Unit 7: Chemical Equilibrium: Dynamic Equilibria, Law of Mass Action & Le Chatelier's Principle

In physical chemistry, a chemical transformation does not generally proceed to complete unidirectional exhaustion of reactants. Instead, as product concentrations accumulate, the reverse reaction accelerates until a state of Dynamic Chemical Equilibrium is established.

The Microscopic Nature of Dynamic Equilibrium

At equilibrium, the macroscopic properties of the system—concentrations, color, pressure, temperature, and refractive index—remain completely invariant with time. However, this macroscopic quiescence masks intense microscopic activity:

Equilibrium is reached when the forward reaction rate $r_f$ equals the reverse reaction rate $r_r$ identically:

Chemical bonds continue to break and form at colossal frequencies, but the net rate of change of every chemical species is zero: $\frac{d[A_i]}{dt} = 0$.

The Principle of Detailed Balance (Microscopic Reversibility): In an isolated system at thermodynamic equilibrium, every microscopic elementary process must be in exact balance with its time-reversed counterpart. A system cannot maintain equilibrium by establishing cyclical reaction flows ($A \rightarrow B \rightarrow C \rightarrow A$).

The Law of Mass Action & Equilibrium Constants ($K_c, K_p, K_x$)

In 1864, Cato Maximilian Guldberg and Peter Waage formulated the Law of Mass Action: For a reversible reaction at constant temperature:

The ratio of the product of the active masses of the products to the product of the active masses of the reactants, each raised to their respective stoichiometric coefficients, is a constant:

1. Concentration Equilibrium Constant ($K_c$):

2. Pressure Equilibrium Constant ($K_p$):

For gas-phase reactions expressed in partial pressures $P_i$:

3. Mole Fraction Equilibrium Constant ($K_x$):

Exact Relationship Between $K_p, K_c$, and $K_x$

For ideal gases, partial pressure is related to molar concentration by $P_i = \left(\frac{n_i}{V}\right) R T = [A_i] R T$. Substituting into $K_p$:

Defining $\Delta n_g \equiv (c + d) - (a + b)$ as the stoichiometric change in moles of gas:

• If $\Delta n_g = 0$ (e.g., $\text{H}_2(g) + \text{I}_2(g) \rightleftharpoons 2\text{HI}(g)$): $K_p = K_c$ identically.
• If $\Delta n_g > 0$ (dissociation): $K_p > K_c$ (at $R T > 1$).
• If $\Delta n_g < 0$ (condensation/synthesis): $K_p < K_c$.

From Dalton's law of partial pressures, $P_i = x_i P_{total}$. Substituting into $K_p$:

While $K_p$ depends only on temperature, $K_x$ depends explicitly on total system pressure whenever $\Delta n_g \neq 0$!

Thermodynamic Foundation: $\Delta G^\circ = -R T \ln K^\circ$

Thermodynamically, equilibrium is the state of minimum Gibbs free energy at constant $T$ and $P$ ($dG = 0$). For any reaction $\sum \nu_i A_i = 0$, the reaction Gibbs energy is:

The chemical potential of species $i$ in a mixture is:

where $a_i$ is the dimensionless thermodynamic activity ($a_i = P_i / P^\circ$ for ideal gases with $P^\circ = 1\text{ bar}$, $a_i \approx [C_i] / C^\circ$ for ideal dilute solutions with $C^\circ = 1\text{ M}$).

Substituting into the reaction Gibbs energy:

Defining the Reaction Quotient $Q \equiv \prod_i a_i^{\nu_i}$:

At thermodynamic equilibrium, the driving force vanishes: $\Delta_r G = 0$, and the reaction quotient reaches its equilibrium value $Q \rightarrow K^\circ$:

Or expressing $K^\circ$ in exponential form:

Because $\Delta_r G^\circ$ is a pure standard-state property evaluated at $P^\circ = 1\text{ bar}$, the thermodynamic equilibrium constant $K^\circ$ is dimensionless and depends strictly upon temperature alone.

The algebraic formulation of the equilibrium constant expression depends upon the number and nature of thermodynamic phases present in the reacting system.

Homogeneous vs Heterogeneous Equilibria

1. Homogeneous Equilibria: All reacting chemical species reside within a single, continuous physical phase:

• Gas phase: $\text{N}_2\text{O}_4(g) \rightleftharpoons 2\text{NO}_2(g)$
• Liquid aqueous phase: $\text{CH}_3\text{COOH}(aq) + \text{H}_2\text{O}(l) \rightleftharpoons \text{CH}_3\text{COO}^-(aq) + \text{H}_3\text{O}^+(aq)$

2. Heterogeneous Equilibria: Reacting species occupy two or more distinct physical phases in contact (solid-gas, solid-liquid, liquid-gas):

• Thermal decomposition of limestone:

• Iron-steam equilibrium:

The Rule of Invariant Condensed Phase Activities

Fundamental Convention of Heterogeneous Equilibria: Pure solids and pure liquids are completely omitted from equilibrium constant expressions. Thermodynamic Proof: The thermodynamic activity of a pure solid or pure liquid in its standard state at pressure $P$ is:

Because the molar volumes of solids and liquids are minuscule ($V_m \sim 10^{-5}\text{ m}^3\cdot\text{mol}^{-1}$), even under several atmospheres of pressure, the Poynting factor correction is negligible:

Applying this rigorous rule to the limestone decomposition:

At a given temperature, the equilibrium partial pressure of carbon dioxide above decomposing calcium carbonate is fixed and independent of the amounts of solid $\text{CaCO}_3$ or $\text{CaO}$ present:

Mathematical Rules for Manipulating Equilibrium Constants

When balancing and combining chemical equilibria:

1. Reversing a Reaction:

If $A \rightleftharpoons B$ has constant $K_1$, then $B \rightleftharpoons A$ has constant $K_2$:

2. Multiplying Stoichiometric Coefficients by a Factor $n$:

If $A \rightleftharpoons B$ has constant $K_1$, then $n A \rightleftharpoons n B$ has constant $K_n$:

3. Adding Consecutive Reactions (Multiple Equilibria):

If Reaction 1 ($A \rightleftharpoons B$, constant $K_1$) is added to Reaction 2 ($B \rightleftharpoons C$, constant $K_2$) to yield the net reaction ($A \rightleftharpoons C$):

When chemical equations are added, their equilibrium constants are multiplied.

Before a chemical system reaches dynamic equilibrium, or when an equilibrium system is subjected to external perturbations, physical chemists must determine the spontaneous direction of reaction.

The Reaction Quotient ($Q$) as a Thermodynamic Compass

For a general reaction $a A + b B \rightleftharpoons c C + d D$, the Reaction Quotient ($Q_c$ or $Q_p$) has an identical mathematical algebraic form to the equilibrium constant, but is evaluated using instantaneous non-equilibrium concentrations or pressures:

Comparing the instantaneous quotient $Q$ with the equilibrium constant $K$ reveals the thermodynamic driving force from $\Delta_r G = R T \ln(Q / K)$:

| Condition | Sign of $\Delta_r G = R T \ln(Q/K)$ | Thermodynamic State | Spontaneous Direction of Shift | |---|---|---|---| | $Q < K$ | $\Delta_r G < 0$ (Negative) | Reactant excess / Product deficit | Forward ($\rightarrow$): Reactants convert into products until $Q = K$. | | $Q = K$ | $\Delta_r G = 0$ (Zero) | Dynamic Thermodynamic Equilibrium | No Net Shift: Forward and reverse rates are equal ($r_f = r_r$). | | $Q > K$ | $\Delta_r G > 0$ (Positive) | Product excess / Reactant deficit | Reverse ($\leftarrow$): Products decompose into reactants until $Q = K$. |

Quantitative Equilibrium Calculations: The ICE Table Method

The systematic solution of equilibrium concentrations relies on the ICE (Initial, Change, Equilibrium) algorithmic framework:

1. I (Initial): Tabulate initial concentrations or pressures of all species $[A]_0, [B]_0, \dots$

2. C (Change): Introduce an unknown extent of reaction $x$. Use stoichiometric coefficients to define changes in moles/concentration: $-a x$ for reactants, $+c x$ for products.

3. E (Equilibrium): Formulate algebraic equilibrium expressions: $[A]_{eq} = [A]_0 - a x, [C]_{eq} = [C]_0 + c x$.

4. Solve: Substitute equilibrium terms into $K_c = \frac{[C]_{eq}^c [D]_{eq}^d}{[A]_{eq}^a [B]_{eq}^b}$ and solve the resulting polynomial (linear, quadratic, or cubic) for physical roots ($x > 0$ and $[A_i]_{eq} \ge 0$).

The 5% Approximation Rule:

When $K_c \ll 1$ (typically $K_c < 10^{-4}$ and initial concentration $[A]_0 / K_c > 1000$), the extent of reaction $x$ is exceedingly small compared to $[A]_0$:

This avoids solving complex quadratic equations. The approximation is scientifically valid if the calculated $x$ is less than $5\%$ of $[A]_0$:

In 1884, Henri Louis Le Chatelier formulated one of the most profound qualitative generalizations in physical chemistry:

Le Chatelier's Principle: If a chemical system at dynamic equilibrium is subjected to an external perturbation or stress (a change in concentration, pressure, volume, or temperature), the system will spontaneously adjust its equilibrium state in such a direction as to partially counteract and relieve the imposed stress.

Rather than treating Le Chatelier's principle as an empirical rule, modern physical chemistry derives all shifts directly from the thermodynamic response of the reaction quotient $Q$ and equilibrium constant $K$.

1. Effect of Concentration Perturbations

• Adding a Reactant ($+[A]$): Denominator of $Q$ increases $\implies Q < K$. The system spontaneously shifts forward ($\rightarrow$), consuming added $A$ to form products until $Q = K$.
• Removing a Product ($- [C]$): Numerator of $Q$ decreases $\implies Q < K$. System shifts forward ($\rightarrow$), replenishing the removed product.
• Industrial Application: In the synthesis of ethyl acetate via esterification ($\text{CH}_3\text{COOH} + \text{C}_2\text{H}_5\text{OH} \rightleftharpoons \text{CH}_3\text{COOC}_2\text{H}_5 + \text{H}_2\text{O}$), continuous distillation of water drives the reaction to near $100\%$ completion.

2. Effect of Volume and Pressure Perturbations (Gas Reactions)

Decreasing container volume by compression ($V \downarrow$) increases total pressure ($P \uparrow$) and inflates every partial pressure $P_i = n_i R T / V$. Recall Eq. (7.6): $K_x = K_p P_{total}^{-\Delta n_g}$.

• If $\Delta n_g > 0$ (e.g., $\text{N}_2\text{O}_4(g) \rightleftharpoons 2\text{NO}_2(g), \Delta n_g = +1$):

$Q_p = \frac{P_{\text{NO}_2}^2}{P_{\text{N}_2\text{O}_4}} \propto \frac{(1/V)^2}{1/V} = \frac{1}{V}$. Compressing ($V \downarrow$) causes $Q_p > K_p$. System shifts reverse ($\leftarrow$) toward fewer moles of gas.

• If $\Delta n_g < 0$ (e.g., $\text{N}_2(g) + 3\text{H}_2(g) \rightleftharpoons 2\text{NH}_3(g), \Delta n_g = -2$):

Compressing causes $Q_p < K_p$. System shifts forward ($\rightarrow$) toward fewer moles of gas.

• If $\Delta n_g = 0$: Pressure changes have zero effect on equilibrium composition.

Addition of an Inert Gas (e.g., Argon):

• At Constant Volume ($V = \text{const}$): Total pressure rises, but individual partial pressures $P_i = n_i R T / V$ are unchanged. $Q_p$ is unchanged $\implies$ Zero effect on equilibrium!
• At Constant Pressure ($P = \text{const}$): The container must expand ($V \uparrow$) to accommodate the inert gas. This dilutes all reacting partial pressures, acting identically to a volume expansion, shifting equilibrium toward the side with more moles of gas.

3. Effect of Temperature: The van 't Hoff Equation

Temperature is the only variable that changes the numerical value of the equilibrium constant $K$.

Differentiating the fundamental thermodynamic relation $\ln K^\circ = -\frac{\Delta_r G^\circ}{R T}$ with respect to temperature at constant pressure:

Using the Gibbs-Helmholtz equation $\frac{d(\Delta G^\circ / T)}{dT} = -\frac{\Delta H^\circ}{T^2}$:

This is the differential van 't Hoff Equation.

1. Endothermic Reactions ($\Delta_r H^\circ > 0$):

Increasing temperature increases $K^\circ$. The system shifts forward ($\rightarrow$), absorbing heat to counteract the temperature increase.

2. Exothermic Reactions ($\Delta_r H^\circ < 0$):

Increasing temperature decreases $K^\circ$. The system shifts reverse ($\leftarrow$), releasing less heat.

Integrating the van 't Hoff equation between $T_1$ and $T_2$ assuming constant $\Delta_r H^\circ$:

A plot of $\ln K$ versus $1/T$ is a straight line with slope $-\frac{\Delta_r H^\circ}{R}$, providing a purely non-calorimetric method to determine reaction enthalpies!

4. Effect of a Catalyst

A catalyst lowers the activation energy of both the forward and reverse reactions by an identical amount: $\Delta E_a^{forward} = \Delta E_a^{reverse}$.

A catalyst has zero effect on the equilibrium constant $K$, zero effect on equilibrium concentrations, and causes zero shift in the position of equilibrium. It merely accelerates the rate at which equilibrium is reached.

Aqueous solutions host vital dynamic equilibria governing biological cells, industrial wet chemistry, and environmental geochemistry.

The Autoionization of Water & The pH Scale

Pure liquid water undergoes dynamic self-ionization:

The thermodynamic equilibrium constant is the Ion Product of Water ($K_w$):

At $298.15\text{ K}$, $K_w = 1.008 \times 10^{-14} \approx 1.00 \times 10^{-14}$. Because autoionization is endothermic ($\Delta H_{auto}^\circ = +55.8\text{ kJ}\cdot\text{mol}^{-1}$), $K_w$ increases steeply with temperature: at $373.15\text{ K}$ ($100^\circ\text{C}$), $K_w \approx 5.4 \times 10^{-13}$, and neutral pH drops to $6.13$!

In 1909, Søren Sørensen introduced the logarithmic pH scale:

Taking the negative logarithm of $K_w$:

Weak Acid Dissociation ($K_a$) & Conjugate Pairs

A Brønsted-Lowry weak acid $\text{HA}$ ionizes reversibly in water:

Its conjugate base $\text{A}^-$ hydrolyzes water:

Multiplying $K_a$ and $K_b$:

Buffer Solutions & The Henderson-Hasselbalch Equation

A buffer solution resists changes in pH upon the addition of small amounts of strong acid or base. It consists of a conjugate acid-base pair in roughly equimolar proportions:

• Acidic buffer: Weak acid $+$ conjugate base salt (e.g., $\text{CH}_3\text{COOH} / \text{CH}_3\text{COONa}$).
• Basic buffer: Weak base $+$ conjugate acid salt (e.g., $\text{NH}_3 / \text{NH}_4\text{Cl}$).

Taking the logarithm of the $K_a$ expression:

Rearranging yields the Henderson-Hasselbalch Equation (1908/1916):

• When $[\text{A}^-] = [\text{HA}]$ (the half-equivalence point in a titration): $\text{pH} = \text{p}K_a$.
• Optimal buffer capacity occurs within the range $\text{pH} = \text{p}K_a \pm 1$.

Solubility Equilibria: The Solubility Product Constant ($K_{sp}$)

For a sparingly soluble ionic solid $M_p X_q$ in equilibrium with its dissolved aqueous ions:

Since the activity of the pure solid is unity ($a_{solid} = 1$):

The Ion Product ($Q_{sp}$) & Precipitation Criteria:

• $Q_{sp} < K_{sp}$: Unsaturated solution; more solid can dissolve.
• $Q_{sp} = K_{sp}$: Saturated dynamic equilibrium solution.
• $Q_{sp} > K_{sp}$: Supersaturated metastable solution; spontaneous precipitation occurs until $Q_{sp} = K_{sp}$.

The Common Ion Effect on Solubility:

According to Le Chatelier's Principle, adding an external soluble salt sharing an ion with the precipitate (e.g., adding $\text{NaCl}$ to a saturated $\text{AgCl}$ solution, adding $\text{Cl}^-$) drives the dissolution equilibrium reverse ($\leftarrow$), drastically decreasing the molar solubility of the sparingly soluble salt.