Extent of reaction and the equilibrium constant \(K_c\)
This note explains how the calculator connects the extent of reaction and the
equilibrium constant for a general reaction carried out at constant volume. It also shows
how to work with a vertical ICE table and how to solve the problem either from a known
equilibrium concentration of one species or from a known value of \(K_c\).
1) Definitions and notation
- General reaction:
\[
\sum_i \nu_i\,\ce{A_i} = 0,\qquad
\nu_i>0\text{ for products, }\nu_i<0\text{ for reactants.}
\]
- At constant volume \(V\), using the concentration-extent \(x\) (units mol·L\(^{-1}\)):
\[
c_i \,=\, c_{i,0} \;+\; \nu_i\,x.
\]
If you prefer the mole-extent \(\xi\) (units mol), then \(x=\xi/V\) and
\(c_i=c_{i,0} + (\nu_i/V)\,\xi\).
- Equilibrium constant in concentration form (activities \(\approx\) concentrations for
dilute solutions):
\[
K_c \;\approx\; \frac{\prod\limits_{\text{products}} [A_i]^{\nu_i}}
{\prod\limits_{\text{reactants}} [A_i]^{|\nu_i|}},
\]
where only (aq) and (g) species appear; pure (s) and (l) are omitted.
2) Vertical ICE table (constant \(V\))
The calculator uses a vertical ICE layout (rows = I/Δ/E; columns = species):
|
\(\ce{A}\) |
\(\ce{B}\) |
\(\ce{C}\) |
\(\ce{D}\) |
| I | \(c_{\ce A,0}\) | \(c_{\ce B,0}\) | \(c_{\ce C,0}\) | \(c_{\ce D,0}\) |
|---|
| Δ | \(\nu_{\ce A}x\) | \(\nu_{\ce B}x\) | \(\nu_{\ce C}x\) | \(\nu_{\ce D}x\) |
|---|
| E | \(c_{\ce A,0}+\nu_{\ce A}x\) | \(c_{\ce B,0}+\nu_{\ce B}x\) | \(c_{\ce C,0}+\nu_{\ce C}x\) | \(c_{\ce D,0}+\nu_{\ce D}x\) |
|---|
Sign convention: for reactants \(\nu_i<0\) so \(c_i\) decreases with \(x\); for products
\(\nu_i>0\) so \(c_i\) increases with \(x\).
3) Two ways to determine \(x\) and \(K_c\)
- Known equilibrium concentration of one species.
If \(c_j^{(\mathrm{eq})}\) is measured for species \(j\),
\[
x \;=\; \frac{c_j^{(\mathrm{eq})} - c_{j,0}}{\nu_j}.
\]
Then compute all \(c_i^{(\mathrm{eq})} = c_{i,0}+\nu_i x\) and finally evaluate \(K_c\) with the
equilibrium concentrations of (aq,g) species.
- Known \(K_c\) (solve for \(x\)).
Insert \(c_i=c_{i,0}+\nu_i x\) into the reaction quotient
\[
Q_c(x) \;=\; \frac{\prod_{\text{prod}} \big(c_{i,0}+\nu_i x\big)^{\nu_i}}
{\prod_{\text{react}} \big(c_{i,0}+\nu_i x\big)^{|\nu_i|}},
\]
and solve the scalar equation \(Q_c(x)=K_c\) for \(x\) on the feasible interval
(see §4). This generally requires a numerical method; the calculator brackets the root and
uses a robust bisection search.
4) Feasible interval for \(x\)
Physical concentrations must remain non-negative:
\[
c_{i,0}+\nu_i x \;\ge\; 0 \quad \forall i \text{ in } (aq,g).
\]
This yields bounds:
- for reactants (\(\nu_i<0\)): \(x \le \dfrac{c_{i,0}}{|\nu_i|}\)
- for products (\(\nu_i>0\)): \(x \ge -\dfrac{c_{i,0}}{\nu_i}\)
The intersection of all bounds defines the interval \([x_{\min}, x_{\max}]\). If the chosen
data imply \(x\) outside this interval, the model is inconsistent (inputs must be revised).
5) Activities vs. concentrations and relation to \(K_p\)
- The rigorous expression uses activities \(a_i=\gamma_i\,\dfrac{c_i}{c^\circ}\) so that
\(K=\dfrac{\prod a_i^{\nu_i}}{\prod a_j^{|\nu_j|}}\) is dimensionless. For sufficiently
dilute solutions, \(\gamma_i\approx 1\) and \(K\approx K_c\).
- For ideal gases, \(K_p\) and \(K_c\) are related by
\[
K_p \;=\; K_c\,(RT)^{\Delta n}, \qquad
\Delta n = \sum_{\text{prod}}\nu_i - \sum_{\text{react}}|\nu_i|.
\]
6) Useful limiting cases (rule of thumb)
- \(K_c \ll 1\): reaction lies far to the left; \(x\) is small compared with typical
\(c_{i,0}\); linearized or quadratic approximations may be adequate.
- \(K_c \gg 1\): reaction lies far to the right; in the absence of products initially,
\(x\) approaches the smallest \(\dfrac{c_{i,0}}{|\nu_i|}\) among reactants.
7) Worked symbolic template: \(a\ce{A}+b\ce{B} \rightleftharpoons c\ce{C}+d\ce{D}\)
Vertical ICE:
| \(\ce{A}\) | \(\ce{B}\) | \(\ce{C}\) | \(\ce{D}\) |
|---|
| 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
\[
K_c \;=\; \frac{(C_0+cx)^{c}\,(D_0+dx)^{d}}{(A_0-ax)^{a}\,(B_0-bx)^{b}},
\]
and either (i) obtain \(x\) from a known \(C^{(\mathrm{eq})}\) or \(D^{(\mathrm{eq})}\), or
(ii) solve \(Q_c(x)=K_c\) for \(x\).
8) Assumptions and limitations
- Constant total volume; temperature fixed; single dominant equilibrium.
- Solution is sufficiently dilute so that activities \(\approx\) concentrations for (aq); for
higher ionic strength, activity coefficients are required.
- Pure solids and liquids are excluded from the equilibrium expression (activity ≈ 1).
