Question

Use the Nernst equation to show that \(E_{\text {cell }}=E_{\text {cell }}^{\circ}\) under standard conditions.

Short answer

Under standard conditions, the activity terms in the Nernst equation each equal 1. Since the natural log of 1 is zero, the cell's EMF equals the standard EMF (\( E_{\text{cell}} = E_{\text{cell}}^\circ \)).

Step-by-step solution

Understand the Nernst Equation

The Nernst equation relates the electromotive force (EMF) of a cell to the standard electrode potential, temperature, charge, and the activities (or concentrations) of the chemical species involved. The equation is given by: \[ E_{\text{cell}} = E_{\text{cell}}^\circ - \frac{RT}{nF} \ln\left(\frac{\text{activity of anode}}{\text{activity of cathode}}\right) \] where: - \( E_{\text{cell}} \) is the EMF of the cell- \( E_{\text{cell}}^\circ \) is the standard EMF of the cell- \( R \) is the universal gas constant- \( T \) is the temperature in Kelvin- \( n \) is the number of moles of electrons transferred- \( F \) is the Faraday's constant- The activity of anode and cathode species usually represent their concentration in moles per liter.

Apply Standard Conditions

Under standard conditions, all solutes are at a concentration of 1 M (molarity), and the temperature is at 298 K (25 °C). The activity of the species involved under standard conditions is 1. Substituting into the Nernst equation with activities both equal to 1 gives: \[\ln\left(\frac{1}{1}\right) = 0\] This implies that the second term of the Nernst equation becomes zero, because the log of one is zero.

Simplify the Equation

With the natural logarithm of one ratio of activities being zero, the entire term after the standard EMF becomes zero. Therefore, the Nernst equation under standard conditions simplifies to: \[ E_{\text{cell}} = E_{\text{cell}}^\circ - \frac{RT}{nF} \times 0 \]\[ E_{\text{cell}} = E_{\text{cell}}^\circ \] This shows that the EMF of the cell is equal to the standard EMF under standard conditions.

Key concepts

Electromotive Force (EMF)

Electromotive force (EMF) is a critical concept in electrochemistry that denotes the voltage generated by an electrochemical cell. This voltage is essentially the driving force that pushes electrons through the circuit and enables a current to flow. It's comparable to water pressure in plumbing; just as higher water pressure can push water through a pipe more effectively, higher EMF means greater potential to drive electrons around a circuit.

Understanding EMF is vital because it tells us how much energy per charge can be gained or lost when chemicals are converted between their oxidized and reduced forms. It's measured in volts (V). In the context of the Nernst equation, EMF represents the electrical potential difference of a reaction under any conditions, compared to under standard conditions, which are 1 M concentration for all solutes and 298 K temperature.

Standard Electrode Potential

The standard electrode potential (\( E_{\text{cell}}^\circ \)) is a benchmark measurement that provides essential information on the intrinsic tendency of a reacting species to lose or gain electrons—essentially, it represents how strongly a chemical species wants to be reduced. Measured under standard conditions (all reactants and products at 1M concentration and a temperature of 298 K), this value is crucial for predicting the direction and force of the electron flow when two species are connected by an electric circuit.

In the Nernst equation, the standard electrode potential acts as the reference point from which deviations are calculated when the actual conditions in the cell drift from the 'standard'. Because of its central role, data for these potentials are tabulated and widely available for various electrode reactions, providing a comparison point for different electrochemical cells.

Chemical Species Concentration

In any electrochemical cell, the concentration of the chemical species—meaning the amount of ions or molecules of the substance present in a given volume—plays a pivotal role in determining the actual EMF for the system. This is because reactions tend to move towards the side with the lower concentration, as per Le Chatelier's principle. A higher concentration of reactants will drive the reaction to produce more products, and vice versa.

The Nernst equation incorporates these concentration differences into its terms. When the concentrations of anode and cathode species differ from the standard 1M condition, it reflects in a change in the EMF of the cell from the standard EMF. This relationship is critical in explaining why batteries lose voltage as they discharge (since the concentration of the chemical species in the battery changes), and in designing cells to have a particular voltage for a specific application.