Cell Potential (Non-Standard Conditions)
Core Concept
The voltage of an electrochemical cell when the concentrations, pressures, or temperatures differ from standard conditions (1 M,1 atm, 25°C).
Significance: Real-world electrochemical reactions rarely occur under standard conditions, so understanding how to adjust for non-standard conditions is critical.
Cell potential depends on concentration, pressure, and temperature.
Use the Nernst equation to adjust $E_{\text{cell}}$ for non-standard conditions.
The reaction quotient (Q) determines whether the cell potential is higher or lower than $E^°_{\text{cell}}$.
Mastery of the Nernst equation is essential for understanding real-world electrochemical systems.
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The Nernst Equation
The Nernst equation calculates the cell potential (E<sub>cell</sub>) under non-standard conditions:
$E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF} \ln Q$
Where:
$E_{\text{cell}}$: Cell potential under non-standard conditions (V)
$E°_{\text{cell}}$: Standard cell potential (V)
R: Gas constant (8.314 J/mol·K)
T: Temperature (Kelvin)
n: Number of moles of electrons transferred
F: Faraday’s constant (96,485 C/mol)
Q: Reaction quotient (ratio of products to reactants)
Key Concepts
Reaction Quotient (Q):
$Q = \frac{[\text{products}]^{\text{coefficients}}}{[\text{reactants}]^{\text{coefficients}}}$
Concentrations of solids and pure liquids are not included in Q.
Effect of Q on $E_{\text{cell}}$:
Q < 1: Reactants dominate; ln Q < 0, $E_{\text{cell}}$ > E°<sub>cell</sub>
Q = 1: Standard conditions; $E_{\text{cell}}$ = E°<sub>cell</sub>
Q > 1: Products dominate; ln Q > 0, $E_{\text{cell}}$ < E°<sub>cell</sub>
Temperature Dependence:
Higher temperatures amplify the effect of ln Q due to the $\frac{RT}{nF}$ term.
Simplified Nernst Equation at 25°C
At 25∘C25^\circ \text{C}25∘C (T=298 KT = 298 \, \text{K}T=298K):
$E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{0.0592}{n} \log Q$
This simplifies calculations when temperature is standard.
| Condition | Standard Conditions | Nonstandard Conditions |
|---|---|---|
| Concentration of Aqueous Solutions | Aqueous solutions of reactant(s) and product(s) are equal to 1.0 M. | At least one of the aqueous solutions is NOT equal to 1.0 M. |
| Partial Pressure of Gases (if gases are present) | Partial pressures of gaseous reactant(s) and/or product(s) are equal to 1.0 atm. | At least one of the gaseous substances has a partial pressure that is NOT equal to 1.0 atm. |
| Temperature | Temperature is 298 K. | Temperature is NOT 298 K. |
| Reaction Quotient \( Q \) | \( Q \) is equal to 1.0. | It is likely that \( Q \) is NOT equal to 1.0. |
Steps to Calculate EcellE_{\text{cell}}Ecell
Write the Half-Reactions:
Identify the oxidation and reduction reactions.
Determine E°_{\text{cell}}$:
Use the standard reduction potential table:
$E°_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}}$
Calculate Q:
Use the concentrations or pressures of the reactants and products.
Apply the Nernst Equation:
Plug values into the Nernst equation to find $E_{\text{cell}}$.
Cu (s) + 2 Ag+ (aq) → Cu2+ (aq) + 2 Ag (s) [$E°_{cell}$ = +0.458 V]
If the value of Ecell = +0.396 V when ([Ag^{+}]=2.56\times10^{-3}M) at 298 K, what is [Cu2+]?
We can apply the Nernst equation find [Cu2+]. Be careful about the form of Q!
Ecell = E°cell - (RT/nF) ln ([Cu2+]/[Ag+]^2)
+0.396 V = +0.458 V - ((8.314 J/(mol•K))(298 K)/(2 mol e^-)(96485 C/mol)) ln ([Cu2+]/(2.56 x 10^-3)^2)
[Cu2+] = 8.25 x 10^-4 M
Calculate the cell potential for the following reaction under non-standard conditions:
Zn(s) + Cu²⁺(aq, 0.010 M) → Zn²⁺(aq, 1.0 M) + Cu(s)
Given:
E°<sub>cell</sub> = +1.10 V
n = 2
Solution:
1. Calculate Q:
Q = [Zn²⁺] / [Cu²⁺] = 1.0 / 0.010 = 100
2. Use the Simplified Nernst Equation:
E<sub>cell</sub> = E°<sub>cell</sub> - (0.0592/n) log Q E<sub>cell</sub> = 1.10 - (0.0592/2) log(100) E<sub>cell</sub> = 1.10 - (0.0296)(2) E<sub>cell</sub> = 1.10 - 0.0592 E<sub>cell</sub> = 1.04 V
Conclusion: The cell potential under these conditions is 1.04 V.