Question

What is the value of \(E^{\circ}\) for a redox reaction involving the transfer of \(2 \mathrm{~mol}\) of electrons if its equilibrium constant is \(1.8 \times 10^{-5}\) ?

Short answer

The standard electrode potential \(E^{\circ}\) is approximately \(-0.14 \, V\).

Step-by-step solution

Understand the Equation

We will use the Nernst equation to relate the standard electrode potential \(E^{\circ}\) to the equilibrium constant \(K_{eq}\). The equation is: \[E^{\circ} = \frac{RT}{nF} \ln K_{eq}.\] Here, \(R\) is the universal gas constant \(8.314\, J/(mol\, K)\), \(T\) is the temperature in Kelvin, \(n\) is the number of moles of electrons transferred (2 in this case), and \(F\) is Faraday's constant \(96485\, C/mol\).

Simplify the Equation

Assume standard temperature conditions, \(T = 298 \ K\). Substituting \(R\), \(T\), and \(F\) into the Nernst equation gives:\[E^{\circ} = \frac{8.314 \times 298}{2 \times 96485} \ln(1.8 \times 10^{-5}).\]

Calculate Natural Logarithm

Calculate the natural logarithm of the equilibrium constant:\[\ln(1.8 \times 10^{-5}) \approx -10.925.\]

Substitute and Solve for Standard Potential

Substitute \(\ln K_{eq}\) into the simplified Nernst equation:\[E^{\circ} = \frac{8.314 \times 298}{2 \times 96485} \times -10.925.\]Calculate to find \(E^{\circ}\):\[E^{\circ} \approx \frac{2476.172}{192970} \times -10.925 \approx -0.1395 \, V.\]

Interpretation

The negative sign of \(E^{\circ}\) indicates that the reaction is non-spontaneous under standard conditions. The standard electrode potential of the redox reaction is approximately \(-0.14 \, V\).

Key concepts

Redox Reactions

Redox reactions, short for reduction-oxidation reactions, are chemical processes in which electrons are transferred between reactants. In these reactions, one substance gets oxidized by losing electrons, while another gets reduced by gaining electrons. This exchange of electrons changes the oxidation states of the reactants.

Understanding redox reactions is crucial because they are involved in various processes:

• Biological systems such as cellular respiration
• Industrial processes like corrosion and electroplating
• Energy generation in batteries

In electrochemistry, redox reactions form the basis of understanding how electricity can be produced through chemical changes. The ability of a reactant to gain electrons determines its potential to function as an oxidizing agent, while losing electrons makes it a reducing agent.

Equilibrium Constant

The equilibrium constant, denoted as \( K_{eq} \), is a crucial parameter in chemical reactions. It reflects the ratio of concentrations of products to reactants at equilibrium. For redox reactions, where the concentration of reactants and products impact the reaction progress, \( K_{eq} \) provides valuable insight.

If \( K_{eq} \) is greater than 1, the products are favored at equilibrium, implying a spontaneous reaction. Conversely, if \( K_{eq} \) is much less than 1, as in our exercise (1.8 \times 10^{-5}), the reactants are favored, suggesting that the reaction is non-spontaneous under standard conditions. This equilibrium constant is critical in using the Nernst equation to determine standard electrode potential, further linking it to the spontaneity of redox reactions.

Standard Electrode Potential

Standard electrode potential \(E^{\circ}\), expressed in volts, is a measure of the intrinsic tendency of a species to be reduced, as measured under standard conditions (1 M concentration, 1 atm pressure, and 25°C temperature). It is connected to the Gibbs free energy change of the reaction and reveals how thermodynamically favorable a reaction is.

To calculate \(E^{\circ}\), we can use the Nernst equation, which quantifies the relation between the potential difference and reactants/product concentrations. A positive \(E^{\circ}\) suggests a spontaneous process, while a negative value, like -0.14 V observed in the step-by-step solution, indicates a non-spontaneous process under standard conditions. Understanding \(E^{\circ}\) is fundamental when assessing the feasibility of electrochemical cells and reactions involving electron transfer.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a mathematical function essential in the field of chemistry, especially when dealing with exponential growth or decay processes. The base of the natural logarithm is the mathematical constant \(e\), approximately equal to 2.718.

In the context of the Nernst equation, \(\ln\) is used to relate the equilibrium constant \(K_{eq}\) to standard electrode potential \(E^{\circ}\). The negative natural logarithm in our example indicates that the equilibrium heavily favors the reactants, as evidenced by the highly negative value (-10.925). The inclusion of natural logarithm calculations is integral to solving for the standard potentials, making it a critical step in understanding redox reaction-based calculations in electrochemistry.