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 \(7 \times 10^{8}\) 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 \(\mathrm{atm} ?(\mathbf{c})\) 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

Find the required coulombs of charge for the electrolysis to produce H2 gas

We know that it will require \(7 \times 10^8\) moles of H2 to provide the buoyancy to lift the ship. We will use Faraday's law of electrolysis, which states that the amount of substance produced/reacted during electrolysis depends on the quantity of electric charge (Q) passed through the cell. For H2 production from water electrolysis: \[ 2H_2O(l) \rightarrow 2H_2(g) + O_2(g) \] From the balanced equation, we see that one mole of H2 gas requires 2 moles of electrons. So, the total moles of electrons required = \(2 \times 7 \times 10^8\) moles Now, we can find the amount of charge in coulombs using the formula: \[ Q = n \times F \] where n is the amount of substance in moles (moles of electrons) and F is the Faraday constant (approximately \(96485 C/mol\)).

02

Calculate the minimum voltage required to generate H2 and O2 gases

We know the pressure of the gases at the depth of the wreckage is 300 atm. Using the ideal gas law, we can find the relationship between the pressures and potentials for H2 and O2: \( PV = nRT \) For hydrogen gas, we have: \( P_hV = n_hRT \) For oxygen gas, we have: \( P_oV = n_oRT \) Since we need both H2 and O2 gases, we need to maintain a minimum voltage that can generate both gases. The minimum voltage (called the cell potential) can be determined from the Nernst equation: Cell potential = standard cell potential - \(\dfrac{RT}{zF}\ln Q\) For the electrolysis reaction (ignoring overpotentials), the standard cell potential is approximately 1.23V. Assuming room temperature (\(298K\)), we can calculate the minimum voltage required.

03

Calculate the minimum electrical energy required to raise the Titanic

Now that we know the required charge (Q) and the minimum potential (V), we can calculate the minimum electrical energy required to generate the necessary amount of H2 gas using the formula: \[ E = QV \]

04

Calculate the minimum cost of the electrical energy

We know the electricity cost is 85 cents per kilowatt-hour. First, we need to convert the energy calculated in step 3 to kilowatt-hours: \[ E_{kWh} = E_{J} \times \dfrac{1 kW}{1000 W} \times \dfrac{1 h}{3600 s} \] Now, we can find the minimum cost of the required energy by multiplying the energy (in kilowatt-hours) by the price: \[ Cost = E_{kWh} \times cost\ per\ kilowatt-hour \]

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