Concentration Cells and Their Voltage
A concentration cell is an electrochemical cell where both half-cells use the same redox couple, but the ion concentration or gas pressure differs between compartments. A concentration cell calculator typically computes the nonstandard cell potential \(E_{\text{cell}}\) from the concentration ratio (or activity ratio) and the electron count \(n\).
Core definitions and essential formulas
\[
E^\circ_{\text{cell}} = 0
\]
\[
E_{\text{cell}} = \frac{R T}{n F}\,\ln Q
\]
\[
E_{\text{cell}} \approx \frac{0.05916}{n}\,\log_{10} Q \qquad (T=298~\mathrm{K})
\]
\(R\) is the gas constant, \(T\) is temperature in kelvin, \(F\) is the Faraday constant, and \(n\) is the number of electrons transferred in the balanced half-reaction. \(Q\) is the overall reaction quotient for the concentration difference and is built from activities; for dilute aqueous solutions, activities are often approximated by concentrations. With identical electrodes and identical standard reduction potentials, the only driving force is the tendency for the system to reduce a concentration gradient.
How to interpret results
A larger \(E_{\text{cell}}\) means a stronger driving force to equalize concentrations and a greater ability to deliver electrical work in the written direction. \(E_{\text{cell}}\) increases when the ratio in \(Q\) moves farther from \(1\); if the two compartments have equal activity, then \(Q=1\) and \(E_{\text{cell}}=0\). Outputs commonly include \(Q\), \(E_{\text{cell}}\), and the sign of \(E_{\text{cell}}\); the unit is volts, where \(1\ \mathrm{V}=1\ \mathrm{J\,C^{-1}}\).
- Inverted ratio: swapping anode and cathode activities changes the sign of \(\ln Q\).
- Wrong \(n\): using the ion charge instead of the electron count from the balanced half-reaction.
- Temperature mix-up: using \(0.05916\ \mathrm{V}\) when \(T\neq 298\ \mathrm{K}\).
- Including solids: placing \( \mathrm{M(s)} \) in \(Q\) even though its activity is \(1\).
Micro example: For \(\mathrm{Cu^{2+} + 2e^- \rightarrow Cu(s)}\), \(n=2\).
If \([\mathrm{Cu^{2+}}]_{\text{anode}}=0.010\ \mathrm{M}\) and \([\mathrm{Cu^{2+}}]_{\text{cathode}}=1.0\ \mathrm{M}\),
then \(Q=\dfrac{0.010}{1.0}=0.010\) and \(E_{\text{cell}}\approx \dfrac{0.05916}{2}\log_{10}(0.010)=-0.059\ \mathrm{V}\)
for that written direction.
Use this tool for metal–ion or gas concentration cells where the electrodes are the same and only activities differ between half-cells. Do not use it for cells with different redox couples (nonzero \(E^\circ_{\text{cell}}\)); a common next step is combining the Nernst equation with full cell reactions and reaction quotients for mixed-couple galvanic cells.
