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 for calculating the cell potential of an electrochemical cell under non-standard conditions, which occur when reactant/product concentrations differ from their standard values, temperature is different than 298 K, or both. (b) Under standard conditions, the reaction quotient Q becomes one because every concentration term is equal to unity. (c) When reactant concentrations are increased, the value of Q decreases, causing the emf (E) of the cell to increase according to the Nernst equation: \(E = E° - \frac{RT}{nF} \ln Q\).

Step-by-step solution

Question (a): Applicability of the Nernst Equation

The Nernst equation is applicable when calculating the cell potential or emf (electromotive force) of an electrochemical cell under non-standard conditions. Non-standard conditions occur when the concentrations of reactants and products in the cell differ from their standard concentrations (1 Molar for solutes, 1 atm for gases), the temperature is different than 298 K, or both.

Question (b): Numerical value of the reaction quotient Q under standard conditions

Under standard conditions, the concentrations of reactants and products in the cell are equal to their standard concentrations (1 Molar for solutes, 1 atm for gases). Thus, the reaction quotient, Q, is equal to the ratio of the concentrations of products raised to their stoichiometric coefficients divided by the concentrations of reactants raised to their stoichiometric coefficients. For any given reaction, when all the concentrations are at their standard values, the reaction quotient Q becomes one, as every concentration term is equal to unity.

Question (c): Effect on emf when reactant concentrations are increased

The Nernst equation for calculating the emf (E) of a cell is given as: \[E = E° - \frac{RT}{nF} \ln Q\] where: - E: the cell potential under non-standard conditions - E°: the cell potential under standard conditions - R: the universal gas constant (8.314 J/mol·K) - T: the temperature in Kelvin - n: the number of moles of electrons transferred in the redox reaction - F: the Faraday constant (96,485 C/mol) - Q: the reaction quotient under non-standard conditions Now, as we increase the concentrations of reactants, the value of Q will decrease (as Q is the ratio of products to reactants). According to the Nernst equation, when Q decreases, the value of \( \ln Q \) becomes more negative, and therefore, the difference between \(E\) and \(E°\) becomes more positive. As a result, the emf (E) of the cell will increase.

Key concepts

Electrochemical Cell

An electrochemical cell is a device capable of either generating electrical energy from chemical reactions or facilitating chemical reactions through the introduction of electrical energy. Electrochemical cells consist of two electrodes (an anode and a cathode) immersed in an electrolyte solution. The anode is where oxidation occurs, while reduction takes place at the cathode.
During operation, electrons flow from the anode to the cathode through an external circuit, while ions in the electrolyte move between the electrodes to balance the charge. Electrochemical cells are essential for various applications, from batteries and fuel cells to sensors and electrolysis systems.

EMF (Electromotive Force)

The electromotive force (EMF) is the energy provided by a cell or a battery per coulomb of charge passing through it. It is the driving force that pushes electrons through the external circuit and is measured in volts (V). Although called 'force', it is actually a potential energy per unit charge, similar to voltage. EMF represents the maximum potential difference between the two electrodes of an electrochemical cell when no current is flowing. It's worth noting that the actual voltage may be less due to internal resistance within the cell.

Reaction Quotient Q

The reaction quotient, Q, is a measure of the relative amounts of products and reactants present during a reaction at a given moment in time. It is calculated similarly to the equilibrium constant but is used for non-equilibrium conditions. The equation for Q is dependent on the reaction at hand, where concentrations of gaseous reactants and products are expressed in pressures (atm) and concentrations for solutions are given in molarity (M).
For a generic reaction aA + bB \rightarrow cC + dD, the reaction quotient Q is given by:\[Q = \frac{[C]^c[D]^d}{[A]^a[B]^b}\]In the context of electrochemical cells, Q helps to determine the direction in which a reaction is proceeding and how far it is from reaching equilibrium.

Cell Potential

Cell potential, often symbolized as E, is the measure of the potential difference between two electrodes of an electrochemical cell. It reflects the cell's ability to produce an electric current when the circuit is complete. The standard cell potential (E°) is measured when all reactants and products are at standard conditions (1M concentration for solutes and 1 atm for gases). It serves as a reference point, offering the potential under ideal circumstances. The Nernst equation allows for the calculation of cell potential under non-standard conditions, which is critical as real-world applications rarely operate under standard conditions.

Non-Standard Conditions

Non-standard conditions refer to situations in which at least one of the parameters affecting an electrochemical cell's behavior deviates from the standard state. These include concentrations of reactants and products that are not at 1 M or pressures that differ from 1 atm, or temperatures other than the standard 298 K. Under these conditions, the cell potential changes from the standard potential. To calculate the cell potential under non-standard conditions, the Nernst equation is used. It takes into account these variations by incorporating the reaction quotient Q and the actual temperature T in its formula, allowing us to measure the cell’s EMF during any stage of a chemical reaction.