Some years ago a unique proposal was made to raise the Titanic. The plan involved placing pontoons within the ship using a surface-controlled submarine-type vessel. The pontoons would contain cathodes and would be filled with hydrogen gas formed by the electrolysis of water. It has been estimated that it would require about \(7 \times 10^{8} \mathrm{~mol}\) of \(\mathrm{H}_{2}\) to provide the buoyancy to lift the ship (J. Chem. Educ, \(1973,\) Vol. 50,61 ). (a) How many coulombs of electrical charge would be required? (b) What is the minimum voltage required to generate \(\mathrm{H}_{2}\) and \(\mathrm{O}_{2}\) if the pressure on the gases at the depth of the wreckage ( \(2 \mathrm{mi}\) ) is \(300 \mathrm{~atm} ?\) (c) What is the minimum electrical energy required to raise the Titanic by electrolysis? (d) What is the minimum cost of the electrical energy required to generate the necessary \(\mathrm{H}_{2}\) if the electricity costs 85 cents per kilowatt-hour to generate at the site?

Step by step solution

01

(a) Find the number of coulombs required to generate the hydrogen gas

To find the number of coulombs required, we must first convert the number of moles of hydrogen gas into individual hydrogen atoms. Then, we'll use Faraday's Law to relate the number of moles to the required electrical charge. To generate hydrogen gas, we need \(2 \times 7 \times 10^8 \mathrm{~mol}\) of hydrogen ions. Using Faraday's constant (\(1 \mathrm{F} = 96,485 \mathrm{~C/mol}\)), we calculate the required coulombs: Charge (Q) = \(\mathrm{Moles~of~Hydrogen~Ions \times F}\) Q = \((2 \times 7 \times 10^8 \mathrm{~mol}) \times (96485 \mathrm{~C/mol})\) Q = \(1.35 \times 10^{13} \mathrm{~C}\) Thus, the number of coulombs of electrical charge required is approximately \(1.35 \times 10^{13} \mathrm{~C}\).

02

(b) Calculate the minimum voltage required for electrolysis

The minimum voltage is given by the Nernst Equation: \(E = E^o - \frac{RT}{nF} \ln \frac{[P(\mathrm{H_2})][P(\mathrm{O_2})^{1/2}]}{[P(\mathrm{H_2O})]}\) Assuming standard conditions (\(E^o = 1.23\) V), E = \(1.23 - \frac{8.314 \times 298}{2 \times 96485} \ln \frac{(300)^{1/2}}{1}\) E = \(1.23 - 0.040 \ln (300)\) E = \(1.01 \mathrm{~V}\) The minimum voltage to generate the hydrogen and oxygen gas at 300 atm pressure is approximately 1.01 V.

03

(c) Calculate the minimum electrical energy required

The energy required (in joules) can be calculated using the formula: Energy (E) = Q * V E = \(1.35 \times 10^{13} \mathrm{~C} \times 1.01 \mathrm{~V}\) E = \(1.36 \times 10^{13} \mathrm{~J}\) The minimum electrical energy required to raise the Titanic by electrolysis is approximately \(1.36 \times 10^{13} \mathrm{~J}\).

04

(d) Compute the minimum cost of electrical energy

Convert the energy in joules to kilowatt-hours: Energy (in kilowatt-hours) = \(\frac{1.36 \times 10^{13} \mathrm{~J}}{3.6 \times 10^6 \mathrm{~J/kWh}}\) Energy = \(3.78 \times 10^6 \mathrm{~kWh}\) Now, calculate the cost of the energy required: Cost ($) = Energy (in kilowatt-hours) * Cost per kilowatt-hour Cost = \(3.78 \times 10^6 \mathrm{~kWh} \times 0.85 \mathrm{~\$ /kWh}\) Cost = \(3.21 \times 10^6 \mathrm{\$}\) The minimum cost of the electrical energy required to generate the necessary hydrogen gas is approximately 3.21 million dollars.

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