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Chapter 18: Problem 24

The equilibrium constant for the reaction $$\operatorname{Sr}(s)+\operatorname{Mg}^{2+}(a q) \rightleftharpoons \operatorname{Sr}^{2+}(a q)+\operatorname{Mg}(s)$$ is \(2.69 \times 10^{12}\) at \(25^{\circ} \mathrm{C}\). Calculate \(E^{\circ}\) for a cell made up of \(\mathrm{Sr} / \mathrm{Sr}^{2+}\) and \(\mathrm{Mg} / \mathrm{Mg}^{2+}\) half- cells.

Short Answer

Expert verified
The standard cell potential \(E°\) for a cell made up of Sr/Sr2+ and Mg/Mg2+ half-cells is 4.70V.

Step by step solution

01

Understand the Nernst Equation

First, note that the Nernst Equation, which connects equilibrium constants with cell potentials, is as follows: \(E° = - \frac{RT}{nF} \ln K\), where \(R\) is the gas constant, \(T\) is the temperature in Kelvin, \(n\) is the number of electrons involved in the reaction, \(F\) is the Faraday’s constant, and \(K\) is the equilibrium constant.
02

Identify the values for the Nernst Equation

It's given that \(K = 2.69 \times 10^{12}\), \(T = 25°C\) (or \(298.15K\); always convert the temperature to Kelvin in these calculations), \(R = \frac{8.3145 \, J} {\, mol·K} (the molar gas constant), and \(F = 96485.33 \, C \, mol^{-1}\) (the Faraday’s constant). The number of electrons can be determined from the balanced chemical equation and it shows that \(n = 2\) electrons are transferred.
03

Substitute values into the equation and calculate

Substitute the values into the formula: \(E° = -\frac{(8.3145 J/mol·K)(298.15K)}{(2)(96485.33C/mol)}* \ln (2.69 \times 10^{12})\). Remember that the ln function is the natural logarithm, which may need to be calculated using a calculator. Note that the constant F is defined in Coulombs per mole, which can be seen as Joules per Volt·mol, so this calculation will give you the result in Volts.

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