DAT General Chemistry: The Nernst Equation

DAT High-Yield Concept

Standard vs. Nonstandard Conditions: Standard cell potential ( $E^\circ_{\text{cell}}$ ) occurs when solute concentrations are $\pu{1.0 M}$ , gas pressures are $\pu{1 atm}$ , and temperature is $\pu{298 K}$ . Nonstandard potential ( $E_{\text{cell}}$ ) is calculated using the Nernst equation.
The Nernst Equation at $\pu{298 K}$ : $E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{\pu{0.0592 V}}{n} \log Q$ , where $n$ is the number of transferred electrons and $Q$ is the reaction quotient.
Le Chatelier Shifts: Increasing reactant concentrations or decreasing product concentrations decreases $Q$ , making $E_{\text{cell}} > E^\circ_{\text{cell}}$ . Increasing product concentrations or decreasing reactant concentrations increases $Q$ , making $E_{\text{cell}} < E^\circ_{\text{cell}}$ .
Concentration Cells: $E^\circ_{\text{cell}} = \pu{0 V}$ . Voltage is driven solely by a concentration gradient from dilute to concentrated compartments until equilibrium ( $E_{\text{cell}} = \pu{0 V}$ ) is established.

1. Nonstandard Cell Conditions and the Nernst Equation

Electrochemical cells rarely operate under standard state conditions. Standard cell potential ( $E^\circ_{\text{cell}}$ ) assumes all aqueous species are at $\pu{1.0 M}$ concentration and all gases are at $\pu{1 atm}$ pressure at $\pu{298 K}$ . When ion concentrations deviate from $\pu{1.0 M}$ , the measured potential ( $E_{\text{cell}}$ ) changes. The relationship between free energy and cell potential (covered in depth in its own lesson) links electrochemical potential directly to the reaction quotient ( $Q$ ).

Concept Definition: Nonstandard State
A cell is in a nonstandard state whenever the concentration of any dissolved ion differs from

\pu{1.0 M}

or gas pressure differs from

\pu{1 atm}

. The driving force for electron flow is quantified by the general Nernst equation:

E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF} \ln Q

At standard temperature (

\pu{298 K}

\pu{25 degC}

), substituting physical constants (

R = \pu{8.314 J/(mol*K)}

F = \pu{96485 C/mol}

) and converting

\ln

\log_{10}

yields the simplified DAT form:

E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{\pu{0.0592 V}}{n} \log Q

Worked Example 1: Calculating Nonstandard Cell Potential

Problem: Calculate the cell potential at $\pu{298 K}$ for the redox reaction: $\ce{Zn(s) + Cu^{2+}(aq, 0.010 M) -> Zn^{2+}(aq, 0.10 M) + Cu(s)}$ given that $E^\circ_{\text{cell}} = \pu{1.10 V}$ .

Step-by-Step Reasoning:

Identify transferred electrons ( $n$ ): Zinc undergoes a two-electron oxidation ( $\ce{Zn(s) -> Zn^{2+}(aq) + 2e^-}$ ) and copper undergoes a two-electron reduction ( $\ce{Cu^{2+}(aq) + 2e^- -> Cu(s)}$ ), so $n = 2$ .
Write the reaction quotient expression: Solvents and solids are omitted. $Q = \frac{[\ce{Zn^{2+}}]}{[\ce{Cu^{2+}}]} = \frac{0.10}{0.010} = 10$ .
Substitute into the Nernst equation: $E_{\text{cell}} = \pu{1.10 V} - \frac{\pu{0.0592 V}}{2} \log(10)$ Since $\log(10) = 1$ : $E_{\text{cell}} = \pu{1.10 V} - \pu{0.0296 V} = \pu{1.0704 V} \approx \pu{1.07 V}$

DAT Exam Trap: Forgetting stoichiometric coefficients as exponents in $Q$ or incorrectly assigning $n$ by ignoring the balanced overall equation.

2. Le Chatelier Reasoning Applied to Cell Potential

Le Chatelier's principle provides a qualitative method to predict how changing ion concentrations moves cell potential. Because electrochemical cells represent reversible systems approaching equilibrium, disturbing the concentration ratio affects the thermodynamic driving force for electron flow.

Increasing Reactants or Decreasing Products: The equilibrium shifts forward, driving the reaction to produce more products. This lowers $Q$ ( $Q < 1$ relative to standard condition $Q=1$ ), making $\log Q$ negative. Consequently, the subtraction term in the Nernst equation becomes positive, increasing cell potential: $E_{\text{cell}} > E^\circ_{\text{cell}}$ .
Increasing Products or Decreasing Reactants: The reaction shifts toward reactants. This increases $Q$ ( $Q > 1$ ), making $\log Q$ positive. The subtracted term reduces voltage: $E_{\text{cell}} < E^\circ_{\text{cell}}$ .
Equilibrium (Dead Battery): As the reaction proceeds, reactants are consumed and products accumulate until $Q = K$ . At this point, $E_{\text{cell}} = \pu{0 V}$ and no net current flows.

Worked Example 2: Qualitative Concentration Shifts

Problem: Consider the spontaneous cell reaction: $\ce{2Ag^+(aq) + Fe(s) -> 2Ag(s) + Fe^{2+}(aq)}$ Predict the effect on $E_{\text{cell}}$ if pure water is added to the anode compartment, diluting its contents.

Step-by-Step Reasoning:

Identify the anode ion: Oxidation occurs at the anode, so the anode compartment contains $\ce{Fe^{2+}(aq)}$ .
Determine the concentration change: Diluting the anode compartment decreases $[\ce{Fe^{2+}}]$ .
Apply $Q$ and Le Chatelier reasoning: $Q = \frac{[\ce{Fe^{2+}}]}{[\ce{Ag^+}]^2}$ . Decreasing product concentration $[\ce{Fe^{2+}}]$ decreases $Q$ . According to Le Chatelier's principle, the system shifts forward to replace products.
Evaluate potential shift: A smaller $Q$ decreases the term subtracted in the Nernst equation, causing $E_{\text{cell}}$ to increase.

DAT Exam Trap: Assuming diluting one half-cell affects both reactant and product concentrations equally. Always trace dilution to the specific compartment specified.

3. Concentration Cells and Membrane Potentials

A concentration cell is a specialized galvanic cell where both half-cells contain the same chemical species, but at different concentrations. Because both electrodes are identical, standard cell potential is zero ( $E^\circ_{\text{cell}} = \pu{0 V}$ ). Voltage is generated strictly by the concentration gradient across the system.

Esquema sobre el potencial de Nernst; cubetas con KCl separadas por una membrana semipermeable. (Source: Retama · CC BY-SA 3.0 · Wikimedia Commons)

As illustrated above, ion separation across a semipermeable membrane creates an electrochemical potential gradient. In concentration cells, oxidation occurs in the more dilute compartment (anode) to generate ions, while reduction occurs in the more concentrated compartment (cathode) to consume ions, continuing until ion concentrations equalize.

Worked Example 3: Concentration Cell Calculation

Problem: A concentration cell consists of two $\ce{Cu/Cu^{2+}}$ half-cells at $\pu{298 K}$ . Compartment A contains $[\ce{Cu^{2+}}] = \pu{0.0010 M}$ and Compartment B contains $[\ce{Cu^{2+}}] = \pu{1.0 M}$ . Calculate $E_{\text{cell}}$ .

Step-by-Step Reasoning:

Assign anode and cathode: Dilute compartment A ( $\pu{0.0010 M}$ ) acts as the anode: $\ce{Cu(s) -> Cu^{2+}(aq, \text{dilute}) + 2e^-}$ . Concentrated compartment B ( $\pu{1.0 M}$ ) acts as the cathode: $\ce{Cu^{2+}(aq, \text{concentrated}) + 2e^- -> Cu(s)}$ .
Write overall reaction and $Q$ : $\ce{Cu^{2+}(aq, 1.0 M) -> Cu^{2+}(aq, 0.0010 M)}$ $Q = \frac{[\ce{Cu^{2+}}]_{\text{dilute}}}{[\ce{Cu^{2+}}]_{\text{concentrated}}} = \frac{0.0010}{1.0} = 1.0 \times 10^{-3}$
Calculate cell potential ( $n = 2, E^\circ_{\text{cell}} = \pu{0 V}$ ): $E_{\text{cell}} = \pu{0 V} - \frac{\pu{0.0592 V}}{2} \log(1.0 \times 10^{-3})$ Since $\log(10^{-3}) = -3$ : $E_{\text{cell}} = \pu{0 V} - (\pu{0.0296 V})(-3) = \pu{+0.0888 V} \approx \pu{0.089 V}$

DAT Exam Trap: Inverting $Q$ as $\frac{\text{concentrated}}{\text{dilute}}$ , which gives a negative cell potential for a spontaneous concentration cell.

Comparison of Cell Types

Cell Type	$E^\circ_{\text{cell}}$	Reaction Quotient ( $Q$ )	Nonstandard Voltage ( $E_{\text{cell}}$ )	Primary Driving Force
Standard Galvanic Cell	$> \pu{0 V}$	$Q = 1$	$E_{\text{cell}} = E^\circ_{\text{cell}}$	Standard reduction potential difference
Nonstandard Galvanic Cell	$> \pu{0 V}$	$Q \neq 1$	$E_{\text{cell}} \neq E^\circ_{\text{cell}}$	Combined reduction potential and concentration ratio
Concentration Cell	$\pu{0 V}$	$Q < 1$ (for spontaneous flow)	$E_{\text{cell}} > \pu{0 V}$	Concentration gradient toward equilibrium

DAT Memory Trick:

References

Sources returned by Google Search grounding during lesson verification.

20.6: Cell Potential Under Nonstandard Conditions - Chemistry LibreTexts — chem.libretexts.org
17.4: The Nernst Equation - Chemistry LibreTexts — chem.libretexts.org

The Nernst Equation

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