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.

Key Tips

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.

Test Yourself

Assorted Multiple Choice

A voltaic cell operates based on the reaction $\text{Zn}(s) + \text{Cu}^{2+}(aq) \rightarrow \text{Zn}^{2+}(aq) + \text{Cu}(s)$, with a standard cell potential $E^\circ_{\text{cell}} = +1.10\text{ V}$. If the concentration of $\text{Zn}^{2+}(aq)$ is increased to $2.0\text{ M}$ while the concentration of $\text{Cu}^{2+}(aq)$ is decreased to $0.10\text{ M}$, how will the new cell potential ($E_{\text{cell}}$) compare to $E^\circ_{\text{cell}}$?

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The Nernst Equation

The Nernst equation calculates the cell potential (Ecell) 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°cell
Q = 1: Standard conditions; $E_{\text{cell}}$ = E°cell
Q > 1: Products dominate; ln Q > 0, $E_{\text{cell}}$ < E°cell

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.

  
  Standard Conditions vs. Nonstandard Conditions
  
      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 $E_{\text{cell}}$

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°cell = +1.10 V
n = 2

Solution:

1. Calculate Q:

Q = [Zn²⁺] / [Cu²⁺] = 1.0 / 0.010 = 100

2. Use the Simplified Nernst Equation:

Ecell = E°cell - (0.0592/n) log Q Ecell = 1.10 - (0.0592/2) log(100) Ecell = 1.10 - (0.0296)(2) Ecell = 1.10 - 0.0592 Ecell = 1.04 V

Conclusion: The cell potential under these conditions is 1.04 V.

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