Problem 74 - Chemistry The Molecular Science

Question

A hydrogen-oxygen fuel cell operates on the simple reaction $$ 2 \mathrm{H}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(\ell) $$ If the cell is designed to produce \(1.5 \mathrm{~A}\) of current, determine how long it can operate if there is an excess of oxygen and only sufficient hydrogen to fill a \(1.0-\mathrm{L}\) tank at 200. bar pressure at \(25^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

The cell can operate for approximately 2.88 hours.

Step 1: Use Ideal Gas Law to find moles of H₂

First, we need to determine the number of moles of hydrogen gas available. Use the ideal gas law equation: \[ PV = nRT \] where:- \( P = 200 \text{ bar} = 200,000 \text{ Pa} \) (converted to Pa),- \( V = 1.0 \text{ L} = 0.001 \text{ m}^3 \) (converted to cubic meters),- \( R = 8.314 \text{ J/mol K} \) (universal gas constant),- \( T = 25 + 273.15 = 298.15 \text{ K} \) (temperature in Kelvin).Rearrange the equation to solve for \( n \):\[ n = \frac{PV}{RT} = \frac{200,000 \times 0.001}{8.314 \times 298.15} \approx 0.0806 \text{ moles of } \text{H}_2 \]

Step 2: Determine moles of H₂O produced

The balanced chemical reaction \( 2 \text{H}_2 + \text{O}_2 \rightarrow 2 \text{H}_2\text{O} \) shows that 2 moles of \( \text{H}_2 \) produce 2 moles of \( \text{H}_2\text{O} \). Thus, 0.0806 moles of \( \text{H}_2 \) will produce 0.0806 moles of \( \text{H}_2\text{O} \).

Step 3: Calculate total charge using Faraday's law

According to Faraday's law of electrolysis, the amount of substance altered at an electrode during electrolysis is proportional to the quantity of electricity that passes through the cell. The total charge \( Q \) required is given by:\[ Q = nF\text{ (mol of electrons)} \]where \( F = 96485 \text{ C/mol} \) is the Faraday constant. For the balanced reaction, 4 moles of electrons are transferred for every 2 moles of \( \text{H}_2 \). Thus:\[ Q = 0.0806 \times 2 \times 96485 = 15549.204 \text{ C} \]

Step 4: Find operating time from current and charge

The current \( I \) is given as 1.5 A. The time \( t \) in seconds for which the cell can operate is given by the formula:\[ I = \frac{Q}{t} \]Rearrange to solve for \( t \):\[ t = \frac{Q}{I} = \frac{15549.204}{1.5} \approx 10366.136 \text{ seconds} \]Convert this into hours: \[ t = \frac{10366.136}{3600} \approx 2.88 \text{ hours} \]

Key Concepts

Fuel Cell

A fuel cell is a device that converts chemical energy directly into electrical energy. It does this by undergoing an electrochemical reaction. Unlike batteries that store energy, fuel cells require a continuous supply of fuel and an oxidizing agent to sustain the reaction.
Fuel cells are used as a cleaner energy alternative, as they produce electricity without combustion. They offer higher efficiency compared to traditional combustion engines.

• They typically operate with hydrogen as the fuel and oxygen as the oxidant.
• One of their primary advantages is the low environmental impact as they mainly emit water vapor as a byproduct.

Fuel cells are versatile and can be used in various applications, including powering vehicles, and even providing electricity in remote areas. Understanding how they work enhances our approach to sustainable energy solutions.

Hydrogen-Oxygen Reaction

The hydrogen-oxygen reaction is the fundamental chemical reaction that occurs in a hydrogen-oxygen fuel cell. In this reaction, hydrogen gas (\( ext{H}_2 \)) and oxygen gas (\( ext{O}_2 \) ) combine to form water (\( ext{H}_2 ext{O} \)). The balanced equation is:\[ 2 ext{H}_2 + ext{O}_2 ightarrow 2 ext{H}_2 ext{O} \] This reaction is exothermic, meaning it releases energy, which is harnessed to generate electricity in a fuel cell.
When this reaction occurs in the fuel cell, electrons are released and flow through an external circuit, providing electrical power. This reaction is efficient and the only byproduct is water, making it an environmentally friendly option for energy production.

• The reaction needs specific conditions: stable temperature, pressure, and proper catalysts to maintain the flow of electrons.
• The efficiency of a fuel cell largely depends on the control of this reaction.

Ideal Gas Law

The ideal gas law is expressed by the equation \( PV = nRT \). It relates the pressure (\( P \)), volume (\( V \)), and temperature (\( T \)) of a gas to the number of moles (\( n \)) and a constant (\( R \)). This law is crucial in calculating the number of moles of gas in a given volume at a specific temperature and pressure.
In the context of the fuel cell exercise, it helps determine how much hydrogen gas is available to sustain the electrochemical reaction.

• The pressure must be in pascals, and the volume in cubic meters for standard calculations.
• The temperature should be converted to Kelvin to align with the universal gas constant.

Using the ideal gas law aids in understanding how gaseous reactants behave under different conditions, allowing for efficient fuel cell design.

Faraday's Law

Faraday's law of electrolysis states that the amount of substance produced or consumed at an electrode during electrolysis is proportional to the electric charge that passes through the circuit. It provides a quantitative relationship between the amount of electric charge and the chemical reaction taking place at the electrodes.
Using Faraday’s law, we calculate how much charge is needed for a specific amount of substance. In a fuel cell context, the number of moles of electrons required to transform hydrogen and oxygen into water indicates the total charge involved.

• The Faraday constant (\( F = 96485 ext{ C/mol} \)) facilitates these calculations by providing the charge per mole of electrons.
• For the given reaction in the problem, proper calculation confirms the cell’s operating time based on available reactants and current.

Understanding Faraday's law is vital for optimizing the efficiency and output of electrochemical cells.