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Chapter 18: Problem 21

Write the equations relating \(\Delta G^{\circ}\) and \(K\) to the standard emf of a cell. Define all the terms.

Short Answer

Expert verified
1. \(\Delta G^{\circ}\) is the standard Gibbs free energy change, \(K\) is the equilibrium constant, and \(E^{\circ}\) is the standard cell potential or emf. 2. The equations relating them are \(\Delta G^{\circ} = -nFE^{\circ}\) and \(\Delta G^{\circ} = -RTlnK\). Combining the two we get \(E^{\circ} = \frac{RT}{nF}lnK\).

Step by step solution

01

Definition of Terms

First, define each term: \n1. \(\Delta G^{\circ}\) refers to the standard Gibbs free energy change. It describes the change in energy when a reaction proceeds at constant temperature and pressure. \n2. \(K\) is the equilibrium constant, which represents the ratio of concentrations or partial pressures of products to reactants at equilibrium. \n3. \(E^{\circ}\) is the standard cell potential or electromotive force (emf). It is the potential difference between two half-cells under standard conditions.
02

Relationship between \(\Delta G^{\circ}\) and \(E^{\circ}\)

The standard Gibbs free energy change (\(\Delta G^{\circ}\)) and the standard cell potential (\(E^{\circ}\)) for a redox reaction are related by the equation: \(\Delta G^{\circ} = -nFE^{\circ}\), where \(n\) is the number of moles of electrons transferred in the redox reaction, and \(F\) is the Faraday's constant (the quantity of electrical charge that corresponds to one mole of electrons, approximately 96485.3 C/mol). This equation describes how the potential energy of a chemical reaction (Gibbs free energy) can be converted to electrical energy in electrochemical cells.
03

Relationship between \(\Delta G^{\circ}\) and \(K\)

The standard Gibbs free energy change \(\Delta G^{\circ}\) is also related to the equilibrium constant \(K\) by the equation: \(\Delta G^{\circ} = -RTlnK\), where \(R\) is the universal gas constant (8.314 J/(mol·K)), and \(T\) is the temperature in Kelvin. This equation tells us that as the reaction becomes more spontaneous (\(\Delta G^{\circ}\) becomes more negative), the equilibrium constant becomes larger (favoring formation of the products).
04

Combining Both Relations

By combining both relations, you can also express \(K\) in terms of \(E^{\circ}\) as follows: \(-nFE^{\circ} = -RTlnK\), so \(E^{\circ} = \frac{RT}{nF}lnK\).

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Most popular questions from this chapter

Consider the electrolysis of molten barium chloride, \(\mathrm{BaCl}_{2}\). (a) Write the half-reactions. (b) How many grams of barium metal can be produced by supplying 0.50 A for 30 min?

Consider a concentration cell made of the following two compartments: \(\mathrm{Cl}_{2}(0.20 \mathrm{~atm}) \mathrm{ICl}^{-}(1.0 \mathrm{M})\) and \(\mathrm{Cl}_{2}(2.0 \mathrm{~atm}) \mid \mathrm{Cl}^{-}(1.0 \mathrm{M}) .\) Platinum is used as the in- ert electrodes. Draw a cell diagram for the cell and calculate the emf of the cell at \(25^{\circ} \mathrm{C}\).

What is the emf of a cell consisting of a \(\mathrm{Pb}^{2+} / \mathrm{Pb}\) half-cell and a \(\mathrm{Pt} / \mathrm{H}^{+} / \mathrm{H}_{2}\) half-cell if \(\left[\mathrm{Pb}^{2+}\right]=0.10 \mathrm{M}\), \(\left[\mathrm{H}^{+}\right]=0.050 \mathrm{M},\) and \(P_{\mathrm{H}}=2.0 \mathrm{~atm} ?\)

Tarnished silver contains \(\mathrm{Ag}_{2} \mathrm{~S}\). The tarnish can be removed by placing silverware in an aluminum pan containing an inert electrolyte solution, such as \(\mathrm{NaCl} .\) Explain the electrochemical principle for this procedure. [The standard reduction potential for the half- cell reaction \(\mathrm{Ag}_{2} \mathrm{~S}(s)+2 e^{-} \longrightarrow 2 \mathrm{Ag}(s)+\) \(\mathrm{S}^{2-}(a q)\) is \(-0.71 \mathrm{~V} .1\)

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