A teapot with a surface area of 630 cm2 is to be plated with silver. It is attached to the negative electrode of an electrolytic cell containing silver nitrate (Ag+ NO3−). The cell is powered by a 12.0-V battery and has a resistance of 1.90 Ω. If the density of silver is 1.05 104 kg/m3, over what time interval does a 0.133-mm layer of silver build up on the teapot?
Question
A teapot with a surface area of 630 cm2 is to be plated with silver. It is attached to the negative electrode of an electrolytic cell containing silver nitrate (Ag+ NO3−). The cell is powered by a 12.0-V battery and has a resistance of 1.90 Ω. If the density of silver is 1.05 104 kg/m3, over what time interval does a 0.133-mm layer of silver build up on the teapot?
Solution
To solve this problem, we need to follow these steps:
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Convert the thickness of the silver layer from millimeters to meters: 0.133 mm = 0.133 x 10^-3 m.
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Calculate the volume of the silver layer: Volume = Surface Area x Thickness = 630 x 10^-4 m^2 x 0.133 x 10^-3 m = 8.379 x 10^-8 m^3.
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Calculate the mass of the silver layer: Mass = Volume x Density = 8.379 x 10^-8 m^3 x 1.05 x 10^4 kg/m^3 = 8.798 x 10^-4 kg.
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Convert the mass of the silver layer to moles: Moles = Mass / Molar Mass. The molar mass of silver (Ag) is approximately 107.87 g/mol, so Moles = 8.798 x 10^-4 kg x (1000 g/1 kg) / 107.87 g/mol = 0.00816 mol.
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Use Faraday's Law of Electrolysis to calculate the charge needed to deposit this amount of silver. Faraday's Law states that the amount of substance deposited at an electrode during electrolysis is directly proportional to the amount of electricity passed through the cell. The charge (Q) needed to deposit one mole of silver is equal to the product of the number of moles (n), Faraday's constant (F = 96485 C/mol), and the number of electrons involved in the reaction (z). For silver, z = 1, so Q = nFz = 0.00816 mol x 96485 C/mol x 1 = 787.6 C.
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Use Ohm's Law (V = IR) to calculate the current (I) flowing through the cell: I = V / R = 12.0 V / 1.90 Ω = 6.32 A.
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Calculate the time needed for this amount of charge to flow: Time = Charge / Current = 787.6 C / 6.32 A = 124.7 s.
So, it would take approximately 125 seconds for a 0.133-mm layer of silver to build up on the teapot.
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