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 \(7 \times 10^{8} \mathrm{~mol}\) of \(\mathrm{H}_{2}\) to provide the buoyancy to lift the ship (J. Chem. Educ., Vol. \(50,1973,61\) ). (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?

Short answer

a) The number of coulombs of electrical charge required is: \(Q = n \times F = (7 \times 10^{8} \mathrm{mol}) \times (96485 \mathrm{C/mol}) = 6.754 \times 10^{13} \mathrm{C}\) b) The minimum voltage required for electrolysis is the standard potential for water electrolysis, which is \(1.23 \mathrm{V}\). c) The minimum electrical energy required is: \(E = Q \times V = (6.754 \times 10^{13} \mathrm{C}) \times (1.23 \mathrm{V}) = 8.307 \times 10^{13} \mathrm{J}\) d) The minimum cost of electrical energy required to generate the necessary hydrogen gas is: \(kWh = \frac{8.307 \times 10^{13} \mathrm{J}}{3.6 \times 10^6 \mathrm{J}} = 23.075 \times 10^6 \mathrm{kWh}\) \(Cost = 23.075 \times 10^6 \mathrm{kWh} \times 0.85 = 1.961 \times 10^7 \mathrm{USD}\)

Step-by-step solution

a) Finding the number of coulombs of charge required

To find the coulombs of electrical charge required to generate the given amount of hydrogen gas, we will use Faraday's law of electrolysis. \(n = \frac{Q}{F}\) Where \(n\) is the number of moles of hydrogen gas, \(Q\) is the amount of electrical charge in coulombs and \(F\) is the Faraday's constant (\(96485 \mathrm{C/mol}\)). Given, \(n = 7 \times 10^{8} \mathrm{mol}\). Rearranging the equation to find \(Q\): \(Q = n \times F\) Now, substitute the values and calculate the electrical charge.

b) Finding the minimum voltage required

To find the minimum voltage required to generate hydrogen and oxygen gases at the given pressure, we first need to find the partial pressures of these gases. The pressure on the gases at the depth of the wreckage was given as \(300 \mathrm{atm}\). This pressure is equally divided between hydrogen and oxygen gas. \(P_{H2} = P_{O2} = \frac{300 \mathrm{atm}}{2} = 150 \mathrm{atm}\) Now, using the Nernst equation, we can determine the minimum voltage required for the electrolysis process: \(E = E^{0} - \frac{RT}{nF} \ln{Q}\) Where \(E\) is the cell potential (voltage), \(E^{0}\) is the standard cell potential (\(1.23 \mathrm{V}\) for water electrolysis), \(R\) is the gas constant (\(8.314 \mathrm{J/mol \cdot K}\)), \(T\) is the temperature in Kelvin (assuming room temperature, \(298 \mathrm{K}\)), \(n\) is the number of electrons transferred, and \(Q\) is the reaction quotient. Since we have equal molar amounts of \(H_{2}\) and \(O_{2}\) produced, we can assume their product over the reactant (water) activities equals 1, which means the logarithmic term vanishes: \(E = E^{0}\) Therefore, the minimum voltage required is the standard potential for water electrolysis, which is \(1.23 \mathrm{V}\).

c) Finding the minimum electrical energy required

To find the minimum electrical energy required, we use the formula for the energy in an electrolysis process: \(E = Q \times V\) Where \(Q\) is the electrical charge in coulombs (from part a), and \(V\) is the voltage (from part b). Calculate the minimum electrical energy required by substituting the known values.

d) Finding the minimum cost of electrical energy

Finally, we need to determine the minimum cost of the electrical energy required to generate the necessary hydrogen gas. Given the cost of electricity is 85 cents per kilowatt-hour. We can convert the electrical energy found in part c into kilowatt-hours by dividing it by \(3.6 \times 10^6 \mathrm{J}\). \(kWh = \frac{E}{3.6 \times 10^6 \mathrm{J}}\) Now, calculate the cost by multiplying this value by the cost of electricity in cents per kilowatt-hour: \(Cost = kWh \times 0.85\) Determine the minimum cost of electrical energy required to generate the hydrogen gas.

Key concepts

Faraday's Law

Faraday's Law of Electrolysis is essential in understanding how much electric charge is needed to drive an electrochemical reaction. The law states that the amount of chemical change produced by an electric current is proportional to the quantity of electricity used. This means that the generation of a specific amount of a substance (like hydrogen gas in our problem) is tied directly to the electrical charge flowing through the system.
The key formula for Faraday's Law is:

• \[ n = \frac{Q}{F} \]

Here, \( n \) represents the moles of a substance produced, \( Q \) is the total charge in coulombs, and \( F \) is the Faraday constant, approximately 96485 C/mol. By rearranging this equation, you can determine the total charge required:

• \[ Q = n \times F \]

In our example, given that you need \( 7 \times 10^8 \) moles of hydrogen gas, the calculation involves multiplying these moles by the Faraday constant to find the charge required.

Nernst Equation

The Nernst Equation plays a pivotal role when conditions deviate from standard states, such as in our example where gases are under high pressure. This equation helps determine the actual cell potential required to drive an electrochemical reaction.The Nernst Equation is given by:

• \[ E = E^0 - \frac{RT}{nF} \ln{Q} \]

Where:

• \( E \) represents the cell potential under non-standard conditions,
• \( E^0 \) is the standard cell potential,
• \( R \) is the ideal gas constant (8.314 J/mol·K),
• \( T \) is the temperature in Kelvin,
• \( n \) is the number of moles of electrons exchanged, and
• \( Q \) is the reaction quotient.
• \( \ln{Q} \) accounts for partial pressures of gases involved.

In this scenario, with equal molar production under 300 atm pressure, we use the equation to verify the minimum voltage needed to split water into hydrogen and oxygen is the standard cell potential of 1.23 V. Since partial pressures balance the equation, \( E \) equals \( E^0 \).

Standard Cell Potential

Standard Cell Potential, denoted as \( E^0 \), is a measure of the voltage or electrical energy created by a chemical reaction under standard conditions, which include 25°C, 1 atmosphere of pressure, and 1 molar concentrations.For water electrolysis, the standard cell potential \( E^0 \) is 1.23 V, which is the sum of potentials from both half-reactions involved:

• \[ 2H_2O(l) \rightarrow 2H_2(g) + O_2(g), \quad E^0_{cell} = 1.23 V \]

This value reflects the energy change required for converting liquid water back into its gaseous components, hydrogen and oxygen.The importance of the standard cell potential lies in its utility for predicting the feasibility of a reaction. Only reactions with a positive \( E^0 \) will proceed spontaneously, while a negative \( E^0 \) indicates a non-spontaneous reaction that requires external energy, which is typical for electrolysis.

Electrical Energy Cost

Understanding the cost of electrical energy is important in applications requiring significant energy input, like the proposed Titanic raising.The calculation involves first finding the total energy consumed during the electrolysis process. This is given by:

• \[ E = Q \times V \]

Where \( E \) is energy in joules, \( Q \) is the total charge calculated previously, and \( V \) is the voltage needed to drive the electrolysis. Once obtained, this energy must be converted to kilowatt-hours, a standard measurement for electrical consumption:

• \[ kWh = \frac{E}{3.6 \times 10^6} \]

This conversion is necessary because electricity costs are typically calculated per kilowatt-hour. Once the energy in kWh is known, multiplying by the venue-specific electricity cost gives you the total cost.At 85 cents per kilowatt-hour, multiplying this rate by the calculated energy consumption provides the expense needed to generate the required hydrogen gas.