Problem 110 - Chemistry The Central Science

Question

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

Answer

Using the given information and solving through the steps, we get: (a) The required coulombs of electrical charge for the electrolysis to produce H2 gas is approximately \(2.7 \times 10^{10}\, C\). (b) The minimum voltage required to generate H2 and O2 gases at the depth of the wreckage at 300 atm pressure is approximately 1.42 V. (c) The minimum electrical energy required to raise the Titanic by electrolysis is approximately \(3.8 \times 10^{10}\, J\). (d) The minimum cost of the electrical energy required to generate the necessary H2 gas to raise the Titanic at an electricity cost of 85 cents per kW-h is about \$162,000.

Step 1: 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 \(96485 C/mol\)).

Step 2: 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.

Step 3: 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: \[ E = QV \]

Step 4: 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: \[ Cost = E_{kWh} \times cost\ per\ kilowatt-hour \]

Key Concepts

Faraday's law

Faraday's law of electrolysis is a principle that helps us understand how electricity can be used to cause a chemical change, specifically in electrolysis processes. In electrolysis, electrical energy is used to drive a non-spontaneous chemical reaction, like splitting water molecules into hydrogen and oxygen gas. Faraday's law states that the amount of chemical change produced by an electric current is proportional to the amount of electricity used. In mathematical terms, this is explained by the formula:\[Q = n imes F\]where:

• \(Q\) is the total electric charge in coulombs,
• \(n\) is the number of moles of electrons required for the reaction, and
• \(F\) is Faraday's constant, approximately \(96,485\) coulombs per mole of electrons.

With Faraday's law, we can calculate how much electrical charge is needed to produce a desired quantity of substance in an electrolysis process. In the Titanic's case, this means ensuring we have enough charge to produce all the hydrogen gas needed to float the ship.

Coulombs

Coulombs are the standard unit of electric charge in the International System of Units (SI). They measure how much electricity is flowing through or is stored in an object. Just like meters measure distance, coulombs measure electric charge.In electrolysis, calculating the number of coulombs is essential because it tells us how much electric charge passes through the electrolytic cell to produce a specific amount of gas, like hydrogen. The formula used to find the quantity of charge is\[Q = n \times F\]where \(Q\) stands for charge in coulombs, \(n\) represents the number of moles of electrons, and \(F\) is the Faraday constant. Knowing the number of coulombs helps scientists and engineers determine the amount of electricity needed to complete a chemical reaction during electrolysis. For example, if one mole of electrons carries approximately 96,485 coulombs, then finding how many moles of electrons are required to produce the needed hydrogen gives us the total charge required.

Cell Potential

Cell potential, or electromotive force (EMF), refers to the potential difference between two electrodes in an electrochemical cell. It indicates the cell’s ability to drive an electric current through an external circuit. In simpler terms, it shows how well the cell can make electricity flow, which is crucial in processes like electrolysis.For electrolysis, the minimum cell potential needed to cause a reaction can be calculated using the Nernst equation. The cell potential's value will vary depending on conditions such as temperature and pressure, particularly in scenarios like deep-water environments with high pressure.Electrolysis requires sufficient voltage to split water into hydrogen and oxygen. The standard cell potential for this reaction is approximately \(1.23\text{V}\) under standard conditions. However, at different pressures and temperatures, the value may change, and more energy might be needed to achieve the necessary split.Understanding cell potential helps ensure that enough voltage is applied to drive the desired chemical reaction effectively in an electrolysis cell.

Nernst Equation

The Nernst equation is a formula that relates the electromotive force (EMF) of an electrochemical cell to the standard electrode potential, temperature, and the reaction quotient. It provides a valuable tool for predicting how the cell potential will change based on different concentrations of reactants and products.The Nernst equation is given by:\[E = E^0 - \dfrac{RT}{nF} \ln Q\]where:

• \(E\) is the cell potential at non-standard conditions,
• \(E^0\) is the standard cell potential,
• \(R\) is the universal gas constant (8.314 J/(mol K)),
• \(T\) is the temperature in Kelvin,
• \(n\) is the number of moles of electrons exchanged in the reaction, and
• \(Q\) is the reaction quotient, representing the concentrations or partial pressures of reactants and products.

Using the Nernst equation, we can pinpoint the precise voltage needed under specific conditions, such as high pressures like those found at the Titanic wreck site. By adjusting for these variables, engineers can calculate the voltage necessary to efficiently produce the required hydrogen and oxygen gas through electrolysis.