Question

Derive an equation that directly relates the standard emf of a redox reaction to its equilibrium constant.

Short answer

The standard emf (E°) of a redox reaction can be directly related to its equilibrium constant (K) using the Nernst equation and the expression for the equilibrium constant in terms of concentrations. At equilibrium, the cell potential (E) becomes zero, and the equation can be rearranged to give: \[E° = \frac{RT}{nF} \ln K\] This equation shows that the standard emf is proportional to the natural logarithm of the equilibrium constant.

Step-by-step solution

1. Relate standard emf to cell potential using the Nernst equation

The Nernst equation relates the cell potential (E) to the standard cell potential (Eº), the temperature (T), the number of moles of electrons transferred (n), the universal gas constant (R), and the concentrations of the species involved in the redox reaction [Q, the reaction quotient]. The equation is: \[E = E° - \frac{RT}{nF} \ln Q\] Where F is the Faraday constant and n is the number of moles of electrons transferred in the reaction.

2. Define equilibrium constant in terms of concentrations

The equilibrium constant (K) is the value of the reaction quotient (Q) when a system has reached equilibrium. In terms of concentrations, it can be expressed as: \[K = \frac{[C]^c[D]^d}{[A]^a[B]^b}\] Where [A], [B], [C], and [D] are the concentrations of the various species involved in the reaction and a, b, c, and d are their stoichiometric coefficients.

3. Combine the Nernst equation and the equilibrium constant expression

At equilibrium, the cell potential (E) becomes zero, as there is no net flow of electrons in the system. Therefore, we can write: \[0 = E° - \frac{RT}{nF} \ln K\]

4. Solve for the standard emf in terms of the equilibrium constant

Rearrange the equation obtained in step 3 to solve for Eº: \[E° = \frac{RT}{nF} \ln K\] This equation directly relates the standard emf (Eº) of a redox reaction to its equilibrium constant (K) and shows that the standard cell potential is proportional to the natural logarithm of the equilibrium constant.

Key concepts

Nernst Equation

The Nernst equation is a fundamental relation in electrochemistry that describes the electromotive force (emf) of a chemical cell under non-standard conditions. It integrates the influence of temperature, the concentrations of the substances involved, and the number of electrons transferred during the reaction to calculate the actual cell potential.

In practical terms, the equation can be used by students to predict the voltage of an electrochemical cell at any point in its reaction, not just under standard conditions. This understanding is crucial for applications such as batteries, where performance depends on varying conditions. When the cell reaches equilibrium, its cell potential equals zero, signifying that there is no further drive for the reaction to proceed in either direction.

Redox Reaction

A redox (reduction-oxidation) reaction involves the transfer of electrons between two substances. It's made up of two half-reactions: one substance is oxidized by losing electrons, and the other is reduced by gaining electrons. These tandem processes are fundamental to electrochemical cells.

Grasping the concept of redox reactions is vital for students as it not only applies to electrochemistry but also to various biochemical processes and industrial applications. Aspects such as stoichiometry and balancing redox reactions serve as a base for further understanding more complex topics such as calculating cell potentials and equilibrium constants.

Cell Potential

Cell potential, denoted as E, is essentially the driving force of an electrochemical cell, measured in volts. It determines how far a redox reaction is from reaching equilibrium. The standard cell potential (E°) is the cell potential under standard conditions (conc. = 1M, pressure = 1 atm, temp = 25°C).

Students can evaluate the feasibility of reactions with this quantity: a positive cell potential indicates a spontaneous reaction, while a negative value suggests non-spontaneity. This idea underpins the practical design and use of electrochemical cells and is closely related to the Nernst equation, providing a bridge to relate this potential to the equilibrium constant.

Equilibrium Constant

The equilibrium constant (K) is a dimensionless number that provides insight into the position of equilibrium for a chemical reaction. It's a ratio of the concentrations of products to reactants, raised to the power of their stoichiometric coefficients, at equilibrium.

A higher value of K indicates that, at equilibrium, products are favored; conversely, a lower value indicates that reactants are favored. Understanding how to calculate and interpret K helps students predict the extent of a reaction and tie together the quantitative relationship between cell potentials and reaction quotients.

Faraday Constant

The Faraday constant (F) represents the total charge of one mole of electrons, approximately equal to 96,485 coulombs per mole. This constant is integral to calculations involving electrochemical processes and appears in both the Nernst equation and the formula linking standard emf to the equilibrium constant.

As students delve into electrochemistry, the Faraday constant emerges as a key value in determining the relationship between chemical change and electric charge, a relationship that underlies commercial applications such as in the charging and discharging of batteries and the electroplating of metals.