19.5 How to Calculate Nonstandard Cell Potential [Nernst Equation]
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Nernst Equation vs Standard Cell Potential
What's the Difference?
General Chemistry 19.5
The Nernst equation relates non-standard cell potential to standard cell potential through a correction term involving the reaction quotient Q, and is derived from the thermodynamic relationship between free energy and cell voltage. At 298 K, the general form simplifies to two equivalent versions—one using natural logarithm (with the constant 0.0257) and one using log base 10 (with the constant 0.0592). Le Chatelier's principle provides a qualitative framework for predicting whether non-standard conditions will increase or decrease the cell potential relative to the standard value.
Deriving the Nernst Equation from Thermodynamics
In the previous lesson, you learned how to calculate standard cell potentials. Now it's time to tackle the next logical step: calculating non-standard cell potentials using the Nernst equation. If this progression feels familiar, it should — thermodynamics followed the same pattern. You first learned to calculate standard ΔG using thermodynamic data, then moved on to non-standard ΔG in a subsequent lesson.
The Nernst equation states that the non-standard cell potential equals the standard value minus a correction factor involving the reaction quotient (Q):
Ecell = E°cell − (RT / nF) · ln Q
You may recall the analogous equation from thermodynamics:
ΔG = ΔG° + RT ln QThe Nernst equation is actually derived from this thermodynamic relationship. The key connection is:
ΔG = −nFEcellThis relationship holds under both standard and non-standard conditions. Under non-standard conditions, ΔG relates to Ecell; under standard conditions, ΔG° relates to E°cell.
Example: The Derivation in Brief
Substitute −nFEcell for ΔG and −nFE°cell for ΔG° into the free energy equation. Factor out −nF from both terms, then divide through by −nF. The result is the Nernst equation, with the −RT/nF coefficient in front of ln Q.
Note: The relationship ΔG = −nFE will be explored in detail in the next lesson. For now, the important takeaway is that the Nernst equation allows you to calculate cell potentials when conditions deviate from standard state.
3 Forms of the Nernst Equation
4 Key Constants: R, F, T, and n
Before simplifying the equation, let's identify each variable:
- R = universal gas constant = 8.314 J/(mol·K)
- F = Faraday's constant = 96,485 C/mol e− (often rounded to 96,500)
- T = temperature in kelvins
- n = number of moles of electrons transferred in the balanced reaction
Of these four quantities, R and F are always constants. When the temperature is fixed at 298 K (the most common scenario in general chemistry), T becomes constant as well. That leaves n as the only variable that changes from reaction to reaction.
Tip: Whether you use 96,485 or 96,500 for Faraday's constant, you'll get a very close answer either way. Use whichever value your professor or textbook provides.
Simplified Form at 298 K (Natural Log Version)
Since a constant times a constant divided by a constant yields another constant, you can combine R, T, and F at 298 K:
(8.314 × 298) / 96,500 = 0.0257This gives the first simplified form:
Ecell = E°cell − (0.0257 / n) · ln QMany professors prefer this version when they only plan to test you at 298 K. It's cleaner and faster to use.
Simplified Form at 298 K (Log Base 10 Version)
Historically, before calculators were widely available, natural logarithms were cumbersome to compute. Scientists preferred log base 10 because those values could be looked up in published tables or estimated with a slide rule. Converting from ln to log10 requires multiplying by 2.303:
0.0257 × 2.303 = 0.0592This produces the third common form:
Ecell = E°cell − (0.0592 / n) · log QAlthough modern calculators handle natural logs effortlessly, you'll still encounter this version in many textbooks and on exams.
| Form | Equation | When to Use |
|---|---|---|
| General | E = E° − (RT/nF) ln Q | Any temperature |
| Simplified (ln) | E = E° − (0.0257/n) ln Q | 298 K only |
| Simplified (log) | E = E° − (0.0592/n) log Q | 298 K only |
Common Mistake: Don't mix up the ln and log10 forms. If your equation uses 0.0257, you must use the natural log. If it uses 0.0592, you must use log base 10. Mixing them up will give you the wrong answer.
Example: Calculating Non-Standard Cell Potential
Setting Up the Problem
Calculate the cell potential for the following reaction at 298 K:
Zn2+(aq) + Co(s) → Zn(s) + Co2+(aq)Given conditions: [Zn2+] = 0.0010 M and [Co2+] = 0.10 M.
These concentrations are clearly not 1 M, so you're not under standard conditions. Standard conditions require all aqueous species at 1 M concentration and all gases at 1 atm partial pressure. The moment you're asked to find a cell potential under non-standard conditions, reach for the Nernst equation.
Tip: Any of the three Nernst equation forms will produce the same answer at 298 K. Use whichever version your instructor provides.
Step 1: Calculate the Standard Cell Potential
To find the non-standard cell potential, you first need E°cell. Using the standard reduction potentials:
- Zn2+ + 2e− → Zn E° = −0.76 V (cathode, reduction)
- Co → Co2+ + 2e− E° = −0.28 V as reduction (anode, oxidation)
Tip: Whether you flip the sign on the anode's reduction potential and then add, or use the "cathode minus anode" formula, the result is the same. Pick whichever method feels more natural to you.
Step 2: Identify R, T, n, and F
- R = 8.314 J/(mol·K)
- T = 298 K
- F = 96,500 C/mol e−
- n = 2 mol e−
How do you determine n? Look at the balanced reaction. Going from Zn2+ to Zn is a two-electron reduction. Going from Co to Co2+ is a two-electron oxidation. Two electrons lost, two electrons gained — so n = 2.
Note: The number of electrons lost must always equal the number of electrons gained in a properly balanced redox reaction. If they don't match, the reaction isn't balanced.
Step 3: Calculate the Reaction Quotient Q
The reaction quotient Q follows the same products-over-reactants format as an equilibrium expression, but it applies to any set of conditions — not just equilibrium:
Q = [Co2+] / [Zn2+]Solids do not appear in the reaction quotient or equilibrium expression. Substituting the given concentrations:
Q = 0.10 / 0.0010 = 100Step 4: Plug into the Nernst Equation and Solve
Example: Full Nernst Equation Calculation
Using the general form of the Nernst equation:
Ecell = −0.48 − [(8.314 × 298) / (2 × 96,500)] × ln(100)
The quantity (8.314 × 298) / 96,500 = 0.0257, so:
Ecell = −0.48 − (0.0257 / 2) × 4.605
Ecell = −0.48 − 0.059
Ecell = −0.54 V
Tip: If you used the log base 10 simplified form instead, you'd compute E = −0.48 − (0.0592 / 2) × log(100). Since log(100) = 2, that gives E = −0.48 − 0.0592 = −0.54 V — the same answer.
Interpreting Non-Standard Cell Potential Results
The standard cell potential for this reaction is −0.48 V, and the non-standard cell potential came out to −0.54 V. What does this tell you?
- A negative Ecell means the reaction is non-spontaneous in the forward direction.
- The reaction was already non-spontaneous under standard conditions (−0.48 V).
- Under these particular non-standard conditions, the cell potential became even more negative, meaning the reaction is even less spontaneous.
As you'll see in the next section, this result could have been predicted qualitatively — without any calculation at all.
Le Chatelier's Principle and Non-Standard Cell Potential
Beyond plugging numbers into the Nernst equation, developing a qualitative intuition about how non-standard conditions affect cell potential is just as important.
The Key Relationship: Equilibrium and Ecell
When an electrochemical reaction reaches equilibrium, Ecell = 0. This is a critical reference point. Any disturbance from equilibrium pushes Ecell away from zero in a predictable direction.
Example: Adding Reactant to a System at Equilibrium
Imagine the reaction has reached equilibrium (Ecell = 0). Now suppose you suddenly add a large amount of Zn2+. Le Chatelier's principle tells you the system will shift to the right to counteract the stress. A rightward shift means the forward reaction becomes spontaneous, and Ecell becomes positive.
The Rule
| Shift Direction | Effect on Ecell | Meaning |
|---|---|---|
| Shift right | Ecell increases (more positive / less negative) | Forward reaction more spontaneous |
| Shift left | Ecell decreases (less positive / more negative) | Forward reaction less spontaneous |
Applying This to the Example
Look back at the reaction conditions: [Zn2+] = 0.0010 M and [Co2+] = 0.10 M. Both concentrations are below the 1 M standard. How does each one affect the equilibrium?
- Zn2+ is a reactant. Lowering its concentration below 1 M causes a shift to the left.
- Co2+ is a product. Lowering its concentration below 1 M causes a shift to the right.
These two effects compete. But Zn2+ at 0.0010 M is 1,000 times below standard, while Co2+ at 0.10 M is only 10 times below standard. Both species have a coefficient of 1, so there's no exponent difference to consider. The leftward shift from the drastically reduced reactant concentration dominates.
Remember: Coefficients in the balanced equation become exponents in the reaction quotient Q expression. When comparing which concentration change has a larger effect, consider both how far each species deviates from standard conditions and the magnitude of its exponent.
The net result is a shift to the left relative to standard conditions, which predicts that the non-standard cell potential should be more negative than E°cell. And indeed, the Nernst equation calculation confirmed exactly that: −0.54 V versus −0.48 V.
7 Key Takeaways
- The Nernst equation calculates non-standard cell potentials by adding a correction factor based on the reaction quotient Q to the standard cell potential.
- The equation is derived from the thermodynamic relationship ΔG = ΔG° + RT ln Q combined with ΔG = −nFE.
- At 298 K, the general Nernst equation simplifies to two convenient forms: one using ln (with 0.0257) and one using log10 (with 0.0592). All three forms give the same answer.
- To use the Nernst equation, you need four things: E°cell, the number of electrons transferred (n), the temperature (T), and the reaction quotient (Q).
- Solids do not appear in the reaction quotient. Only aqueous and gaseous species are included, raised to the power of their stoichiometric coefficients.
- A shift to the right (Le Chatelier's principle) causes Ecell to increase; a shift to the left causes Ecell to decrease.
- You can qualitatively predict whether the non-standard cell potential will be more or less positive than E°cell by analyzing which direction the reaction shifts under the given non-standard conditions.