Chemical Equilibrium II: Calculating and Applying Equilibrium Constants

Chemical Equilibrium II

Equilibrium Constants and Their Calculation

Chemical equilibrium occurs when the rates of the forward and reverse reactions are equal, resulting in constant concentrations of reactants and products. The equilibrium constant (K) quantifies the ratio of product to reactant concentrations at equilibrium for a given reaction.

• Kc is the equilibrium constant in terms of concentration (mol/L).

• Kp is the equilibrium constant in terms of partial pressure (atm).

• For a general reaction: $aA + bB \rightleftharpoons cC + dD$ $K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}$ $K_p = \frac{(P_C)^c(P_D)^d}{(P_A)^a(P_B)^b}$

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

• Conversion between Kc and Kp: $K_p = K_c(RT)^{\Delta n}$ where $\Delta n =$ (moles of gaseous products) $-$ (moles of gaseous reactants).

Comparing Kc and Kp

The relationship between Kc and Kp depends on the change in the number of moles of gas ($\Delta n$):

• If $\Delta n > 0$, $K_p > K_c$

• If $\Delta n = 0$, $K_p = K_c$

• If $\Delta n < 0$, $K_p < K_c$

ICE Table Method for Equilibrium Calculations

The ICE (Initial, Change, Equilibrium) table is a systematic way to calculate equilibrium concentrations when given initial concentrations and either the equilibrium constant or one equilibrium concentration.

• Step 1: Write the balanced chemical equation.

• Step 2: Set up the ICE table with initial concentrations, changes (using variables like x), and equilibrium concentrations.

• Step 3: Substitute equilibrium concentrations into the equilibrium expression and solve for x.

• Step 4: Calculate all equilibrium concentrations using the value of x.

Example: Calculating K from Equilibrium Concentrations

Given: $H_2(g) + I_2(g) \rightleftharpoons 2 HI(g)$ at 445°C. If $[H_2] = 0.11$ M, $[I_2] = 0.11$ M, $[HI] = 0.78$ M at equilibrium:

• $K_c = \frac{[HI]^2}{[H_2][I_2]} = \frac{0.78^2}{0.11 \times 0.11} = 50$

Reaction Quotient (Q) and Predicting Direction of Change

The reaction quotient (Q) is calculated using the same expression as K, but with concentrations at any point (not necessarily equilibrium).

• If $Q < K$, the reaction proceeds to the right (toward products).

• If $Q > K$, the reaction proceeds to the left (toward reactants).

• If $Q = K$, the system is at equilibrium.

Example: Using Q to Predict Reaction Direction

For $I_2(g) + Cl_2(g) \rightleftharpoons 2 ICl(g)$, with $K_p = 81.9$ and partial pressures $P_{I_2} = 0.114$ atm, $P_{Cl_2} = 0.102$ atm, $P_{ICl} = 0.355$ atm:

• $Q_p = \frac{P_{ICl}^2}{P_{I_2}P_{Cl_2}} = \frac{0.355^2}{0.114 \times 0.102} = 10.8$

• Since $Q_p < K_p$, the reaction will proceed to the right.

Solving for Equilibrium Concentrations Using Quadratic Equations

When the equilibrium expression leads to a quadratic equation, use the quadratic formula:

• General form: $ax^2 + bx + c = 0$

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

• Choose the physically meaningful (positive) solution for x.

Summary Table: ICE Table Example

Species	Initial (M)	Change	Equilibrium (M)
N2	0.200	–x	0.200 – x
O2	0.200	–x	0.200 – x
NO	0.000	+2x	2x

Key Takeaways

• The equilibrium constant K is a fundamental quantity for predicting the composition of a reaction mixture at equilibrium.

• ICE tables and the reaction quotient Q are essential tools for solving equilibrium problems and predicting the direction of reaction shifts.

• Always check units and ensure concentrations/pressures are in the correct form for Kc or Kp calculations.