The question of whether an equilibrium constant has units is a common point of confusion in chemistry. The true, thermodynamic equilibrium constant (\(K\)) is always dimensionless—it has no units. \(K\) represents the ratio of products to reactants when a chemical reaction reaches equilibrium. Apparent units often appear in simple calculations due to a common simplification that ignores the full thermodynamic definition.
Defining the Equilibrium Constant
The equilibrium constant, \(K\), is a numerical value describing the ratio of product concentrations to reactant concentrations for a reversible reaction at equilibrium. This ratio remains constant at a specific temperature. For a generic reversible reaction, \(aA + bB \rightleftharpoons cC + dD\), the expression for \(K\) is a quotient of product concentrations raised to their stoichiometric coefficients, divided by reactant concentrations raised to theirs.
When concentrations are measured in moles per liter (molarity, \(M\)), the constant is denoted as \(K_c\). The expression is \(K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b}\), where the brackets denote molar concentration. For reactions involving gases, concentrations are expressed using partial pressures, typically in atmospheres (atm) or bars, and the constant is called \(K_p\). The value of the equilibrium constant provides a measure of the extent to which a reaction proceeds; a large \(K\) means products are favored, while a small \(K\) means reactants are favored.
Apparent Units in Basic Calculations
The unit question is confusing because introductory calculations use measured concentrations or pressures, which have units like \(M\) or \(atm\). When these values are substituted into the \(K_c\) or \(K_p\) expression, the resulting constant often appears to have units based on the reaction’s stoichiometry. The presence or absence of an apparent unit depends entirely on \(\Delta n\), the difference between the total moles of products and reactants.
For a reaction where the total moles of products and reactants are equal, such as \(H_2(g) + I_2(g) \rightleftharpoons 2HI(g)\), the units cancel out completely. The \(K_c\) expression \(\frac{[HI]^2}{[H_2][I_2]}\) results in units of \(\frac{M^2}{M \cdot M} = M^0\), making the constant unitless. However, in a reaction like \(N_2O_4(g) \rightleftharpoons 2NO_2(g)\), the expression \(\frac{[NO_2]^2}{[N_2O_4]}\) yields units of \(M\).
The resulting apparent unit, such as \(M\) or \(M^{-1}\), is arbitrary and specific only to the balanced chemical equation used. This dependency on stoichiometry makes these apparent units chemically uninformative. The unit of \(K_c\) for one reaction might be \(M\), while for another it is \(M^{-2}\), highlighting that the unit is merely a mathematical artifact of the simplified calculation.
The Role of Standard States and Activity
The definitive explanation for why the equilibrium constant is dimensionless lies in the thermodynamic concept of activity (\(a\)). The true equilibrium constant, \(K\), is defined by the activities of the products and reactants. Activity represents the effective concentration or pressure of a species, accounting for non-ideal behavior.
Activity is a dimensionless quantity because it is defined as the ratio of a substance’s actual concentration or pressure to its standard state concentration or pressure. For solutes, the standard state concentration (\(C^\circ\)) is defined as \(1 \text{ M}\). For gases, the standard state partial pressure (\(P^\circ\)) is typically defined as \(1 \text{ bar}\).
To calculate the true, dimensionless \(K\), every concentration term in the \(K_c\) expression must be written as \(\frac{[X]}{C^\circ}\), and every pressure term in \(K_p\) must be written as \(\frac{P_X}{P^\circ}\). Since the standard state values (\(1 \text{ M}\) or \(1 \text{ bar}\)) have the same units as the measured values, this division mathematically cancels the units for every term. The simplified \(K_c\) and \(K_p\) values used in basic calculations are approximations of the true thermodynamic constant \(K\). \(K\) is the only constant directly related to the change in Gibbs Free Energy (\(\Delta G^\circ\)) via the equation \(\Delta G^\circ = -RT \ln K\).