Equilibrium Constant \(K_p\) — Theory & Guide
\(K_p\) quantifies the position of a chemical equilibrium using partial pressures.
For a gas-phase reaction
\[
\sum_j \nu_j^{(\mathrm{R})}\,\ce{Reactant_j(g)} \;\ce{<=>}\; \sum_i \nu_i^{(\mathrm{P})}\,\ce{Product_i(g)},
\]
the thermodynamically correct definition is in terms of activities
\[
K_p \;=\; \frac{\displaystyle\prod_{i}\,a_i^{\nu_i^{(\mathrm{P})}}}{\displaystyle\prod_{j}\,a_j^{\nu_j^{(\mathrm{R})}}},\qquad
a_k \equiv \frac{P_k}{P^\circ}.
\]
Here \(P_k\) is the partial pressure of species \(k\) and \(P^\circ\) is the standard pressure (usually 1 bar, sometimes 1 atm in textbooks).
What actually goes into \(K_p\)?
- Only gaseous species appear. Pure solids \((\mathrm{s})\) and liquids \((\mathrm{l})\) have activity \(a\approx 1\) and are omitted.
- Aqueous species \((\mathrm{aq})\) are also omitted in a \(K_p\) expression (they belong to \(K_c\) or activity-based \(K\) with molalities).
- Stoichiometric coefficients become exponents in the expression.
Writing the expression
If all participating species are gases, the symbolic form is
\[
K_p \;=\; \frac{\displaystyle\prod_{\text{products}}\left(\frac{P_i}{P^\circ}\right)^{\nu_i}}{\displaystyle\prod_{\text{reactants}}\left(\frac{P_j}{P^\circ}\right)^{\nu_j}}
\;=\; \left(P^\circ\right)^{-\Delta n}\;
\frac{\displaystyle\prod_{\text{products}} P_i^{\nu_i}}{\displaystyle\prod_{\text{reactants}} P_j^{\nu_j}},
\qquad \Delta n \equiv \sum_{\text{products}}\nu_i - \sum_{\text{reactants}}\nu_j.
\]
Because each activity \(a=P/P^\circ\) is a ratio, \(K_p\) is dimensionless. Some texts show a “formal” pressure unit when the \((P^\circ)^{-\Delta n}\) factor is not written explicitly; this tool uses the activity form so your reported \(K_p\) is unitless.
Relation between \(K_p\) and \(K_c\)
For ideal gases at temperature \(T\) and for concentrations in mol·L\(^{-1}\),
\[
K_p \;=\; K_c\,(R T)^{\Delta n}\;\times\;\Big(\frac{1}{P^\circ}\Big)^{\Delta n},
\]
If \(P^\circ=1\ \mathrm{bar}\), it is convenient to use \(R=0.08314\ \mathrm{L\,bar\,mol^{-1}\,K^{-1}}\);
if \(P^\circ=1\ \mathrm{atm}\), use \(R=0.082057\ \mathrm{L\,atm\,mol^{-1}\,K^{-1}}\). This calculator avoids unit pitfalls by working directly with activities \(P/P^\circ\).
How the calculator computes \(K_p\)
- You enter the reaction table. Give each species, its state, coefficient, and partial pressure with units.
- The tool omits non-gases. Any \((\mathrm{s}),(\mathrm{l}),(\mathrm{aq})\) is treated with \(a=1\).
- Pressures are normalized. It converts each gaseous partial pressure to the selected base (bar or atm), then forms
\(a_i=P_i/P^\circ\).
- It builds and evaluates the expression. It substitutes the activities with the proper exponents and computes
\(\text{numerator}\), \(\text{denominator}\), and \(K_p\).
Worked outline
For \(\ce{N2(g) + 3H2(g) <=> 2NH3(g)}\) at some \(T\) with partial pressures \(P_{\ce{N2}},P_{\ce{H2}},P_{\ce{NH3}}\):
\[
K_p \;=\; \frac{\left(\dfrac{P_{\ce{NH3}}}{P^\circ}\right)^2}{\left(\dfrac{P_{\ce{N2}}}{P^\circ}\right)\left(\dfrac{P_{\ce{H2}}}{P^\circ}\right)^3}
\;=\; \frac{P_{\ce{NH3}}^2}{P_{\ce{N2}}\,P_{\ce{H2}}^{3}}\;\left(P^\circ\right)^{-2+1+3}
\;=\; \frac{P_{\ce{NH3}}^2}{P_{\ce{N2}}\,P_{\ce{H2}}^{3}}\;\left(P^\circ\right)^{2}.
\]
If you use activities \(P/P^\circ\) directly (the calculator’s default), the \(P^\circ\) factor is already embedded and the final \(K_p\) is dimensionless.
Interpreting \(K_p\)
- \(K_p \gg 1\): products favored at equilibrium.
- \(K_p \ll 1\): reactants favored.
- \(K_p \approx 1\): comparable amounts.
\(K_p\) depends on temperature. As \(T\) changes, so does \(K_p\) (van ’t Hoff relation). The standard pressure \(P^\circ\) selection (1 bar vs 1 atm) only affects the normalization of activities, not the underlying equilibrium.
Common pitfalls
- Including non-gases. Do not include \((\mathrm{s}),(\mathrm{l}),(\mathrm{aq})\) in \(K_p\).
- Mixing units without normalization. Always express partial pressures relative to the same \(P^\circ\) (the tool does this for you).
- Forgetting exponents. Coefficients in the balanced equation are exponents in the equilibrium expression.
- Non-ideal gases. At high pressures or strong interactions, replace \(P\) with fugacity \(f\). This tool assumes ideal behavior.
Using this tool
- Enter the number of reactants and products; choose the standard pressure \(P^\circ\) (1 bar recommended).
- For each species, set the state. If not \(g\), it will be omitted and “locked” to activity 1.
- Provide coefficients and partial pressures (any common unit). The tool converts, forms \(a=P/P^\circ\), and evaluates \(K_p\).
- Read the step-by-step derivation and interpretation shown under the result.
