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Chapter 18: Problem 127

A galvanic cell using \(\mathrm{Mg} / \mathrm{Mg}^{2+}\) and \(\mathrm{Cu} / \mathrm{Cu}^{2+}\) halfcells operates under standard-state conditions at \(25^{\circ} \mathrm{C}\) and each compartment has a volume of \(218 \mathrm{~mL}\). The cell delivers 0.22 A for \(31.6 \mathrm{~h}\). (a) How many grams of \(\mathrm{Cu}\) are deposited? (b) What is the \(\left[\mathrm{Cu}^{2+}\right]\) remaining?

Short Answer

Expert verified
a) To calculate the mass of deposited copper, calculate the total charge passed through the cell, convert this into moles of electrons transferred using Faraday's constant, and then convert this to grams of copper using the stoichiometry of the copper half-cell reaction and the atomic mass of copper. b) The remaining concentration of \([Cu^{2+}]\) is determined by calculating the initial moles of \([Cu^{2+}]\) assuming a 1 M concentration, subtracting the moles equivalent to the grams of copper deposited, and then dividing this by the volume of the cell compartments.

Step by step solution

01

Calculate the total charge

The total charge passed, Q can be calculated using the formula: Q = I.t. Here, 'I' is the current in Amps, and 't' is the time in seconds. The current, I given in the Problem is 0.22 Amps and the time, t is given as 31.6 hours. We first need to convert this time to seconds as the base unit for time in SI units is seconds. So, t = 31.6 * 60 * 60 seconds.
02

Calculate the moles of electrons

The moles of electrons transferred can be calculated using the formula, n = Q/F where 'F' is the Faraday constant, which equals \(96485 \, C/mol^{−1}\). Substituting 'Q' calculated from step 1, we get the number of moles of electrons transferred.
03

Calculate the mass of Cu deposited

The reaction at the copper electrode is \(Cu^{2+} + 2e^{-} → Cu\). From this, we can understand that for every mole of copper deposited, two moles of electrons will be used up. Therefore, the moles of copper deposited are equal to the moles of electrons transferred in Step 2 divided by 2. The mass of copper deposited in grams can then be calculated by multiplying the moles of copper deposited by the atomic mass of copper 'M', given as 63.5 g/mol.
04

Calculate the remaining concentration of \([Cu^{2+}]\)

The remaining concentration of \([Cu^{2+}]\) can be calculated by subtracting the initial moles of \([Cu^{2+}]\) (which can be calculated assuming a 1 M initial concentration for standard-state conditions) by the moles of copper deposited calculated in the previous step. The molar concentration is then obtained by dividing the resulting number of moles of \([Cu^{2+}]\) by the total volume of both compartments (436 mL total, converted to liters).

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Most popular questions from this chapter

Steel hardware, including nuts and bolts, is often coated with a thin plating of cadmium. Explain the function of the cadmium layer.

Show a sketch of a galvanic concentration cell. Each compartment consists of a Co electrode in a \(\mathrm{Co}\left(\mathrm{NO}_{3}\right)_{2}\) solution. The concentrations in the compartments are \(2.0 \mathrm{M}\) and \(0.10 \mathrm{M}\), respectively. Label the anode and cathode compartments. Show the direction of electron flow. (a) Calculate the \(E_{\text {cell }}\) at \(25^{\circ} \mathrm{C}\). (b) What are the concentrations in the compartments when the \(E_{\text {cell }}\) drops to 0.020 V? Assume volumes to remain constant at \(1.00 \mathrm{~L}\) in each compartment.

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