Problems and Solution Strategies

• Calculating Kc from Concentration Data:

  • Write the balanced equation.

  • Set up a reaction table (ICE table).

  • Calculate Kc using equilibrium concentrations.

• Determining Equilibrium Concentrations from Kc:

  • Write the balanced equation and Kc expression.

  • Convert units to M or atm as needed.

  • Set up an ICE table and solve for unknowns using Kc.

• Determining Equilibrium Concentrations from Initial Concentrations and Kc:

  • Write the balanced equation and Kc expression.

  • Set up an ICE table with initial values.

  • Substitute values into the Kc expression and solve for the unknown.

Summary Table: Steps for Solving Equilibrium Problems

Step	Description
1	Write the balanced chemical equation.
2	Write the Kc or Qc expression.
3	Set up an ICE table with initial, change, and equilibrium concentrations.
4	Substitute known values and solve for unknowns.
5	Check assumptions (e.g., small x) if used.
6	Interpret results and answer the question.

Example

For the reaction $2A + B \rightleftharpoons 2C$ with initial concentrations [A] = 1.0 M, [B] = 1.0 M, [C] = 0 M, and Kc = 4.0:

• Set up ICE table.

• Let x = change in [C] at equilibrium.

• Write Kc expression: $K_c = \frac{[C]^2}{[A]^2 [B]}$

• Solve for x using the quadratic formula if necessary.

Additional info: The notes also reference using the quadratic formula and the small x assumption for solving equilibrium problems, as well as the importance of checking the validity of assumptions by comparing the calculated change to the initial concentration (good if less than 5%).