13.2 Equilibrium Constants

Chemical equilibrium is a dynamic state where forward and reverse reactions occur at equal rates. Equilibrium constants quantify this balance, telling you whether products or reactants are favored once the system settles. Understanding these constants helps you predict which direction a reaction will shift and connects to thermodynamic ideas like Gibbs free energy.

Equilibrium Constants

Reaction quotients for chemical reactions

The reaction quotient ($Q$) is a snapshot of where a reaction stands right now, before it necessarily reaches equilibrium. You calculate it using the current concentrations (or partial pressures) of reactants and products, and then compare it to $K$ to figure out which direction the reaction needs to shift.

For the general reaction $aA + bB \rightleftharpoons cC + dD$, the reaction quotient is:

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

Products go in the numerator, reactants in the denominator, and the stoichiometric coefficients become exponents.

Homogeneous vs. heterogeneous reactions:

• In homogeneous reactions, all species are in the same phase. For gaseous reactions, you can use partial pressures instead of concentrations:

$Q_p = \frac{P_C^c \cdot P_D^d}{P_A^a \cdot P_B^b}$

• In heterogeneous reactions, species are in different phases. Solids and pure liquids are excluded from the expression because their concentrations are constant and don't change during the reaction. Only gaseous and aqueous species appear in $Q$.

Calculation of equilibrium constants

The equilibrium constant ($K$) is just the value of $Q$ when the system has reached equilibrium. You calculate it the same way, but using equilibrium concentrations or pressures.

For $aA + bB \rightleftharpoons cC + dD$:

• Concentration-based: $K_c = \frac{[C]_{eq}^c[D]_{eq}^d}{[A]_{eq}^a[B]_{eq}^b}$
• Pressure-based: $K_p = \frac{P_{C,eq}^c \cdot P_{D,eq}^d}{P_{A,eq}^a \cdot P_{B,eq}^b}$

Converting between $K_c$ and $K_p$:

These two are related by:

$K_p = K_c(RT)^{\Delta n}$

where $\Delta n$ = (moles of gaseous products) − (moles of gaseous reactants), $R = 0.08206 \text{ L atm mol}^{-1}\text{ K}^{-1}$, and $T$ is temperature in Kelvin. If $\Delta n = 0$ (equal moles of gas on both sides), then $K_p = K_c$.

A few important points to remember:

• The balanced equation's coefficients become the exponents in the $K$ expression. If you double the equation, $K$ gets squared. If you reverse the equation, the new $K$ is $1/K$.
• $K$ changes only with temperature. Changing concentration, pressure, or adding a catalyst does not change the value of $K$.

Significance of equilibrium constant values

The size of $K$ tells you how far a reaction goes toward completion:

• Large $K$ ($K >> 1$): Products are heavily favored. At equilibrium, you'll find mostly products.
• Small $K$ ($K << 1$): Reactants are heavily favored. The reaction barely proceeds forward.
• $K \approx 1$: Significant amounts of both reactants and products are present at equilibrium.

Connection to Gibbs free energy:

The equilibrium constant is linked to the standard Gibbs free energy change by:

$\Delta G^\circ = -RT \ln K$

• When $K > 1$, $\Delta G^\circ$ is negative (thermodynamically favorable in the forward direction).
• When $K < 1$, $\Delta G^\circ$ is positive (the reverse direction is favored under standard conditions).

Using $Q$ vs. $K$ to predict reaction direction:

This comparison is one of the most useful tools in equilibrium:

• $Q < K$: Too few products relative to equilibrium. The reaction shifts right (toward products).
• $Q > K$: Too many products relative to equilibrium. The reaction shifts left (toward reactants).
• $Q = K$: The system is at equilibrium. No net change occurs.

Relationship between Equilibrium, Kinetics, and Thermodynamics

Equilibrium sits at the intersection of kinetics and thermodynamics, and it's worth understanding what each one controls:

• Kinetics tells you how fast a reaction reaches equilibrium. A reaction can have a large $K$ (strongly favoring products) but still be extremely slow without a catalyst.
• Thermodynamics tells you where the equilibrium lies and determines the value of $K$. It answers the question "how far will this reaction go?" but says nothing about speed.
• The reaction mechanism (the step-by-step molecular pathway) influences the rate laws and kinetics, but the overall equilibrium constant depends only on the balanced equation and the temperature.