1. Equilibrium Constant Kc
The equilibrium constant \(K_c\) quantifies the composition of a chemical system at equilibrium
in terms of species activities. For ideal mixtures we approximate activities by molar
concentrations (mol·L⁻¹) for dissolved species and by partial pressures for gases, with appropriate
standards. This page summarizes the definitions, what to include (and exclude), and how the calculator
builds the expression and shows the steps.
1) Definition and general form
For a balanced reaction
\[
\begin{aligned}
\nu_1\ce{A1} + \nu_2\ce{A2} + \cdots &\rightleftharpoons \mu_1\ce{B1} + \mu_2\ce{B2} + \cdots
\end{aligned}
\]
\[
\begin{aligned}
K &= \frac{\prod_i a(\ce{B_i})^{\mu_i}}{\prod_j a(\ce{A_j})^{\nu_j}}
\end{aligned}
\]
where \(a(\ce{X})\) is the activity of species \(\ce{X}\). In many classroom problems we take
\(a(\ce{X}) \approx [\ce{X}]/c^\circ\) for solutes (with \(c^\circ=1\,\mathrm{mol\,L^{-1}}\)),
giving the familiar concentration form
\[
\begin{aligned}
K_{\mathrm{c}} &= \frac{\prod_i [\ce{B_i}]^{\mu_i}}{\prod_j [\ce{A_j}]^{\nu_j}}
\end{aligned}
\]
Strictly, \(K\) is dimensionless because activities are dimensionless; using concentrations introduces
“formal units” that cancel when referenced to the standard state. The calculator reports the numerical
value based on concentrations and explains the formal power \(\Delta\nu\) when helpful.
2) What is included in \(K_c\) (and what is not)
- Include species in the gas (g) or aqueous (aq) phases.
- Omit pure solids (s) and pure liquids (l):
their activity is defined as \(a=1\), so they do not appear in the expression.
Examples:
\[
\begin{aligned}
\ce{CaCO3(s)} &\rightleftharpoons \ce{CaO(s)} + \ce{CO2(g)} \\[4pt]
K_{\mathrm{c}} &= [\ce{CO2}]
\end{aligned}
\]
\[
\begin{aligned}
\ce{H2(g)} + \ce{I2(s)} &\rightleftharpoons \ce{2HI(g)} \\[4pt]
K_{\mathrm{c}} &= \frac{[\ce{HI}]^2}{[\ce{H2}]}
\end{aligned}
\]
In the calculator, choosing “s” or “l” automatically fixes the species’ activity to 1 and excludes it
from the \(K_c\) expression (the input box is locked to 1 to reflect this).
3) Relation between \(K_c\) and \(K_p\) (gas-phase)
For a gas-only reaction at temperature \(T\),
\[
\begin{aligned}
K_p &= K_c\,(RT)^{\Delta n}\,,\qquad
\Delta n = \sum(\text{stoich. coeffs of gaseous products}) - \sum(\text{gaseous reactants})
\end{aligned}
\]
Here \(R\) is the gas constant and \(T\) is the absolute temperature in kelvin.
4) How the calculator constructs the steps
- Balanced reaction. It draws the reaction line you enter, showing each species and its state.
- Exclude (s) and (l). Pure solids and liquids are listed as “omitted (activity = 1)”.
- Write \(K_c\). It forms
\(\displaystyle K_{\mathrm{c}} = \frac{\prod [\text{products}]^{\mu_i}}{\prod [\text{reactants}]^{\nu_j}}\)
using only gases/aqueous species.
- Substitute numbers. Each concentration is inserted, keeping exponents equal to stoichiometric coefficients.
- Evaluate. Numerator and denominator are computed separately, then \(K_c\) is reported and briefly interpreted
(products favored if \(K_c>1\), reactants favored if \(K_c<1\)).
5) Mini worked examples
(a) Haber process (all gases): \(\ce{N2(g) + 3H2(g) <=> 2NH3(g)}\)
\[
\begin{aligned}
K_{\mathrm{c}} &= \frac{[\ce{NH3}]^{2}}{[\ce{N2}]\,[\ce{H2}]^{3}}
\end{aligned}
\]
(b) Heterogeneous equilibrium: \(\ce{CaCO3(s) <=> CaO(s) + CO2(g)}\)
\[
\begin{aligned}
K_{\mathrm{c}} &= [\ce{CO2}]
\end{aligned}
\]
6) Practical notes & common pitfalls
- Use coefficients as exponents. If the coefficient is 2, the concentration is squared, etc.
- Don’t include pure solids or pure liquids in \(K_c\); the calculator handles this automatically when you choose the state.
- Zero concentrations. A zero in the denominator makes \(K_c\to\infty\). Physically, true zero at equilibrium is unusual; check inputs.
- Temperature dependence. \(K\) depends only on temperature; changing initial amounts at fixed \(T\) changes the equilibrium
composition but not \(K\).
- Units remark. With concentrations, a “formal unit” \((\mathrm{mol\,L^{-1}})^{\Delta\nu}\) may appear. In rigorous terms
with activities and standard states, \(K\) is dimensionless.
7) Using the calculator efficiently
- Enter the number of reactants and products, then click Next.
- Fill each row: species formula, choose the physical state, coefficient, and concentration (mol·L⁻¹).
When you choose “s” or “l”, the concentration locks to 1 and the species is omitted from the expression.
- Click Calculate Kc to see the aligned derivation and result. Use Fill example to try a preset.
