Chemical Equilibrium: Equilibrium Constants, Reaction Quotient, and Calculating Equilibrium Concentrations

Chapter 14: Chemical Equilibrium

Introduction to Chemical Equilibrium

Chemical equilibrium occurs when the rates of the forward and reverse reactions are equal, resulting in constant concentrations of reactants and products. Understanding equilibrium is essential for predicting the outcome of chemical reactions and for industrial applications.

Relationships Between the Equilibrium Constant and the Chemical Equation

The equilibrium constant (K) quantifies the ratio of product and reactant concentrations at equilibrium. The value of K depends on how the chemical equation is written.

• Multiplying the Equation by a Factor: If a chemical equation is multiplied by a factor n, the new equilibrium constant is the original constant raised to the power of n. Example: $A(g) \rightarrow B(g)$ $K_1 = \frac{P_B}{P_A}$ $2A(g) \rightarrow 2B(g)$ $K_2 = \frac{P_B^2}{P_A^2} = K_1^2$

• Reversing the Equation: If the equation is reversed, the new equilibrium constant is the reciprocal of the original. Example: $A(g) \rightarrow B(g)$ $K_1 = \frac{P_B}{P_A}$ $B(g) \rightarrow A(g)$ $K_2 = \frac{P_A}{P_B} = \frac{1}{K_1}$

• Adding Equations: When two or more equations are added, the equilibrium constant for the overall reaction is the product of the individual constants. Example: $A(g) \rightarrow C(g)$, $K_1 = \frac{P_C}{P_A}$ $B(g) \rightarrow D(g)$, $K_2 = \frac{P_D}{P_B}$ $A(g) + B(g) \rightarrow C(g) + D(g)$, $K_3 = \frac{P_C P_D}{P_A P_B} = K_1 \times K_2$

Calculating Equilibrium Constants for Combined Reactions

When given multiple reactions and their equilibrium constants, you can determine the constant for a new reaction by manipulating the given equations (multiplying, reversing, or adding) and applying the rules above.

• Example:

  • (a) $FeO(s) + CO(g) \rightarrow Fe(s) + CO_2(g)$, $K_1 = 4.17$

  • (b) $2CO(g) + O_2(g) \rightarrow 2CO_2(g)$, $K_2 = 1.01 \times 10^{15}$

  • Find $K$ for $2Fe(s) + O_2(g) \rightarrow 2FeO(s)$

  • Manipulate equations: reverse (a), multiply by 2, add to (b), and apply rules: $K = (1/K_1)^2 \times K_2$

The Reaction Quotient (Q)

The reaction quotient (Q) is used to determine the direction a reaction will proceed to reach equilibrium. It has the same algebraic form as K, but uses current (not necessarily equilibrium) concentrations or pressures.

• Definition: For the reaction $aA + bB \rightarrow cC + dD$: $Q = \frac{[C]^c [D]^d}{[A]^a [B]^b}$ or for gases, $Q_p = \frac{P_C^c P_D^d}{P_A^a P_B^b}$

• Application: Substitute any values (not just equilibrium) into the Q expression. Compare Q to K to predict the direction of the reaction:

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

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

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

• Example: For $N_2O_4(g) \rightleftharpoons 2NO_2(g)$, given pressures, calculate Q and compare to K to predict direction.

Calculating Equilibrium Concentrations (ICE Tables)

Determining equilibrium concentrations is crucial in industrial chemistry to assess feasibility and profitability. The ICE (Initial, Change, Equilibrium) table is a systematic method for solving equilibrium problems.

• Steps for ICE Table Calculations:

  1. Tabulate known initial and equilibrium concentrations/pressures for all species.

  2. Determine the direction of the reaction using Q and K.

  3. If initial and equilibrium values are given for a species, calculate the change.

  4. Use stoichiometry to relate changes for all species.

  5. Deduce equilibrium values for all species and/or solve for K as required.

• Example:

  • For $2SO_2(g) + O_2(g) \rightarrow 2SO_3(g)$, given initial and equilibrium pressures, use ICE table to find K:

  	2 SO2 (g)	O2 (g)	2 SO3 (g)
  Initial	2.43	0	0
  Change	-1.33	+0.665	+1.33
  Equilibrium	1.10	0.665	1.33

  Equilibrium constant expression: $K = \frac{P_{SO_3}^2}{P_{SO_2}^2 P_{O_2}} = \frac{(1.33)^2}{(1.10)^2 \times 0.665} = 0.972$

Quadratic Equations in Equilibrium Calculations

Sometimes, solving for equilibrium concentrations requires solving a quadratic equation, especially when the change in concentration is not negligible compared to the initial amount.

• Example:

  • For $H_2(g) + I_2(g) \rightarrow 2HI(g)$, $K = 49.7$:

  	H2 (g)	I2 (g)	2 HI (g)
  Initial	1.00	2.00	0
  Change	-x	-x	+2x
  Equilibrium	1.00-x	2.00-x	2x

  $K = \frac{(2x)^2}{(1.00-x)(2.00-x)} = 49.7$ Solving quadratic: $x = 2.33$ and $x = 0.93$. Only $x = 0.93$ is valid because $x = 2.33$ would result in negative concentrations.

Special Considerations in Equilibrium Calculations

• If the change in concentration (x) is much smaller than the initial concentration, the 5% rule can be applied to simplify calculations. If $K$ is very small ($K < 10^{-3}$), the change is often negligible.

• For aqueous reactions, use molarity instead of pressure in ICE tables.

• Sometimes, a quadratic equation yields two solutions; only the physically meaningful (non-negative) solution is valid.

Summary Table: ICE Table Example

Step	Description
Initial	Record initial concentrations/pressures of all species.
Change	Determine change using stoichiometry and direction of reaction.
Equilibrium	Calculate equilibrium values by adding change to initial.
Expression	Write equilibrium constant expression and substitute values.
Solve	Solve for unknowns (concentration, pressure, or K).

Key Terms

• Equilibrium Constant (K): A number that expresses the ratio of concentrations or pressures of products to reactants at equilibrium.

• Reaction Quotient (Q): A value calculated like K, but using current (not equilibrium) concentrations or pressures.

• ICE Table: A tabular method for organizing initial, change, and equilibrium values in equilibrium calculations.

Additional info:

• Quadratic equations may arise in equilibrium problems when the change in concentration is not negligible compared to the initial concentration.

• Negative concentrations are physically impossible; always check the validity of solutions.

• Industrial applications of equilibrium include synthesis of chemicals, optimization of reaction conditions, and assessment of process feasibility.