Law of mass action and Kc, Kp

WHAT is the Law of Mass Action?

For an elementary reaction $aA + bB \rightarrow cC + dD$: $\text{rate}_{forward} = k_f [A]^a [B]^b$

WHY the powers? Because a reaction event needs $a$ molecules of A and $b$ of B to meet simultaneously. The chance of that meeting scales like $[A]^a[B]^b$ — same logic as multiplying probabilities of independent events.

HOW we derive $K_c$ (kinetic route)

Consider the reversible reaction: $aA + bB \rightleftharpoons cC + dD$

Step 1 — Forward rate (why: A,B collide to form products): $r_f = k_f [A]^a [B]^b$

Step 2 — Backward rate (why: C,D collide to reform reactants): $r_b = k_b [C]^c [D]^d$

Step 3 — At equilibrium the rates are equal (why: no net change means forward = backward): $k_f [A]^a [B]^b = k_b [C]^c [D]^d$

Step 4 — Rearrange (why: gather the two rate constants, both constant at fixed $T$): $\frac{k_f}{k_b} = \frac{[C]^c [D]^d}{[A]^a [B]^b} = K_c$

HOW we get $K_p$ (for gases)

For gases it is easier to measure partial pressures. Define with partial pressures: $K_p = \frac{p_C^{\,c}\, p_D^{\,d}}{p_A^{\,a}\, p_B^{\,b}}$

Relate $K_p$ to $K_c$ using the ideal gas law $pV = nRT$:

Step 1 — For each gas, partial pressure (why: pressure ∝ concentration for ideal gas): $p_i = \frac{n_i}{V}RT = [i]\,RT \quad\text{where } [i]=\tfrac{n_i}{V}$

Step 2 — Substitute $p_i = [i]RT$ into $K_p$: $K_p = \frac{([C]RT)^c\,([D]RT)^d}{([A]RT)^a\,([B]RT)^b}$

Step 3 — Separate the $RT$ powers (why: collect the exponents of $RT$): $K_p = \underbrace{\frac{[C]^c[D]^d}{[A]^a[B]^b}}_{K_c}\cdot (RT)^{(c+d)-(a+b)}$

Step 4 — Let $\Delta n_g = (c+d)-(a+b)$ = moles of gaseous products − moles of gaseous reactants:

Important rules (WHY they hold)

Manipulating equilibria:

• Reverse a reaction → $K' = 1/K$ (numerator and denominator swap).
• Multiply coefficients by $n$ → $K' = K^n$ (each term raised to $n$).
• Add two reactions → $K = K_1 \times K_2$ (rates chain multiply).

Worked examples

Common mistakes

Flashcards

Connections

• Chemical Equilibrium – Dynamic Nature
• Le Chatelier's Principle
• Reaction Quotient Q vs K
• Gibbs Free Energy and K ($\Delta G^\circ = -RT\ln K$)
• Rate Constants and Arrhenius Equation
• Ionic Equilibrium – Ka, Kb, Kw

Concept Map

Hinglish (regional understanding)