Question

For the electrochemical cell, $$ \mathrm{Mg}(\mathrm{s})\left|\mathrm{Mg}^{2+}(\mathrm{aq}, 1 \mathrm{M}) \| \mathrm{Cu}^{2+}(\mathrm{aq}, 1 \mathrm{M})\right| \mathrm{Cu}(\mathrm{s}) $$ the standard emf of the cell is \(2.70 \mathrm{~V}\) at \(300 \mathrm{~K}\). When the concentration of \(\mathrm{Mg}^{2+}\) is changed to \(x \mathrm{M}\), the cell potential changes to \(2.67 \mathrm{~V}\) at \(300 \mathrm{~K}\). The value of \(x\) is (given, \(\frac{F}{R}=11500 \mathrm{~K} \mathrm{~V}^{-1}\), where \(F\) is the Faraday constant and \(R\) is the gas constant, \(\ln (10)=2.30)\)

Short answer

After all calculations, the value of \( x \) is found to be approximately 2.47 M.

Step-by-step solution

Write the Nernst Equation

Calculate the potential of an electrochemical cell under non-standard conditions using the Nernst equation: \[ E = E^\circ - \frac{RT}{nF}\ln\frac{[\text{products}]}{[\text{reactants}]} \] where \( E \) is the cell potential under non-standard conditions, \( E^\circ \) is the standard cell potential, \( R \) is the gas constant, \( T \) is the temperature in Kelvin, \( n \) is the number of moles of electrons transferred, and \( F \) is Faraday's constant.

Identify Known Values

Substitute the known values into the Nernst equation. We know the standard cell potential \( E^\circ = 2.70 \) V, the temperature \( T = 300 \) K, and the cell potential after the change \( E = 2.67 \) V. We also need to express \( R \) and \( F \) in terms of the given ratio, \( \frac{F}{R} = 11500 \) K V^-1.

Calculate the value of \( \frac{RT}{nF} \)

Using the given ratio and the temperature, compute the value of \( \frac{RT}{nF} \). Since magnesium releases two electrons upon oxidation (\( n = 2 \)), the equation simplifies to: \[ \frac{RT}{nF} = \frac{T}{2 \cdot 11500} \]

Rearrange the Nernst Equation to Solve for \( x \)

Rearrange the Nernst equation to solve for \( [Mg^{2+}] = x \) M. The concentration of \( Cu^{2+} \) is constant at 1 M, so the ratio of the concentrations becomes: \[ \ln\frac{[\text{products}]}{[\text{reactants}]} = \ln\frac{1}{x} \] Then insert this into the rearranged Nernst equation to express \( x \) in terms of the cell potentials and \( \frac{RT}{nF} \) value.

Insert Known Values and Solve for \( x \)

Place the given and calculated values into the Nernst equation and solve for \( x \). Remember that \( E - E^\circ \) is negative because the cell potential decreased.

Use Natural Logarithm Properties to Find \( x \)

Use the properties of logarithms to isolate \( x \) and solve the equation. Since \( \ln(10) = 2.30 \), you will convert the natural logarithm to base 10 to simplify calculations.

Key concepts

Electrochemical Cell

An electrochemical cell is a device that generates electrical energy from chemical reactions or facilitates chemical reactions through the introduction of electrical energy. It consists of two electrodes made of different metals or metal compounds known as the anode and cathode. These electrodes are immersed in electrolytes where ions are present. The driving force behind the electrical current is the chemical potential difference between the electrodes.

In the given exercise, we have a cell comprising a magnesium electrode and a copper electrode with their respective ions in solution. The movement of electrons from the more reactive metal, magnesium, to the less reactive copper, generates a potential difference, or cell potential, which can be measured as voltage. Understanding the specifics of how these reactions occur and the factors that influence the voltage produced is crucial for students aiming to master electrochemistry.

Standard EMF

Standard electromotive force (EMF) is the voltage or potential difference between two half-cells in an electrochemical cell when all reactants and products are at standard conditions, typically 1 molar concentration and 1 atmosphere pressure at a temperature of 25 degrees Celsius. The standard EMF is a measurable quantity that provides us with a reference to predict the direction of the reaction and calculate the cell's maximum potential under these standard conditions.

In the exercise, the standard EMF of the cell made from Mg and Cu is given as 2.70 V. This value indicates the driving force available for the transfer of electrons from magnesium to copper when both ions are at a concentration of 1M. It serves as a baseline to determine how changes in conditions, such as ion concentration, will affect the cell's potential.

Cell Potential

Cell potential, also known as electromotive force (EMF) of a cell, is the measure of the potential energy difference per unit charge between two half-cells in an electrochemical cell. It's the force that drives electrons through the circuit, allowing the cell to do electrical work. Cell potential can be calculated under standard conditions to give standard EMF or under non-standard conditions using the Nernst equation.

In our example, the standard cell potential is given, but when the concentration of Mg²⁺ changes, this potential shifts. By applying the Nernst equation, we can calculate the new cell potential and understand how cell potential is affected by changes in conditions, such as the chemical concentration of the cell's components.

Chemical Concentration

Chemical concentration refers to the amount of a substvance within a given volume of solution, typically expressed in molarity (M), which is the number of moles of solute per liter of solution. The concentration of reactants and products in an electrochemical cell is directly related to the cell potential, as the Nernst equation shows.

As indicated in the exercise, when the concentration of Mg²⁺ ions changes from the standard 1M, it affects the potential of the electrochemical cell. Our task is to calculate the new concentration (\(x\text{M}\)) that results in a cell potential of 2.67 V. The concentration gradient, as dictated by the Nernst equation, influences the movement of ions across the cell and consequently impacts the voltage produced. Understanding this relationship is vital for students, as it connects the abstract concept of concentration with the tangible outcome of voltage in electrochemical cells.