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), used for gaseous reactions.

• 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 omitted from the equilibrium expression.

Conversion between Kc and Kp:

• $K_p = K_c(RT)^{\Delta n}$, where $\Delta n$ is the change in moles of gas ($\Delta n =$ moles of gaseous products $-$ moles of gaseous reactants).

Using ICE Tables to Calculate Equilibrium Concentrations

The ICE (Initial, Change, Equilibrium) table is a systematic method for tracking concentration changes as a reaction approaches equilibrium.

• List initial concentrations of all species.

• Define changes in concentration using variables (e.g., $-x$, $+2x$).

• Express equilibrium concentrations in terms of initial values and changes.

• Substitute equilibrium concentrations into the equilibrium expression to solve for unknowns.

Example: For $N_2(g) + O_2(g) \rightleftharpoons 2 NO(g)$ with initial concentrations $[N_2] = 0.200$ M, $[O_2] = 0.200$ M, $[NO] = 0.000$ M:

	[N2]	[O2]	[NO]
Initial	0.200	0.200	0.000
Change	-x	-x	+2x
Equilibrium	0.200-x	0.200-x	2x

Substitute into $K_c = \frac{[NO]^2}{[N_2][O_2]}$ to solve for $x$.

Reaction Quotient (Q) and Predicting Direction of Change

The reaction quotient (Q) is calculated using the same expression as K, but with current (not necessarily equilibrium) concentrations or pressures.

• 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.

Solving Quadratic Equations in Equilibrium Problems

Some equilibrium problems require solving quadratic equations. For $ax^2 + bx + c = 0$:

• Discriminant: $\Delta = b^2 - 4ac$

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

• Choose the physically meaningful (positive) solution for concentration changes.

Summary Table: ICE Table Example

Step	Description
1	Set up an ICE table with initial, change, and equilibrium rows.
2	Write the equilibrium expression for K.
3	Solve for unknowns using algebra (quadratic if necessary).
4	Check that calculated concentrations are physically reasonable.

Key Equations

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

• $K_p = \frac{(P_C)^c(P_D)^d}{(P_A)^a(P_B)^b}$

• $K_p = K_c(RT)^{\Delta n}$

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

Visual Representation: Simple Equilibrium Shift

Below is a visual representation of a simple equilibrium system where the reaction can shift to the right or left depending on the initial composition:

Additional info: The blue circles can be interpreted as schematic representations of reactant and product molecules in a simple equilibrium system, illustrating the concept of shifting between reactants and products.