Extent of reaction and the equilibrium constant \(K_p\) (gas-phase)
This note explains how the calculator links the reaction extent \(x\) (in pressure units for a constant-\(T,V\) gas mixture) with the equilibrium constant \(K_p\). It uses a vertical ICE table built from partial pressures, supports unit conversion (Pa, kPa, MPa, bar, mbar, atm, Torr, mmHg, psi), and provides two workflows: (i) using one known equilibrium partial pressure; (ii) using a known \(K_p\).
1) Definitions and notation
- General gas reaction (coefficients \(\nu_i>0\) for products, \(\nu_i<0\) for reactants):
\[
\sum_i \nu_i \,\ce{A_i(g)} = 0.
\]
- At constant \(T,V\), define a pressure-extent \(x\) so that for each gaseous species
\[
P_i \;=\; P_{i,0} + \nu_i\,x,\quad
\text{with } \nu_i<0 \text{ (reactant) and } \nu_i>0 \text{ (product).}
\]
(This is the pressure analogue of \(c_i=c_{i,0}+\nu_i x\) used for \(K_c\).)
- Activities for gases: \(a_i=\dfrac{P_i}{P^\circ}\) with a chosen standard pressure \(P^\circ\) (1 bar by IUPAC; 1 atm is common in textbooks). Using activities makes \(K_p\) dimensionless.
2) Vertical ICE table in partial pressures
Only (g) species appear in \(K_p\). Pure (s), (l), and (aq) are omitted (activity \(\approx 1\)).
|
\(\ce{A(g)}\) |
\(\ce{B(g)}\) |
\(\ce{C(g)}\) |
\(\ce{D(g)}\) |
| I | \(P_{\ce A,0}\) | \(P_{\ce B,0}\) | \(P_{\ce C,0}\) | \(P_{\ce D,0}\) |
|---|
| Δ | \(\nu_{\ce A}x\) | \(\nu_{\ce B}x\) | \(\nu_{\ce C}x\) | \(\nu_{\ce D}x\) |
|---|
| E | \(P_{\ce A,0}+\nu_{\ce A}x\) | \(P_{\ce B,0}+\nu_{\ce B}x\) | \(P_{\ce C,0}+\nu_{\ce C}x\) | \(P_{\ce D,0}+\nu_{\ce D}x\) |
|---|
3) Equilibrium constant \(K_p\) and the reaction quotient \(Q_p(x)\)
With activities \(a_i=P_i/P^\circ\),
\[
K_p \;=\; \frac{\prod_{\text{products}} a_i^{\nu_i}}{\prod_{\text{reactants}} a_i^{|\nu_i|}}
\;=\;
\frac{\prod_{\text{products}} \left(\dfrac{P_{i,0}+\nu_i x}{P^\circ}\right)^{\nu_i}}
{\prod_{\text{reactants}} \left(\dfrac{P_{i,0}-|\nu_i| x}{P^\circ}\right)^{|\nu_i|}}.
\]
The corresponding reaction quotient as a function of \(x\) is
\[
Q_p(x) \;=\; \frac{\prod_{\text{products}} \big(P_{i,0}+\nu_i x\big)^{\nu_i}}
{\prod_{\text{reactants}} \big(P_{i,0}-|\nu_i|x\big)^{|\nu_i|}}
\;\;\Big/\;\;(P^\circ)^{\Delta n},\quad
\Delta n=\sum_{\text{prod}}\nu_i-\sum_{\text{react}}|\nu_i|.
\]
In practice, the calculator chooses a base unit equal to \(P^\circ\) (bar or atm), so numerically \(a_i=P_i\) in that base and the \((P^\circ)^{\Delta n}\) factor is implicitly handled.
4) Two workflows supported by the calculator
- Known equilibrium partial pressure of one gas.
If a species \(j\) has measured \(P_j^{(\mathrm{eq})}\), then
\[
x \;=\; \frac{P_j^{(\mathrm{eq})} - P_{j,0}}{\nu_j}.
\]
This \(x\) fixes all equilibrium pressures \(P_i^{(\mathrm{eq})}=P_{i,0}+\nu_i x\), after which
\[
K_p \;=\; \frac{\prod_{\text{prod}} \big(P_i^{(\mathrm{eq})}/P^\circ\big)^{\nu_i}}
{\prod_{\text{react}} \big(P_i^{(\mathrm{eq})}/P^\circ\big)^{|\nu_i|}}.
\]
- Known \(K_p\) (solve for \(x\)).
Insert \(P_i=P_{i,0}+\nu_i x\) into \(Q_p(x)\) and solve the scalar equation
\[
Q_p(x)\;=\;K_p
\quad\Longleftrightarrow\quad
\ln Q_p(x)\;=\;\sum_i \nu_i \ln\!\big(P_{i,0}+\nu_i x\big) \;-\; \Delta n \ln P^\circ \;=\; \ln K_p.
\]
The tool brackets a root on the feasible interval (below) and uses a robust bisection search.
5) Feasible interval for the extent \(x\)
Physical partial pressures must remain non-negative:
\[
P_{i,0}+\nu_i x \;\ge\; 0\quad \text{for every gaseous species}.
\]
This yields bounds:
- for reactants (\(\nu_i<0\)): \(x \le \dfrac{P_{i,0}}{|\nu_i|}\)
- for products (\(\nu_i>0\)): \(x \ge -\dfrac{P_{i,0}}{\nu_i}\)
The intersection over all gases defines \([x_{\min},\,x_{\max}]\). Any \(x\) outside this range would make at least one \(P_i^{(\mathrm{eq})}\le 0\), which is unphysical.
6) Units and standard pressure
- The calculator accepts: Pa, kPa, MPa, bar, mbar, atm, Torr, mmHg, psi, and converts them internally.
- Choose \(P^\circ\) (1 bar or 1 atm) consistently. With the base unit set to \(P^\circ\), numerical activities equal the numeric pressures in that unit.
7) Relation to \(K_c\)
For ideal gases,
\[
K_p \;=\; K_c\,(RT)^{\Delta n},\qquad \Delta n=\sum_{\text{prod}}\nu_i-\sum_{\text{react}}|\nu_i|.
\]
If \(\Delta n=0\) then \(K_p=K_c\).
8) Worked symbolic template: \(a\ce{A(g)}+b\ce{B(g)}\rightleftharpoons c\ce{C(g)}+d\ce{D(g)}\)
| \(\ce{A}\) | \(\ce{B}\) | \(\ce{C}\) | \(\ce{D}\) |
|---|
| I | \(A_0\) | \(B_0\) | \(C_0\) | \(D_0\) |
|---|
| Δ | \(-ax\) | \(-bx\) | \(+cx\) | \(+dx\) |
|---|
| E | \(A_0-ax\) | \(B_0-bx\) | \(C_0+cx\) | \(D_0+dx\) |
|---|
Then
\[
K_p \;=\; \frac{\big((C_0+cx)/P^\circ\big)^{c}\,\big((D_0+dx)/P^\circ\big)^{d}}
{\big((A_0-ax)/P^\circ\big)^{a}\,\big((B_0-bx)/P^\circ\big)^{b}},
\]
and you can either compute \(x\) from a known \(P^{(\mathrm{eq})}\) or solve \(Q_p(x)=K_p\).
9) Assumptions and limitations
- Ideal-gas behavior; temperature and volume are constant.
- Only gaseous species contribute to \(K_p\); condensed phases are omitted (activity \(\approx 1\)).
- For high pressures or non-ideal mixtures, activities should be expressed via fugacities \(a_i=f_i/P^\circ\) rather than partial pressures.
