Topic 7.6 Properties of the Equilibrium Constant: determine how K changes when a reaction is reversed (reciprocal), scaled (power) or combined with another reaction (product), and relate Kc to Kp.

What this topic is asking

The College Board (Topic 7.6) wants you to determine how the equilibrium constant changes when a reaction is reversed, scaled or combined with another reaction, and to relate $K_c$ to $K_p$. These rules let you build the equilibrium constant of a target reaction from known constants, the equilibrium analogue of Hess's law.

Reversing a reaction

So if a forward reaction has $K = 100$, the reverse has $K = 0.01$. A product-favored forward reaction (large $K$) becomes a reactant-favored reverse reaction (small $K$), which makes physical sense: if products dominate going one way, reactants dominate going the other.

Scaling a reaction

Note the contrast with enthalpy, which scales linearly. The equilibrium constant scales as a power because the coefficients are exponents in its expression, not multipliers. Getting this distinction right is a common exam discriminator.

Adding reactions and relating Kc to Kp

When two reactions are added, their equilibrium constants multiply: $K_\text{overall} = K_1 \times K_2$. The intermediate species cancel, and the equilibrium expressions combine multiplicatively, leaving the overall expression. This is the equilibrium parallel of adding enthalpies in Hess's law (where they add rather than multiply).

For gas-phase reactions, the pressure-based constant $K_p$ relates to the concentration-based $K_c$ by

where $\Delta n$ is the change in the number of moles of gas (moles of gaseous products minus moles of gaseous reactants). When $\Delta n = 0$, $K_p = K_c$.

These rules are the equilibrium counterpart of the manipulations you learned for enthalpy in Hess's law, but with one important difference in the arithmetic: enthalpies are added and scaled linearly, whereas equilibrium constants are multiplied, raised to powers and inverted. The reason is structural. Enthalpy appears as a plain sum of terms, so reversing flips a sign and scaling multiplies; the equilibrium constant appears as a product of concentrations raised to exponents, so reversing inverts the whole expression, scaling raises it to a power, and combining multiplies the expressions. Keeping this contrast in mind prevents the most common error on these problems, which is treating $K$ the way you would treat $\Delta H$. When in doubt, write out the equilibrium expression explicitly and apply the manipulation to it, rather than relying on a remembered rule.

Try this

Q1. A reaction has $K = 9.0$. Determine $K$ for the same reaction multiplied by $\tfrac{1}{2}$. [2 points]

• Cue. $K^{1/2} = \sqrt{9.0} = 3.0$.

Q2. For a gas reaction with $\Delta n = 0$, state the relationship between $K_p$ and $K_c$. [1 point]

• Cue. $K_p = K_c$, because $(RT)^0 = 1$.