A chemist wishes to determine the concentration of CrO4^2- electrochemically. A cell is construc

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A chemist wishes to determine the concentration of CrO4^2-...

A chemist wishes to determine the concentration of $\mathrm{CrO}_{4}^{2-}$ electrochemically. A cell is constructed consisting of a saturated calomel electrode (SCE; see Exercise 115$)$ and a silver wire coated with $\mathrm{Ag}_{2} \mathrm{Cr} \mathrm{O}_{4}$. The $\mathscr{C}^{\circ}$ value for the following halfreaction is $+0.446 \mathrm{~V}$ relative to the standard hydrogen electrode: $$\mathrm{Ag}_{2} \mathrm{CrO}_{4}+2 \mathrm{e}^{-} \longrightarrow 2 \mathrm{Ag}+\mathrm{CrO}_{4}^{2-}$$ a. Calculate $\mathscr{E}_{\text {cell }}$ and $\Delta G$ at $25^{\circ} \mathrm{C}$ for the cell reaction when $\left[\mathrm{CrO}_{4}^{2-}\right]=1.00 \mathrm{~mol} / \mathrm{L}$ b. Write the Nernst equation for the cell. Assume that the SCE concentrations are constant. c. If the coated silver wire is placed in a solution (at $25^{\circ} \mathrm{C}$ ) in which $\left[\mathrm{CrO}_{4}^{2-}\right]=1.00 \times 10^{-5} M$, what is the expected cell potential? d. The measured cell potential at $25^{\circ} \mathrm{C}$ is $0.504 \mathrm{~V}$ when the coated wire is dipped into a solution of unknown $\left[\mathrm{Cr} \mathrm{O}_{4}{ }^{2-}\right]$. What is $\left[\mathrm{CrO}_{4}^{2-}\right]$ for this solution? e. Using data from this problem and from Table $18.1$, calculate the solubility product $\left(K_{\mathrm{sp}}\right)$ for $\mathrm{Ag}_{2} \mathrm{CrO}_{4}$.

A chemist wishes to determine the concentration of $\mathrm{CrO}_{4}^{2-}$ electrochemically. A cell is constructed consisting of a saturated calomel electrode (SCE; see Exercise 115$)$ and a silver wire coated with $\mathrm{Ag}_{2} \mathrm{Cr} \mathrm{O}_{4}$. The $\mathscr{C}^{\circ}$ value for the following halfreaction is $+0.446 \mathrm{~V}$ relative to the standard hydrogen electrode:
$$\mathrm{Ag}_{2} \mathrm{CrO}_{4}+2 \mathrm{e}^{-} \longrightarrow 2 \mathrm{Ag}+\mathrm{CrO}_{4}^{2-}$$
a. Calculate $\mathscr{E}_{\text {cell }}$ and $\Delta G$ at $25^{\circ} \mathrm{C}$ for the cell reaction when $\left[\mathrm{CrO}_{4}^{2-}\right]=1.00 \mathrm{~mol} / \mathrm{L}$
b. Write the Nernst equation for the cell. Assume that the SCE concentrations are constant.
c. If the coated silver wire is placed in a solution (at $25^{\circ} \mathrm{C}$ ) in which $\left[\mathrm{CrO}_{4}^{2-}\right]=1.00 \times 10^{-5} M$, what is the expected cell potential?
d. The measured cell potential at $25^{\circ} \mathrm{C}$ is $0.504 \mathrm{~V}$ when the coated wire is dipped into a solution of unknown $\left[\mathrm{Cr} \mathrm{O}_{4}{ }^{2-}\right]$. What is $\left[\mathrm{CrO}_{4}^{2-}\right]$ for this solution?
e. Using data from this problem and from Table $18.1$, calculate the solubility product $\left(K_{\mathrm{sp}}\right)$ for $\mathrm{Ag}_{2} \mathrm{CrO}_{4}$.

Key Concepts

Electrochemical Cells

Electrochemical cells are systems where chemical energy is converted into electrical energy through redox reactions occurring at two electrodes. In these cells, a spontaneous reaction drives electron flow from the anode to the cathode, generating a measurable cell potential. Understanding how each half?cell contributes to the overall cell potential is crucial for analyzing and designing cells used for sensing or energy conversion.

Standard Electrode Potentials

Standard electrode potentials provide a reference for the intrinsic tendency of a chemical species to acquire electrons and be reduced under standard conditions. They are used in conjunction with half-reactions to predict the direction of spontaneous reactions and to calculate the overall cell potential. This concept is fundamental to understanding how different electrodes interact within an electrochemical cell.

Nernst Equation

The Nernst equation relates the electrode potential of a half-cell to its standard electrode potential and the activities (or concentrations) of the chemical species involved in the reaction. It shows how deviations from standard conditions (such as changes in ion concentration) affect the potential. This relationship is key for quantitatively analyzing sensors and electrochemical cells under non-standard conditions.

Gibbs Free Energy and its Relation to Cell Potential

Gibbs free energy change (?G) indicates the spontaneity of a reaction and is directly related to the cell potential via the equation ?G = -nFE, where n is the number of electrons transferred and F is Faraday's constant. This concept connects thermodynamics and electrochemistry by showing how electrical work can be harnessed from chemical reactions.

Solubility Product (Ksp)

The solubility product, Ksp, is a constant that quantifies the extent to which a sparingly soluble salt dissolves in water. It governs the relationship between the concentrations of the ions produced when the salt dissolves, and is often determined by combining electrochemical measurements with equilibrium calculations. Understanding Ksp is essential in analyzing precipitation, dissolution equilibria, and the stability of compounds in solution.

Reference Electrodes

Reference electrodes, such as the saturated calomel electrode (SCE), serve as stable and reproducible reference points for measuring electrode potentials. Their known and constant potential allows for accurate description and comparison of the potentials of other electrodes in an electrochemical cell. This is critical in electrochemical analysis where accurate potential measurements dictate the reliability of concentration or kinetic studies.

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