2.5 Nernst Equation

In the previous sections, we calculated cell potentials under standard conditions (1 M concentration, 1 atm pressure, 25°C). However, in real-world applications like batteries, the concentrations of the reactants and products change as the cell operates. To predict the actual cell potential under these non-standard conditions, chemists use the Nernst equation.

Definition

The Nernst equation is a mathematical expression that relates the standard electrode potential (E^°) to the actual electrode potential (E) under non-standard conditions, considering the concentrations of the reactants and products.

Core Concept

The core concept is that the potential of a half-cell (or a full cell) depends on the concentration of the ions involved in the redox reaction. If the concentration of the ions changes, the position of equilibrium shifts, which in turn changes the voltage (potential) of the cell. The Nernst equation quantitatively predicts these changes. It helps us calculate how the voltage of a battery decreases as it is used up and the reactant concentrations drop.

Detailed Explanation

The Nernst Equation for a Half-Cell

The relationship between the standard electrode potential and the actual electrode potential is given by the Nernst equation. The electrode potential is influenced by the concentrations of the ions involved in the redox reaction.

The mathematical form of the Nernst equation is:

E = E^° + RTnF ln [oxidized species][reduced species]

• E = actual electrode potential under non-standard conditions
• E^° = standard electrode potential
• R = universal gas constant (8.314 J/mol·K)
• T = Kelvin temperature
• n = number of moles of electrons transferred in the half-reaction
• F = Faraday constant (charge of one mole of electrons = 96485 C/mol)
• [oxidized species] = concentration of species with the higher oxidation state
• [reduced species] = concentration of species with the lower oxidation state

The Simplified Nernst Equation

In many textbooks, the equation is converted to base-10 logarithm (log) by multiplying the natural logarithm (ln) by 2.303 (since ln(x) = 2.303 log(x)). At a standard temperature of 298 K (25^°C), the constants 2.303 RTF can be calculated:

2.303 × 8.314 × 29896485 ≈ 0.0591 V

Thus, the simplified Nernst equation at 25^°C becomes:

E = E^° + 0.0591n log [oxidized species][reduced species]

(Note: The textbook uses the approximation 0.059 in its calculations).

How Concentration Affects Electrode Potential

The textbook explains the effect of concentration using Le Chatelier's Principle. Consider the copper half-cell equilibrium: Cu^{2+} + 2e^- \rightleftharpoons Cu

• An increase in the concentration of ions (e.g., Cu^{2+}): According to Le Chatelier's Principle, the equilibrium shifts to the right (to the reduced side) to counteract the change. This makes reduction easier because more ions are available to gain electrons. As a result, the electrode potential increases.
• A decrease in the concentration of ions (e.g., Cu^{2+}): The equilibrium shifts to the left. Reduction becomes harder, meaning the electrode potential decreases.

Solved Example from the Textbook

Question: The standard electrode potential of the Cu^{2+}/Cu system is +0.34 V. What is the electrode potential of a solution containing 0.5 M Cu^{2+} ions?

Solution:

The Nernst equation for the Cu^{2+}/Cu system is: E_Cu^{2+}/Cu = E^°_Cu^{2+}/Cu + 0.059n log [oxidized species][reduced species]

• E^° = +0.34 V
• n = 2 (since two electrons are transferred)
• [Oxidized species] = [Cu^{2+}] = 0.5 M
• [Reduced species] = [Cu] = 1 (the concentration of a pure solid is taken as 1)

Substitution: E = 0.34 + 0.0592 log 0.51 E = 0.34 + 0.0295 × log(0.5) E = 0.34 + 0.0295 × (-0.301) E = 0.34 - 0.0089 E = 0.331 V

Final Answer: The electrode potential of the solution is 0.331 V. (Notice that it is slightly lower than the standard potential of 0.34 V because the concentration of Cu^{2+} is less than 1 M, making reduction slightly harder).

Intermediate Level

The Nernst equation allows you to calculate the actual cell potential under non-standard conditions. The key is to correctly identify n (the number of electrons transferred) and the concentrations of the oxidized and reduced species. Remember that the concentration of pure solids and pure liquids is taken as 1. The equation shows that the potential changes logarithmically with concentration, not linearly. At 25^°C, the simplified equation with the factor 0.059 is very useful for quick calculations. A positive term in the equation means the potential is higher than standard; a negative term means it is lower.

Important Points

• - Nernst Equation: Relates standard potential to non-standard potential.
• - Formula at 25^°C: E = E^° + 0.059n log [oxidized][reduced].
• - n: Number of electrons transferred in the half-reaction.
• - Concentration of solids and liquids: Taken as 1 (they are constant).
• - Effect of concentration: Higher concentration of oxidized species → higher potential. Lower concentration → lower potential.
• - The Nernst equation is used to calculate the potential of a half-cell or a full cell under non-standard conditions.

Remember

• - Nernst equation: E = E^° + 0.059n log [Ox][Red] (at 25^°C).
• - n = number of electrons in the half-reaction.
• - Pure solids and liquids = 1 in the concentration term.
• - [Oxidized] is the concentration of the species being reduced.
• - [Reduced] is the concentration of the product of reduction.
• - An increase in [Oxidized] increases the potential.
• - A decrease in [Oxidized] decreases the potential.
• - The Nernst equation is for non-standard conditions.

Common Mistakes

• - Forgetting to include the stoichiometric coefficients in the concentration term. For example, in Pb^{2+} + 2e^- \rightarrow Pb, the concentration term is just [Pb^{2+}]. But for a reaction like Sn^{4+} + 2e^- \rightarrow Sn^{2+}, it is [Sn^{4+}]/[Sn^{2+}].
• - Using the wrong value of n. Make sure you identify the number of electrons transferred in the balanced half-reaction.
• - Including the concentration of pure solids or liquids. The concentration of a solid metal electrode (e.g., Cu(s)) or a pure liquid (e.g., H_2O) is constant and is assigned a value of 1.
• - Confusing the sign of the log term. If the concentration of the oxidized species is less than 1, the log term will be negative, and the potential will be lower than E^°. If it is greater than 1, the log term will be positive, and the potential will be higher.
• - Using the Nernst equation for a full cell without combining half-cells. For a full cell, E_cell = E_cathode - E_anode, and you can apply the Nernst equation to each half-cell or to the overall cell reaction.

Quick Revision

The Nernst equation relates the standard electrode potential (E^°) to the actual electrode potential (E) under non-standard conditions. The simplified form at 25^°C is E = E^° + 0.059n log [Oxidized][Reduced]. The term n is the number of moles of electrons transferred. Pure solids and liquids are assigned a concentration of 1. The equation shows that changes in ion concentration affect the electrode potential. A higher concentration of the oxidized species increases the potential, while a lower concentration decreases it.

60-Second Revision

The Nernst equation allows you to calculate the actual cell potential when conditions are not standard: E = E^° + 0.059n log [Oxidized][Reduced] at 25^°C. n is the number of electrons transferred. Pure solids and liquids are not included (they are 1). The potential increases if the concentration of the oxidized species increases. It is used to predict the effect of changing concentration on the voltage of a cell.

Frequently Asked Questions (FAQs)

Q1. What is the Nernst equation? Answer: The Nernst equation is a mathematical expression that relates the standard electrode potential (E^°) to the actual electrode potential (E) under non-standard conditions. It accounts for the concentrations of the reactants and products in the redox reaction.

• E = actual electrode potential
• E^° = standard electrode potential
• R = universal gas constant (8.314 J/mol·K)
• T = temperature in Kelvin
• n = number of electrons transferred
• F = Faraday constant (96485 C/mol)
• [oxidized species] = concentration of the oxidized form
• [reduced species] = concentration of the reduced form

• Q3. How does the Nernst equation simplify at 25°C?
• Answer: At 25°C (298 K), the constants can be combined to give the simplified Nernst equation:
• E = E^° + 0.059n log [oxidized species][reduced species]
• This is the form most commonly used in calculations.

Q4. What happens to the electrode potential if the concentration of the oxidized species decreases? Answer: If the concentration of the oxidized species decreases, the ratio [oxidized][reduced] becomes smaller. The logarithm of a value less than 1 is negative, so the term 0.059n log [Ox][Red] becomes negative. This decreases the overall electrode potential E. Physically, this is because fewer ions are available to be reduced, making the reduction harder.

Q5. Why are solids and liquids not included in the concentration term of the Nernst equation? Answer: The concentration of a pure solid or a pure liquid is constant and is not affected by the amount of substance present. Therefore, their activity (effective concentration) is taken as 1. They do not appear in the reaction quotient because they are not dissolved species.

Q6. What does a positive or negative value of the term 0.059n log [Ox][Red] mean? Answer: A positive value means that the concentration of the oxidized species is greater than 1, which increases the electrode potential above the standard value. A negative value means the concentration of the oxidized species is less than 1, which decreases the electrode potential below the standard value.

• Q7. Can the Nernst equation be used for a full cell?
• Answer: Yes. You can apply the Nernst equation to the overall cell reaction. The equation becomes:
• E_cell = E^°_cell - 0.059n log Q
• where Q is the reaction quotient for the overall cell reaction. This is very useful for calculating the cell potential under non-standard conditions.

Q8. How is the Nernst equation related to Le Chatelier's Principle? Answer: The Nernst equation is a quantitative expression of Le Chatelier's Principle. As the concentration of a species changes, the equilibrium of the half-reaction shifts. If you increase the concentration of the oxidized species, the equilibrium shifts to the right (reduction), increasing the potential. If you decrease it, the equilibrium shifts to the left, decreasing the potential.

Multiple Choice Questions (MCQs)

Q1. What is the purpose of the Nernst equation? A. To calculate the standard cell potential. B. To calculate the electrode potential under non-standard conditions. C. To determine the equilibrium constant. D. To measure the current in a circuit. Correct Answer: B Explanation: The Nernst equation relates the standard potential to the actual potential under non-standard conditions, taking into account the concentrations of the reactants and products.

• Q2. In the Nernst equation, what does n represent?
• A. The number of moles of the reactant.
• B. The number of electrons transferred in the half-reaction.
• C. The number of moles of the product.
• D. The number of ions in the solution.
• Correct Answer: B
• Explanation: n is the number of moles of electrons transferred in the balanced half-reaction.

• Q3. What is the simplified form of the Nernst equation at 25°C?
• A. E = E^° + 0.059n log [Ox][Red]
• B. E = E^° - 0.059n log [Ox][Red]
• C. E = E^° + n0.059 log [Ox][Red]
• D. E = E^° + 0.059n ln [Ox][Red]
• Correct Answer: A
• Explanation: At 25°C, the equation is simplified by combining the constants and converting to base-10 log, giving the factor 0.059.

• Q4. In the Nernst equation, the concentration of a pure solid is taken as:
• A. 0
• B. 1
• C. Its molar mass
• D. Its density
• Correct Answer: B
• Explanation: The concentration of a pure solid or liquid is constant and is assigned a value of 1.

Q5. What happens to the electrode potential if the concentration of the oxidized species increases? A. It decreases. B. It remains the same. C. It increases. D. It becomes zero. Correct Answer: C Explanation: Increasing the concentration of the oxidized species increases the ratio [Ox]/[Red], making the log term positive and increasing the potential.

• Q6. In the Nernst equation, the term [reduced species] refers to the concentration of:
• A. The reactant that is oxidized.
• B. The product that is formed by reduction.
• C. The electrons transferred.
• D. The solvent.
• Correct Answer: B
• Explanation: [reduced species] is the concentration of the species with the lower oxidation state (the product of the reduction half-reaction).

Q7. If the standard electrode potential of a half-cell is +0.34 V, and the concentration of the oxidized species is 0.5 M, what is the new potential? (Assume n = 2 and use the simplified Nernst equation). A. 0.34 V B. 0.331 V C. 0.349 V D. 0.330 V Correct Answer: B Explanation: Using E = 0.34 + 0.0592 log(0.5), we get E = 0.34 + 0.0295(-0.301) = 0.34 - 0.0089 = 0.331 V.

Q8. What is the relationship between the Nernst equation and Le Chatelier's Principle? A. The Nernst equation is a contradiction of Le Chatelier's Principle. B. The Nernst equation is a mathematical expression of Le Chatelier's Principle for electrochemical cells. C. They are unrelated. D. The Nernst equation only applies to equilibrium, not to changes in concentration. Correct Answer: B Explanation: The Nernst equation describes how changes in concentration shift the equilibrium and thus change the potential, which is exactly what Le Chatelier's Principle predicts.

Q9. In the Nernst equation, the ratio [Ox]/[Red] for the half-reaction Fe^{3+} + e^- \rightarrow Fe^{2+} is: A. [Fe^{3+}]/[Fe^{2+}] B. [Fe^{2+}]/[Fe^{3+}] C. [Fe]/[Fe^{3+}] D. [Fe^{2+}]/[Fe] Correct Answer: A Explanation: The oxidized species is Fe^{3+} and the reduced species is Fe^{2+}, so the ratio is [Fe^{3+}]/[Fe^{2+}].

Q10. Why does the electrode potential change when the concentration of ions changes? A. Because the number of electrons transferred changes. B. Because the position of equilibrium of the half-reaction shifts. C. Because the temperature changes. D. Because the Faraday constant changes. Correct Answer: B Explanation: A change in concentration shifts the equilibrium position of the half-reaction (Le Chatelier's Principle), which changes the potential. The Nernst equation quantifies this shift.

Multiple Choice Questions (MCQs)

• Q1. What is the Nernst equation used for?
• A. To calculate the standard potential of a cell.
• B. To calculate the potential of a cell under non-standard conditions.
• C. To measure the resistance of a solution.
• D. To determine the equilibrium constant.
• Correct Answer: B
• Explanation: The Nernst equation relates the standard potential to the actual potential under non-standard conditions.

• Q2. In the simplified Nernst equation at 25°C, what is the value of the constant in the numerator?
• A. 0.059
• B. 0.0591
• C. 0.0592
• D. 0.06
• Correct Answer: B
• Explanation: The correct constant is approximately 0.0591 V (often rounded to 0.059 in textbooks).

• Q3. What does n represent in the Nernst equation?
• A. The number of moles of the reactant.
• B. The number of electrons transferred in the half-reaction.
• C. The number of ions in the solution.
• D. The number of moles of the product.
• Correct Answer: B
• Explanation: n is the number of electrons transferred in the balanced half-reaction.

• Q4. In the Nernst equation, what is the concentration of a pure solid?
• A. 0
• B. 1
• C. Its molarity
• D. Its density
• Correct Answer: B
• Explanation: The concentration of a pure solid is constant and is assigned a value of 1.

• Q5. If the concentration of the oxidized species increases, what happens to the electrode potential?
• A. It decreases.
• B. It increases.
• C. It remains the same.
• D. It becomes zero.
• Correct Answer: B
• Explanation: Increasing the concentration of the oxidized species increases the value of the log term, which increases the electrode potential.

• Q6. For the half-reaction Cu^{2+} + 2e^- \rightarrow Cu, what is the value of n?
• A. 1
• B. 2
• C. 3
• D. 4
• Correct Answer: B
• Explanation: Two electrons are transferred in the reduction of Cu^{2+} to Cu, so n = 2.

Q7. What is the Nernst equation for the half-reaction Zn^{2+} + 2e^- \rightarrow Zn? A. E = E^° + 0.0592 log [Zn^{2+}] B. E = E^° - 0.0592 log [Zn^{2+}] C. E = E^° + 0.0591 log [Zn^{2+}] D. E = E^° + 0.0592 ln [Zn^{2+}] Correct Answer: A Explanation: The concentration of the reduced species (Zn) is 1, so the ratio is just [Zn^{2+}]. The correct equation is E = E^° + 0.0592 log [Zn^{2+}].

• Q8. In the Nernst equation, the term [oxidized species] refers to the concentration of:
• A. The reactant that is oxidized.
• B. The reactant that is reduced.
• C. The product that is formed by oxidation.
• D. The product that is formed by reduction.
• Correct Answer: B
• Explanation: The oxidized species is the one that gets reduced (gains electrons), so it is the reactant in the reduction half-reaction.

Q9. What is the relationship between the Nernst equation and Le Chatelier's Principle? A. The Nernst equation is a mathematical expression of Le Chatelier's Principle for electrochemical cells. B. They are unrelated. C. The Nernst equation contradicts Le Chatelier's Principle. D. The Nernst equation is only valid at equilibrium. Correct Answer: A Explanation: The Nernst equation describes how changes in concentration affect the equilibrium position and thus the potential, which is a direct application of Le Chatelier's Principle.

Q10. If the standard electrode potential of a half-cell is +0.34 V and the concentration of the oxidized species is 0.1 M (with n=2), what is the new potential? (Use 0.059 for the constant). A. 0.31 V B. 0.34 V C. 0.37 V D. 0.40 V Correct Answer: A Explanation: E = 0.34 + 0.0592 log(0.1) = 0.34 + 0.0295(-1) = 0.34 - 0.0295 = 0.3105 V, which is approximately 0.31 V.