Chapter 16: Chemical Equilibrium – Principles, Calculations, and Applications

Chemical Equilibrium

Introduction to Chemical Equilibrium

Chemical equilibrium occurs when the rates of the forward and reverse reactions in a reversible chemical process are equal, resulting in constant concentrations of reactants and products over time. This state is dynamic, meaning that reactions continue to occur, but there is no net change in the concentrations of substances involved.

• Dynamic Equilibrium: Both forward and reverse reactions continue at equal rates.

• Independence of Starting Point: The final equilibrium state is the same regardless of whether the reaction is started with reactants or products.

• Example: $2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g)$

Equilibrium Constant ($K_c$)

Definition and Interpretation

The equilibrium constant ($K_c$) quantifies the ratio of product concentrations to reactant concentrations at equilibrium, each raised to the power of their stoichiometric coefficients. It is a measure of the extent to which a reaction proceeds to completion.

• General Form: For a reaction $aA + bB \rightleftharpoons cC + dD$:

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

• Interpretation of $K_c$:

  • $K_c \gg 1$: Products are favored at equilibrium.

  • $K_c \ll 1$: Reactants are favored at equilibrium.

  • $K_c \approx 1$: Neither reactants nor products are strongly favored.

• Concentration Terms: Only concentrations at equilibrium are used in the $K_c$ expression.

• Pure Solids and Liquids: These are omitted from the $K_c$ expression.

• Temperature Dependence: $K_c$ is constant at a given temperature but changes if the temperature changes.

Deriving Equilibrium Constant Expressions

Writing $K_c$ for Specific Reactions

To write the equilibrium constant expression, use the balanced chemical equation and include only aqueous or gaseous species.

• Example 1: $A(aq) \rightleftharpoons 2C(aq)$

$K_c = \frac{[C]^2}{[A]}$

• Example 2: $H_2O(l) \rightleftharpoons H_3O^+(aq) + OH^-(aq)$

$K_w = [H_3O^+][OH^-]$

• Note: Water is a pure liquid and is not included in the $K_w$ expression.

ICE Tables (Initial, Change, Equilibrium)

Using ICE Tables to Calculate Equilibrium Concentrations

ICE tables are a systematic way to organize and calculate the changes in concentrations of reactants and products as a reaction approaches equilibrium.

• Structure:

  • I: Initial concentrations

  • C: Change in concentrations (based on stoichiometry)

  • E: Equilibrium concentrations

• Example Table:

	2A	+	3B	⇌	Products
I	0.10 M		0.10 M
C	-2x		-3x
E	0.10-2x		0.10-3x

• All substances in this reaction are aqueous.

• Changes in concentration should be based on the stoichiometric coefficients.

Calculating Equilibrium Concentrations

Solving for Equilibrium Using $K_c$ and ICE Tables

To find equilibrium concentrations, set up the $K_c$ expression using the equilibrium values from the ICE table and solve for the unknown variable, often requiring the quadratic formula.

• Example: $C(aq) + D(aq) \rightleftharpoons A(aq) + B(aq)$, $K_c = 49.0$

	A(aq)	+	B(aq)	⇌	C(aq)	+	D(aq)
I	0.10 M		0.30 M		0		0
C	-x		-x		+x		+x
E	0.10-x		0.30-x		x		x

$K_c = \frac{[C][D]}{[A][B]} = \frac{x^2}{(0.10-x)(0.30-x)} = 49.0$

• This leads to a quadratic equation in $x$:

$49.0(0.10-x)(0.30-x) = x^2$

$0.03 - 0.10x - 0.30x + x^2 = \frac{x^2}{49.0}$

$0.03 - 0.40x + x^2 = \frac{x^2}{49.0}$

$0.03 - 0.40x + x^2 - \frac{x^2}{49.0} = 0$

• Use the quadratic formula to solve for $x$:

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

• Only physically meaningful (positive, real) roots are used.

• Check your answer by substituting back into the $K_c$ expression.

Reaction Quotient ($Q_c$) and Direction of Shift

Comparing $Q_c$ and $K_c$

The reaction quotient ($Q_c$) is calculated using the same expression as $K_c$, but with initial (not necessarily equilibrium) concentrations. Comparing $Q_c$ to $K_c$ predicts the direction the reaction will proceed to reach equilibrium.

• If $Q_c < K_c$: The reaction proceeds forward (toward products).

• If $Q_c > K_c$: The reaction proceeds in reverse (toward reactants).

• If $Q_c = K_c$: The system is at equilibrium.

• Example: For $2HI(g) \rightleftharpoons H_2(g) + I_2(g)$, $K_c = 65.0$ at a given temperature.

Temperature Dependence of Equilibrium Constants

Van't Hoff Equation

The equilibrium constant changes with temperature. The Van't Hoff equation relates the change in $K$ to the change in temperature and the standard enthalpy change ($\Delta H^\circ$) of the reaction.

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

• $K_1$ = equilibrium constant at temperature $T_1$

• $K_2$ = equilibrium constant at temperature $T_2$

• $R$ = 8.314 \; \text{J} / (\text{mol} \cdot \text{K})$

• $\Delta H^\circ$ = standard enthalpy change (J/mol)

• Example: If $K_1 = 4.4 \times 10^1$ at $298\;K$ and $\Delta H^\circ = 180,500\;J/mol$, calculate $K_2$ at $2400\;K$.

$\ln\left(\frac{K_2}{4.4 \times 10^1}\right) = -\frac{180,500}{8.314}\left(\frac{1}{2400} - \frac{1}{298}\right)$

$K_2 = 2.2 \times 10^2$

• As temperature increases, the equilibrium constant may increase or decrease depending on whether the reaction is endothermic or exothermic.

Summary Table: Key Concepts in Chemical Equilibrium

Concept	Description	Key Formula
Equilibrium Constant ($K_c$)	Ratio of product to reactant concentrations at equilibrium	$K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}$
ICE Table	Organizes initial, change, and equilibrium concentrations	See above for structure
Quadratic Formula	Used to solve for unknowns in equilibrium problems	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Reaction Quotient ($Q_c$)	Predicts direction of reaction shift	Same as $K_c$ expression, but with initial concentrations
Van't Hoff Equation	Relates $K$ to temperature	$\ln\left(\frac{K_2}{K_1}\right) = -\frac{\Delta H^\circ}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)$

Additional info: These notes are based on lecture slides and cover the fundamental principles and calculations related to chemical equilibrium, including the use of ICE tables, the quadratic formula, and the temperature dependence of equilibrium constants. The content is suitable for General Chemistry college students preparing for exams or seeking a concise review of equilibrium concepts.