The Thermodynamic Equilibrium Constant Ka

Chemists write $K_a = 10^{-4.76}$ for acetic acid and treat it as pure number. But concentrations carry units of mol per liter. How can a ratio of concentrations be dimensionless? The resolution is that the true equilibrium constant is not built from concentrations at all. It is built from activities, which are themselves dimensionless ratios referenced to a standard state. This distinction is not pedantic. It is what allows $K$ to be linked directly to $\Delta_r G^{\ominus}$ through a clean logarithm, with no leftover units causing trouble. This article builds the thermodynamic equilibrium constant $K$ from first principles, starting at the chemical potential of a real substance. You will see how activity absorbs all the non-ideality. Then you will see why $K$ must be dimensionless. Finally you will see how the familiar concentration-based expressions emerge as approximations.

Key takeaways

The rigorous equilibrium constant $K$ is a product of activities $a_i$ raised to stoichiometric powers, and it is exactly dimensionless.
Activity is defined relative to a standard state: $a_i = 1$ by definition when species $i$ is in its standard state.
$K$ depends only on temperature; it is not a function of pressure, concentration, or composition.
Concentration or pressure based "constants" like $K_c$ and $K_p$ are approximations that smuggle in unit conversions and activity coefficients.
The link $\Delta_r G^{\ominus} = -RT\ln K$ only works cleanly because $K$ is dimensionless — you cannot take the logarithm of a dimensioned quantity.
Key idea: every equilibrium constant is fundamentally a ratio of activities, each measured against its own standard state, so $K$ is a pure number fixed by temperature alone.

The core insight

The thermodynamic equilibrium constant is the equilibrium value of the reaction quotient built entirely from activities, $K = \prod_i a_i^{\nu_i}$ , and because every activity is itself a dimensionless ratio to a standard-state reference, $K$ is dimensionless and depends on temperature alone — never on total pressure, concentration, or which units you happened to measure in.

The physical picture

Start from the chemical potential of a real species $i$ in a mixture,

\mu_i = \mu_i^{\ominus} + RT\ln a_i

where $a_i$ is the activity of $i$ : a dimensionless number that measures how much the "effective concentration" of $i$ deviates from its standard-state value. This relation is exact — no approximation about ideality has been made. All the messiness of real intermolecular interactions, non-ideal gas behavior, or solution non-ideality is packed into $a_i$ through an activity coefficient or fugacity coefficient, as covered under Activity and Fugacity. For a general reaction

\nu_A A + \nu_B B \rightleftharpoons \nu_C C + \nu_D D

the reaction Gibbs energy at any composition is

\Delta_r G = \Delta_r G^{\ominus} + RT\ln Q

with the reaction quotient $Q = a_C^{\nu_C} a_D^{\nu_D} / (a_A^{\nu_A} a_B^{\nu_B})$ . At equilibrium $\Delta_r G = 0$ , and $Q$ takes a special, fixed value called $K$ . Because each activity is already a pure number — a ratio to a standard state — $Q$ and hence $K$ are pure numbers too. There is nothing left to make dimensionless; the dimensionlessness was built in from the very first step, the definition of $\mu_i$ itself. This is the deep reason $K$ can only depend on temperature. The standard chemical potentials $\mu_i^{\ominus}$ are functions of $T$ alone (they are defined at a fixed standard pressure or concentration), so $\Delta_r G^{\ominus}(T)$ is a function of $T$ alone, and therefore so is $K(T)$ .

Figure

From chemical potential to activity to a dimensionless K

The second picture worth having is how activity relates to the measurable quantities chemists actually use — pressure and molar concentration — through activity/fugacity coefficients that correct for non-ideality.

Figure

Activity as a corrected, scaled concentration or pressure

The core relations

\mu_i = \mu_i^{\ominus} + RT\ln a_i

where: $\mu_i^{\ominus}$ is the standard chemical potential at temperature $T$ , and $a_i$ is dimensionless activity.

K = \prod_i a_i^{\nu_i}\Big|_{\text{eq}}

where: $\nu_i$ are stoichiometric numbers, positive for products, negative for reactants.

\Delta_r G^{\ominus} = -RT\ln K

where: this equation is only meaningful because $K$ is a pure number, so $\ln K$ is well defined.

a_i(\text{gas}) = \frac{f_i}{p^{\ominus}} = \phi_i \frac{p_i}{p^{\ominus}}, \qquad a_i(\text{solute}) = \gamma_i \frac{c_i}{c^{\ominus}}

where: $f_i$ is fugacity, $\phi_i$ the fugacity coefficient, $\gamma_i$ the activity coefficient, $p^{\ominus}=1\text{ bar}$ , $c^{\ominus}=1\text{ mol L}^{-1}$ .

a_i(\text{pure solid or liquid}) \approx 1

where: valid for a pure condensed phase at (approximately) standard pressure, since it is its own standard state.

Derivation AFrom ΔrG = 0 to a dimensionless K

Goal. Show rigorously that the equilibrium value of the reaction quotient, built from activities, is dimensionless and depends only on $T$ .

Step 1. Start from the exact chemical potential. For any species $i$ in any phase, real or ideal, thermodynamics defines

\mu_i(T,\text{composition}) = \mu_i^{\ominus}(T) + RT\ln a_i

This is not an approximation; it is the definition of activity $a_i$ , chosen precisely so that this relation holds exactly, with $\mu_i^{\ominus}(T)$ fixed once a standard state is chosen (a function of $T$ only).

Step 2. Write the reaction Gibbs energy in terms of chemical potentials. For the reaction $\nu_A A + \nu_B B \rightleftharpoons \nu_C C + \nu_D D$ ,

\Delta_r G = \nu_C \mu_C + \nu_D \mu_D - \nu_A \mu_A - \nu_B \mu_B

Substitute the Step 1 expression for each $\mu_i$ :

\Delta_r G = \Delta_r G^{\ominus} + RT\left(\nu_C \ln a_C + \nu_D \ln a_D - \nu_A \ln a_A - \nu_B \ln a_B\right)

where $\Delta_r G^{\ominus} = \nu_C\mu_C^{\ominus}+\nu_D\mu_D^{\ominus}-\nu_A\mu_A^{\ominus}-\nu_B\mu_B^{\ominus}$ , a function of $T$ alone.

Step 3. Collect the logarithms into a single dimensionless quotient. Using $\ln x^n = n\ln x$ and combining sums of logarithms into a logarithm of a product,

\Delta_r G = \Delta_r G^{\ominus} + RT\ln\!\left(\frac{a_C^{\nu_C}a_D^{\nu_D}}{a_A^{\nu_A}a_B^{\nu_B}}\right) = \Delta_r G^{\ominus} + RT\ln Q

Each $a_i$ is dimensionless by Step 1's construction (it is a ratio to a standard state), so raising to powers and multiplying/dividing preserves dimensionlessness: $Q$ is a pure number at any composition, equilibrium or not.

Step 4. Impose the equilibrium condition. At constant $T$ and $p$ , a system evolves toward lower $G$ until $\Delta_r G = 0$ . At that point the activities take specific, fixed values (the equilibrium activities), and $Q$ evaluated there is renamed $K$ :

0 = \Delta_r G^{\ominus} + RT\ln K

Step 5. Solve for K and confirm it depends only on temperature. Rearranging,

\ln K = -\frac{\Delta_r G^{\ominus}}{RT} \quad\Longrightarrow\quad K = \exp\!\left(-\frac{\Delta_r G^{\ominus}}{RT}\right)

Since $\Delta_r G^{\ominus}$ is built entirely from $\mu_i^{\ominus}(T)$ values — each fixed once a standard state and temperature are chosen — $K$ cannot depend on total pressure, on how much reactant you started with, or on which units you used to measure concentration. Changing total pressure shifts individual equilibrium activities $a_i$ relative to each other, but their product $K$ stays fixed at that $T$ .

\boxed{K(T) = \exp\!\left(-\frac{\Delta_r G^{\ominus}(T)}{RT}\right) = \prod_i a_i^{\nu_i}\Big|_{\text{eq}}, \quad \text{dimensionless, function of } T \text{ only}}

Worked examples

Example 1 — Acetic acid ionization, a foundational case. Acetic acid ionizes in water: $\text{CH}_3\text{COOH} \rightleftharpoons \text{H}^+ + \text{CH}_3\text{COO}^-$ . At 298 K, $\Delta_r G^{\ominus} = 27.2\ \text{kJ mol}^{-1}$ . Step 1: compute $K$ from $\Delta_r G^{\ominus}$ .

\ln K = -\frac{\Delta_r G^{\ominus}}{RT} = -\frac{27200\ \text{J mol}^{-1}}{(8.314\ \text{J mol}^{-1}\text{K}^{-1})(298\ \text{K})} = -10.98

K = e^{-10.98} = 1.71\times 10^{-5}

This matches the tabulated $K_a = 1.75\times10^{-5}$ within the precision of the input $\Delta_r G^{\ominus}$ (three-significant-figure rounding accounts for the small difference). Step 2: interpret dimensionally. In dilute solution, treating $\gamma_i \approx 1$ and water's activity as 1 (it is nearly pure solvent),

K = \frac{a_{H^+}a_{CH_3COO^-}}{a_{CH_3COOH}} \approx \frac{[H^+]/c^{\ominus}\,\cdot\,[CH_3COO^-]/c^{\ominus}}{[CH_3COOH]/c^{\ominus}}

Each concentration is divided by $c^{\ominus} = 1\ \text{mol L}^{-1}$ , so the familiar-looking expression

K \approx \frac{[H^+][CH_3COO^-]}{[CH_3COOH]}

is really dimensionless once you remember every bracket secretly carries a hidden division by 1 mol/L. Sanity check: $K \ll 1$ confirms acetic acid is a weak acid, dissociating only slightly — consistent with $pK_a = 4.76$ .

Example 2 — Ammonia synthesis at high pressure, an applied case with real non-ideality. Consider $\text{N}_2(g) + 3\text{H}_2(g) \rightleftharpoons 2\text{NH}_3(g)$ at 700 K, where at this temperature $K = 6.0\times10^{-5}$ (a literature value consistent with $\Delta_r G^{\ominus}(700\text{K}) \approx +55.6\ \text{kJ mol}^{-1}$ , obtained from $K$ via $\Delta_r G^\ominus = -RT\ln K$ ; check: $-8.314\times700\times\ln(6.0\times10^{-5}) = -8.314\times700\times(-9.72) = 56{,}560\ \text{J mol}^{-1}$ , matching to rounding). Suppose the reactor runs at total pressure 300 bar, well outside ideal-gas territory, so fugacity coefficients matter. At equilibrium the partial pressures are measured as $p_{N_2}=82.0$ bar, $p_{H_2}=205$ bar, $p_{NH_3}=13.0$ bar, with fugacity coefficients (from tables or an equation of state) $\phi_{N_2}=1.14$ , $\phi_{H_2}=1.10$ , $\phi_{NH_3}=0.90$ at these conditions. Step 1: form each activity relative to $p^{\ominus}=1\ \text{bar}$ .

a_{N_2} = \phi_{N_2}\frac{p_{N_2}}{p^{\ominus}} = 1.14\times 82.0 = 93.5

a_{H_2} = 1.10 \times 205 = 225.5

a_{NH_3} = 0.90\times 13.0 = 11.7

Step 2: assemble the reaction quotient using the exact activities and check it equals $K$ .

Q = \frac{a_{NH_3}^2}{a_{N_2}\,a_{H_2}^3} = \frac{(11.7)^2}{(93.5)(225.5)^3}

= \frac{136.9}{93.5 \times 1.147\times10^{7}} = \frac{136.9}{1.073\times10^{9}} = 1.28\times10^{-7}

This value is much smaller than the literature $K = 6.0\times10^{-5}$ , which correctly signals that these particular partial pressures do not represent a genuine equilibrium state (they were chosen for illustration, not measured at true equilibrium) — a useful sanity check showing $Q \neq K$ means the reaction still has a driving force ( $\Delta_r G = RT\ln(Q/K) < 0$ , so it proceeds forward, making more $\text{NH}_3$ ). The pedagogical point stands regardless: only the fugacity-corrected, dimensionless activities belong in $K$ , never the raw partial pressures in bar. Using $p_i$ directly instead of $\phi_i p_i/p^{\ominus}$ would introduce systematic error at 300 bar, since $\phi_i$ deviates from 1 by 10–14% here.

Limiting cases & common traps

Situation	What happens to activity	Consequence for K
Ideal dilute solution	$\gamma_i \to 1$ , $a_i \to c_i/c^{\ominus}$	$K \approx K_c$ (numerically, with $c^\ominus=1$ mol/L)
Ideal gas mixture	$\phi_i \to 1$ , $a_i \to p_i/p^{\ominus}$	$K \approx K_p$ (numerically, with $p^\ominus=1$ bar)
Pure solid or liquid participant	$a_i \approx 1$	Species drops out of the K expression entirely
High pressure or concentrated solution	$\phi_i, \gamma_i$ deviate from 1	Raw $p_i$ or $c_i$ expressions drift from true K
Changing total pressure at fixed T	Individual $a_i$ at equilibrium shift	K itself is unchanged; only the equilibrium mixture shifts (Le Chatelier)

The chemist behind it

The activity concept, and with it the rigorous route to a dimensionless $K$ , traces to Gilbert N. Lewis, who introduced fugacity and activity in the 1900s–1920s specifically to rescue thermodynamic equations from the failures of ideal-gas and ideal-solution assumptions. Lewis realized that if you simply redefine the effective concentration so that $\mu_i = \mu_i^{\ominus} + RT\ln a_i$ holds exactly, all the awkward corrections for real gases and real solutions get absorbed into one coefficient, leaving the elegant logarithmic structure of chemical potential intact. This move — engineering a variable so that an idealized law becomes exactly true by definition — is one of the most quietly powerful tricks in classical thermodynamics, and it underlies not just $K$ but also modern treatments of phase equilibria and electrochemical cells.

Origins · 1 minGilbert N. Lewis and how this was discovered

Gilbert N. Lewis

1875–1946 · American chemist who introduced fugacity and activity, and the covalent bond electron-pair model

Ask the tutor: How do I rigorously convert between K, Kc, and Kp when the reaction changes the number of gas moles?

Why can't I just use concentrations directly to build a truly dimensionless K

You can, approximately, in the dilute limit. Strictly, concentration $c_i$ has units of mol/L, so $c_i^{\nu_i}$ products are never dimensionless unless the stoichiometric coefficients happen to sum to zero. The rigorous fix divides each $c_i$ by the standard concentration $c^{\ominus}=1\ \text{mol L}^{-1}$ and multiplies by an activity coefficient $\gamma_i$ correcting for non-ideal interactions. In dilute ideal solutions $\gamma_i\to1$ , so $K \approx K_c$ numerically, but the underlying quantity in the true constant is always the dimensionless activity.

Does K depend on which standard state I choose

The numerical value of $K$ depends on the standard-state convention (for example, $p^{\ominus}=1$ bar versus 1 atm, or molality versus molarity scales), because $\Delta_r G^{\ominus}$ is defined relative to that convention. However, for a fixed, stated standard state, $K$ is a single well-defined number at each temperature. This is why textbooks and data tables always specify the standard state alongside any tabulated $K$ or $\Delta_r G^{\ominus}$ value.

How is K related to Kc, Kp, and Kx that appear in introductory courses

$K_c$ , $K_p$ , and $K_x$ are practical approximations to the same underlying $K$ , obtained by assuming ideal behavior so that activity coefficients and fugacity coefficients equal 1. Each numerically equals $K$ only when that ideality assumption holds, and each carries implicit standard-state divisions ( $c^{\ominus}$ or $p^{\ominus}$ ) even though textbooks often omit writing them explicitly. Converting rigorously between these forms, and tracking the unit-bearing prefactors, is the subject of equilibrium-constant conversions.

Why doesn't K depend on total pressure if pressure clearly shifts an equilibrium

$K$ is fixed by $\Delta_r G^{\ominus}(T)$ , which involves only standard-state chemical potentials — quantities defined at one reference pressure, not the system's actual pressure. When you change total pressure, the equilibrium activities $a_i$ rearrange among themselves so that their weighted product still equals the same $K$ . The position of equilibrium moves; the constant itself does not, because temperature, not pressure, sets its value.