Question

(a) Under what circumstances is the Nernst equation applicable? (b) What is the numerical value of the reaction quotient, \(Q\), under standard conditions? (c) What happens to the emf of a cell if the concentrations of the reactants are increased?

Short answer

(a) The Nernst equation is applicable when the involved species are in equilibrium, the temperature is reasonably constant, and the reaction involves electron transfer (redox reaction). It is given by: \(E = E^\circ - \dfrac{RT}{nF} \ln Q_\mathrm{r}\). (b) Under standard conditions, the numerical value of the reaction quotient, \(Q\), is 1. (c) When the concentrations of the reactants are increased, the cell potential (emf) will also increase, indicating that the cell can provide more electrical energy with higher concentrations of reactants.

Step-by-step solution

(a) The Nernst equation and its applicability

The Nernst equation is used to determine the reduction potential of a half-cell in an electrochemical cell, taking into account the concentration or partial pressure of the involved species. It can be written as: \(E = E^\circ - \dfrac{RT}{nF} \ln Q_\mathrm{r}\) Where: - \(E\) is the cell potential (or emf) at non-standard conditions - \(E^\circ\) is the cell potential at standard conditions - \(R\) is the gas constant (\(8.314\, \mathrm{J} \mathrm{K}^{-1} \mathrm{mol}^{-1}\)) - \(T\) is the temperature in Kelvin - \(n\) is the number of moles of electrons transferred in the redox reaction - \(F\) is the Faraday's constant (\(9.6485 \times 10^4\, \mathrm{C} \mathrm{mol}^{-1}\)) - \(Q_\mathrm{r}\) is the reaction quotient The Nernst equation is applicable when: 1. The involved species are in equilibrium. 2. The temperature is reasonably constant. 3. The reaction involves electron transfer (redox reaction).

(b) Reaction quotient under standard conditions

Under standard conditions, the concentration of each species is 1 M, and the partial pressure of the involved gases is 1 atm. The reaction quotient Q can be defined as: \(Q = \dfrac{\prod([\text{products}]^{\text{stoichiometric coefficients}})}{\prod([\text{reactants}]^{\text{stoichiometric coefficients}})}\) Under standard conditions, the concentrations of all species are the same (1 M). Therefore, the reaction quotient Q becomes: \(Q = \dfrac{1^{\text{stoichiometric coefficients of products}}}{1^{\text{stoichiometric coefficients of reactants}}}\) Since any number raised to any power is always one, the numerical value of Q under standard conditions is: \(Q = 1\)

(c) Effect of increasing reactants concentrations on emf

According to the Nernst equation, the cell potential is related to the natural logarithm of the reaction quotient Q: \(E = E^\circ - \dfrac{RT}{nF} \ln Q_\mathrm{r}\) When the concentrations of the reactants are increased, the reaction quotient Q decreases (less than 1) because Q is the ratio of the concentrations of products and reactants. As a result, the value of the natural logarithm \(\ln Q_\mathrm{r}\) becomes negative. This causes the second term (\(\dfrac{RT}{nF} \ln Q_\mathrm{r}\)) in the Nernst equation to become positive. Therefore, as the concentrations of the reactants increase, the cell potential (emf) will also increase. This indicates that the cell can provide more electrical energy when the concentrations of reactants are increased.