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 \(7 \times 10^{4} \mathrm{~mol}\) 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 \- - the electricity costs 85 cents per kilowatt-

Step-by-Step Solution

Answer

(a) The required electrical charge (Q) to produce the hydrogen gas is \(Q = 1.35 \times 10^9 \mathrm{~C}\). (b) The minimum voltage (V) required to generate hydrogen and oxygen gases is \(E \approx 0.019 \mathrm{~V}\). (c) The minimum electrical energy (E) required to raise the Titanic by electrolysis is \(E \approx 2.57 \times 10^7 \mathrm{~J}\). (d) The total cost of electricity to raise the Titanic is approximately $578.45.

Step 1: (a) Finding the Required Electrical Charge

To find the required electrical charge to produce the necessary hydrogen gas, we need to multiply the moles of hydrogen gas required by the charge of one mole of electron (Faraday's constant). Here, we are given the necessary moles of hydrogen gas, \(7 \times 10^{4} \mathrm{~mol}\), and the Faraday constant, \(F = 96485 \mathrm{~C/mol}\). Let's denote the required electrical charge as \(Q\). The reaction to produce hydrogen gas is: \[2 H^{+} + 2 e^{-} \rightarrow H_{2}\] From the stoichiometry of the reaction, we can see that 2 moles of electrons are required to produce 1 mole of hydrogen gas. Therefore, we can determine the charge needed using the following equation: \[Q = \frac{(7 \times 10^{4} \mathrm{~mol})\times ( 2 \mathrm{~mol~of~electrons}) \times ( 96485 \mathrm{~C/mol})}{(1 \mathrm{~mol~of~H_{2}})}\]

Step 2: (b) Finding the Minimum Voltage

To find the minimum voltage required to generate both hydrogen and oxygen gas, we need to first calculate the cell potential using the Nernst equation: \[E = E^{0} - \frac{RT}{nF} \ln Q\] At the depth of the wreckage, we are given the pressure, \(P = 300 \mathrm{~atm}\). We are interested in the generation of hydrogen gas, so we will analyze the reaction half-cell: \[2 H^{+} + 2 e^{-} \rightarrow H_{2}\] For this reaction, the standard cell potential, \(E^{0} = 0\). The number of electrons being transferred, \(n = 2\), and we assume the temperature to be \(T = 298 \mathrm{~K}\). The gas constant, \(R = 8.314 \mathrm{~J/(mol \cdot K)}\). Finally, the reaction quotient, \(Q = \frac{[H_{2}]}{[H^{+}]^{2}}\). We will assume [\(H^{+}\)] = 1 M and [\(H_{2}\)] = 300 atmospheres. (since the pressure on the gases is 300 atm) Then we can plug these values into the Nernst equation to find the minimum cell potential: \[E = 0 - \frac{(8.314 \mathrm{~J/(mol \cdot K)})\times (298 \mathrm{~K})}{(2 \times 96485 \mathrm{~C/mol})} \ln \frac{300}{1^{2}}\]

Step 3: (c) Finding the Minimum Electrical Energy

To calculate the minimum electrical energy needed to raise the Titanic using electrolysis, we will use the following equation (where energy is denoted by \(E\), charge by \(Q\), and voltage by \(V\)): \[E = Q \times V\] We already found \(Q\) and \(V\), so we can plug those values into this equation.

Step 4: (d) Calculating the Electricity Cost

Finally, we need to calculate the cost of the electricity needed to raise the Titanic. We are given that the cost is 85 cents per kilowatt-hour. First, we need to convert the energy we found earlier in (c), which is in Joules, to kilowatt-hours by using the following conversion factors: \[1 \mathrm{~kW} = 1000 \mathrm{~W}\] \[1 \mathrm{~Wh} = 3600 \mathrm{~J}\] Considering these conversion factors, we can convert the energy as follows: \[\mathrm{Energy~(kWh)} = \frac{ \mathrm{Energy~(J)} }{(1000 \mathrm{~W/kW}\ ) \times (3600 \mathrm{~J/Wh})}\] Now that we have the energy in kilowatt-hours, we can multiply it by the given cost of electricity to find the total cost: \[ \mathrm{Total~Cost} = \mathrm{Energy~(kWh)} \times 0.85 \mathrm{~$/kWh}\]

Key Concepts

Faraday's Constant

Understanding the electrolysis of water is underpinned by a key concept known as Faraday's constant. This is a fundamental value in electrochemistry, representing the amount of electrical charge carried by one mole of electrons. Faraday's constant, denoted as \(F\), has a value of approximately \(96,485 \mathrm{~C/mol}\).

During electrolysis, electrical current causes chemical reactions at the electrodes. For instance, the electrolysis of water produces hydrogen and oxygen gases. When quantifying the amount of charge required to produce a certain amount of gas, Faraday's constant is used to translate moles of electrons into coulombs of charge.

For example, when considering the proposal to raise the Titanic, the calculated moles of \(\mathrm{H}_{2}\) needed to be converted into the total charge required using Faraday's constant in this manner: \[\begin{equation}Q = \text{moles of } \mathrm{H}_{2} \times \text{moles of electrons per mole of } \mathrm{H}_{2} \times F\end{equation}\]Here, the stoichiometry reveals that it takes two moles of electrons to produce one mole of hydrogen gas, which means Faraday's constant is fundamental in linking the chemical reaction happening at the molecular level to the electrical charge needed for the reaction to occur.

Nernst Equation

The Nernst equation is a key principle in electrochemistry that allows us to calculate the electric potential of an electrochemical cell under non-standard conditions. It takes into consideration the effect of ion concentration on the voltage of the cell.

The equation is expressed as:\[\begin{equation}E = E^{0} - \frac{RT}{nF} \ln Q\end{equation}\]where \(E\) is the cell potential, \(E^{0}\) is the standard cell potential, \(R\) is the gas constant, \(T\) is the temperature in Kelvin, \(n\) is the number of moles of electrons transferred in the reaction, \(F\) is Faraday's constant, and \(Q\) is the reaction quotient, which reflects the ratio of the concentrations of the products to reactants.

Applying the Nernst equation in our Titanic example is vital for determining the minimum voltage needed to conduct electrolysis deep underwater. We need to adjust the equation for the increased pressure, which affects the concentration of gases, and ultimately, the required voltage to initiate the electrolysis process to produce hydrogen and oxygen gases.

Electrical Energy

Electrical energy is the energy derived from electric potential energy or kinetic energy. It is essentially harnessed and used to produce various forms of work, in this case, driving the electrolysis of water. When calculating the minimum electrical energy required to raise the Titanic via electrolysis, we consider the voltage and the charge necessary for the reaction.

The formula used is:\[\begin{equation}E = Q \times V\end{equation}\]where \(E\) is the energy in Joules, \(Q\) is the total charge in coulombs, and \(V\) is the voltage in volts. Calculating the electrical energy is crucial in this context because it translates the electrochemical reaction into tangible work - lifting the Titanic using hydrogen gas generated from electrolysis. By understanding the electrical energy involved, we also pave the way for calculating the costs associated with such an endeavor.

Stoichiometry

Stoichiometry is the study of the quantitative relationships or ratios between reactants and products in chemical reactions. In the context of electrolysis, stoichiometry allows us to predict the amount of reactants required and the amount of products formed.

The electrolysis of water involves a specific stoichiometric ratio, where two moles of water yield two moles of \(\mathrm{H}_{2}\) and one mole of \(\mathrm{O}_{2}\). Thus, for every mole of \(\mathrm{H}_{2}\) produced, two moles of electrons are consumed. Understanding this ratio is vital when calculating the total charge needed for the reaction.

The exercise to calculate the charge needed to produce the hydrogen gas for the Titanic project relies on a clear grasp of stoichiometry to ensure accurate estimation of the required materials and the consequent cost. It's the foundation that bridges our understanding of the principles of chemistry with practical applications and outcomes.