Cell Potential: The Nernst Equation: Videos & Practice Problems

Understanding cell potential is crucial in electrochemistry, particularly when examining how it correlates with the Nernst equation. The reaction quotient (Q) plays a pivotal role in this relationship, as it reflects the ratio of product to reactant concentrations at a given moment, excluding solids and liquids. This quotient is vital for determining the maximum potential of an electrochemical cell at the instant the circuit is established, as voltage tends to decrease over time due to cell degradation.

The standard cell potential (E°_cell) is calculated under standard conditions, where ion concentrations are 1 molar, pressure is at 1 atmosphere, and pH is 7. However, the Nernst equation becomes essential when conditions deviate from these standards. The Nernst equation, given by:

$E_{\mathrm{cell}}=E_{°}{\mathrm{cell}}_{}-\frac{0.05916}{n}\times log\left(Q\right)$

where n is the number of moles of electrons transferred in the redox reaction, allows for the calculation of non-standard cell potentials. This equation adjusts the cell potential for different ion concentrations, providing a more accurate measure of a cell's voltage under varying conditions.

The Reaction Quotient

Understanding the reaction quotient, denoted as q, is crucial when exploring its relationship with electrochemical cells and the Nernst equation. The reaction quotient represents the ratio of the concentrations of products to reactants at a specific moment, calculated by an expression that excludes solids and liquids. This is similar to the equilibrium constant, K, which also focuses solely on gaseous and aqueous species.

In the context of electrochemical cells, the reaction quotient is particularly significant as it allows us to determine the maximum potential of the cell at the precise moment the circuit is connected. This is important because, over time, the voltage produced by the electrochemical cell diminishes as the cell degrades. By utilizing the reaction quotient, we can identify the point at which the voltage is at its peak, providing valuable insight into the cell's performance and efficiency.

The Reaction Quotient Example

In a redox reaction involving lead(II) ions and potassium, the reaction can be represented as follows:

Pb²⁺ + 2 K(s) → Pb(s) + 2 K⁺

To determine the reaction quotient (Q), we focus on the concentrations of the products and reactants, while excluding solids and liquids from the expression. The general formula for the reaction quotient is:

Q =  [Products] / [Reactants]

In this case, the relevant species are the potassium ions (K⁺) and lead(II) ions (Pb²⁺). The expression for Q becomes:

Q = [K⁺]² / [Pb²⁺]

Given the concentrations:

• [K⁺] = 0.0015 M
• [Pb²⁺] = 0.0880 M

Substituting these values into the expression, we calculate:

Q = (0.0015)² / (0.0880)

Calculating this gives:

Q = 2.6 × 10^-5

This value of Q indicates the ratio of the concentrations of products to reactants at a given moment in the reaction, providing insight into the direction and extent of the reaction under the specified conditions.

The Nernst Equation

The standard cell potential is determined under specific conditions where the concentrations of ions in half-cells are 1 molar, the pressure is 1 atmosphere, and the pH is 7. However, in real-world scenarios, these conditions often vary, necessitating the use of the Nernst equation to calculate nonstandard cell potentials.

The Nernst equation is expressed as:

E_{\text{cell}} = E^{\circ}_{\text{cell}} - \frac{0.05916}{n} \log(Q)

In this equation, E_cell represents the nonstandard cell potential, while E^°_cell denotes the standard cell potential. The variable n indicates the number of moles of electrons transferred in the redox reaction, and Q is the reaction quotient, which reflects the ratio of the concentrations of products to reactants at any given moment.

By applying the Nernst equation, one can effectively determine the cell potential under varying concentrations, allowing for a more accurate representation of electrochemical behavior in practical applications.

The Nernst Equation Example

To calculate the cell potential for a redox reaction at 25 degrees Celsius, we utilize the Nernst equation, which is essential for determining the non-standard cell potential. The equation is expressed as:

\( E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{0.05916}{n} \log(Q) \)

In this scenario, we are given the concentrations of cobalt(III) ions (\( \text{Co}^{3+} \)) at 1 M and magnesium ions (\( \text{Mg}^{2+} \)) at 0.0033 M, along with their standard reduction potentials. The first step is to calculate the reaction quotient \( Q \), which is defined as the ratio of the concentrations of the products to the reactants, excluding solids and liquids. For this reaction, \( Q \) is calculated as:

\( Q = \frac{[\text{Mg}^{2+}]^3}{[\text{Co}^{3+}]^2} \)

Substituting the given concentrations, we find:

\( Q = \frac{(0.0033)^3}{(1)^2} = 3.5937 \times 10^{-8} \)

Next, we determine \( n \), the number of electrons transferred in the reaction. The half-reactions indicate that cobalt is reduced (gaining electrons) while magnesium is oxidized (losing electrons). The lowest common multiple of the electrons transferred in both half-reactions is 6, meaning \( n = 6 \).

To find the standard cell potential \( E^\circ_{\text{cell}} \), we identify the cathode and anode. The cathode is where reduction occurs, which is cobalt in this case, and the anode is where oxidation occurs, which is magnesium. The standard cell potential is calculated as:

\( E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}} \)

Given the standard reduction potentials, we have:

\( E^\circ_{\text{cell}} = 1.82 \, \text{V} - (-2.37 \, \text{V}) = 4.19 \, \text{V} \)

Now, substituting \( E^\circ_{\text{cell}} \), \( n \), and \( Q \) back into the Nernst equation gives:

\( E_{\text{cell}} = 4.19 \, \text{V} - \frac{0.05916}{6} \log(3.5937 \times 10^{-8}) \)

Calculating this yields:

\( E_{\text{cell}} \approx 4.26 \, \text{V} \)

This value represents the non-standard cell potential for the given redox reaction, illustrating the importance of both the concentrations of the reactants and the standard reduction potentials in determining the overall cell potential.

If [Br^–] = 0.010 M and [Al³⁺] = 0.022 M, predict whether the following reaction would proceed spontaneously as written at 25ºC:
Al (s) + Br₂ (l) ⇌ Al³⁺ (aq) + Br^– (aq)
Standard Reduction Potentials
Al³⁺ (aq) + 3 e^– → Al (s)       E°_red = –1.66 V
Br₂ (l) + 2 e^– → Br^– (aq)     E°_red = +1.09 V

• A. 2.90 V
• B. 2.60 V
• C. 3.03 V
• D. 2.75 V

Answer: (A)

Determine [Fe²⁺] for the following galvanic cell at 25ºC if given [Sn²⁺] = 0.072 M, [Fe³⁺] = 0.0219 M, and [Sn⁴⁺] = 0.00345 M. 
Sn²⁺ (aq) + 2 Fe³⁺ (aq)  ⇌. Sn⁴⁺ (aq) + 2 Fe²⁺ (aq)     E_cell = + 0.68 V
Standard Reduction Potentials
Sn⁴⁺ (aq) + 2 e^–  →. Sn²⁺ (aq)      E°_red = + 0.151 V
Fe³⁺ (aq) + e^–  →  Fe²⁺ (aq)         E°_red = + 0.771 V

• A. 0.13150 M
• B. 0.00968 M
• C. 0.08180 M
• D. 0.01936 M

Answer: (B)

Here’s what students ask on this topic:

What is the Nernst equation?

The Nernst equation is a fundamental equation in electrochemistry that relates the reduction potential of a chemical reaction (specifically a half-cell in an electrochemical cell) to the standard electrode potential, temperature, and activities (often approximated by concentrations) of the chemical species undergoing reduction and oxidation. It was named after the German physical chemist Walther Nernst.

The equation is written as:

$E=E_{°}-\frac{RT}{nF}ln\left(Q\right)$

where:

• E is the cell potential (electrode potential) at non-standard conditions,
• E° is the standard electrode potential,
• R is the universal gas constant (8.314 J/(mol*K)),
• T is the temperature in Kelvin,
• n is the number of moles of electrons exchanged in the electrochemical reaction,
• F is the Faraday constant (approximately 96485 C/mol), and
• Q is the reaction quotient, which is the ratio of the activities (or concentrations) of the products to the reactants, raised to the power of their stoichiometric coefficients.

The Nernst equation allows you to calculate the potential of an electrochemical cell under different conditions, and it helps to understand how changes in concentration or temperature can affect the voltage of a battery or the efficiency of an electrochemical process.

How can the Nernst equation be used?

The Nernst equation is a fundamental equation in electrochemistry that relates the reduction potential of a chemical reaction to the standard electrode potential, temperature, and the activities (or concentrations) of the reactants and products. It is given by the formula:

$E=E_{°}-\left(\frac{RT}{nF}\right)*ln\left(Q\right)$

where E is the cell potential, E^° is the standard electrode potential, R is the gas constant, T is the temperature in Kelvin, n is the number of moles of electrons exchanged in the electrochemical reaction, F is the Faraday constant, and Q is the reaction quotient.

You can use the Nernst equation to:

1. Calculate the cell potential of an electrochemical cell under non-standard conditions.
2. Determine the equilibrium constant for a redox reaction at a given temperature.
3. Predict the direction of an electrochemical reaction by comparing the cell potential to zero.
4. Estimate the concentration of ions in a solution when given the cell potential and other ion concentrations.
5. Understand the effects of temperature and concentration on the voltage of an electrochemical cell.

This equation is crucial for understanding and predicting the behavior of batteries, fuel cells, and natural biochemical processes involving redox reactions.

When do we use the Nernst equation?

The Nernst equation is used to determine the electric potential of a cell under non-standard conditions. In other words, it's applied in electrochemistry to calculate the reduction potential of an electrochemical cell when the concentrations of ions participating in the redox reaction are not at standard conditions, which typically means 1 molar concentration and 1 atmosphere of pressure for gases.

You'll use the Nernst equation when you want to find out the potential of a half-cell or full cell when temperature, ion concentration, or pressure deviates from standard-state conditions. It's particularly useful for understanding how the potential will change with varying concentrations of reactants and products, which is essential in predicting the direction of the reaction and its equilibrium position. The equation is also instrumental in determining the pH of a solution, as it relates to the hydrogen ion concentration.