Car manufacturers are developing engines that use \(\mathrm{H}_{2}\) as fuel. In Iceland, Sweden, and other parts of Scandinavia, where hydroelectric plants produce inexpensive electric power, the \(\mathrm{H}_{2}\) can be made industrially by the electrolysis of water. (a) How many coulombs are needed to produce \(3.5 \times 10^{6} \mathrm{~L}\) of \(\mathrm{H}_{2}\) gas at \(12.0 \mathrm{~atm}\) and \(25^{\circ} \mathrm{C} ?\) (Assume that the ideal gas law applies.) (b) If the coulombs are supplied at \(1.44 \mathrm{~V}\), how many joules are produced? (c) If the combustion of oil yields \(4.0 \times 10^{4} \mathrm{~kJ} / \mathrm{kg},\) what mass of oil must be burned to yield the number of joules in part (b)?

Step by step solution

01

- Use the Ideal Gas Law to find moles of \text{H}_{2}

First, use the Ideal Gas Law equation, \[ PV = nRT \] to find the number of moles of \text{H}_{2}. Given: \( P = 12.0 \text{ atm} \), \( V = 3.5 \times 10^{6} \text{ L} \), \( R = 0.0821 \text{ atm·L·mol}^{-1}\text{K}^{-1} \), and \( T = 25^{\text{\circ}} \text{C} = 298 \text{ K} \). Rearrange the equation to solve for \(n\): \[ n = \frac{PV}{RT} \].

02

- Calculate moles of \text{H}_{2}

Substitute the known values into the rearranged Ideal Gas Law: \[ n = \frac{(12.0 \text{ atm})(3.5 \times 10^{6} \text{ L})}{(0.0821 \text{ atm·L·mol}^{-1}\text{K}^{-1})(298 \text{ K})} \]. Calculate to find the moles of \text{H}_{2}.

03

- Use Faraday's Law to find the coulombs needed

Use Faraday's Law, \[ Q = nFN \], where \( n \) is the number of moles of electrons required and \( F = 96485 \text{ C/mol} \). For the electrolysis of water, 2 moles of electrons are needed to produce 1 mole of \text{H}_{2}. Calculate \( n_F = 2n \) and then substitute into the equation.

04

- Calculate the number of coulombs

Calculate the total charge in coulombs based on the moles of \text{H}_{2} found in Step 2. \[ Q = 2n \times 96485 \text{ C/mol} \]

05

- Calculate the number of joules

Use the relation \[ E = QV \], where \( Q \) is the charge in coulombs found in Step 4, and \( V = 1.44 \text{ V} \) to find the energy in joules.

06

- Calculate the mass of oil needed

Given the energy yield from combustion of oil, \( 4.0 \times 10^{4} \text{ kJ/kg} \), convert the energy from joules to kilojoules, and then find the mass of oil required. Rearrange the formula, \[ \text{Mass} = \frac{E \text{ (in kJ)}}{4.0 \times 10^{4} \text{ kJ/kg}} \], to solve for the mass.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ideal Gas Law

Understanding the Ideal Gas Law is crucial when dealing with gases in various conditions. It states that for a given amount of gas, the product of its pressure (P) and volume (V) is proportional to its temperature (T). The formula is \[ PV = nRT \] where

• P is the pressure
• V is the volume
• n is the number of moles
• R is the universal gas constant, 0.0821 atm·L·mol^-1K^-1
• T is the temperature in Kelvin

For the given problem, the pressure (P) is 12.0 atm, the volume (V) is 3.5 x 10^6 L, and the temperature (T) after converting from 25°C to Kelvin is 298 K. By substituting these values into the Ideal Gas Law formula, you can calculate the number of moles of hydrogen gas, n, present.

Faraday's Law of Electrolysis

Faraday's Law of Electrolysis helps determine the number of coulombs needed to produce a substance via electrolysis. The formula is \[ Q = nFN \] where

• Q is the total charge in coulombs
• n is the number of moles of the substance (from the Ideal Gas Law calculation)
• F is Faraday's constant, 96485 C/mol
• N is the number of moles of electrons per mole of substance

For breaking down water into hydrogen gas, two moles of electrons are needed to produce one mole of hydrogen gas (H₂). Therefore, you multiply the moles of hydrogen gas by 2 to get the total moles of electrons, and then multiply by Faraday's constant to get the total charge in coulombs.

Energy Conversion

In these calculations, we often convert between different types of energy. To convert the electrical energy (in coulombs) into joules, we use the formula: \[ E = QV \] where

• E is the energy in joules
• Q is the charge in coulombs (calculated using Faraday's Law)
• V is the voltage supplied, which is given as 1.44 V in the problem

Multiplying these values gives you the energy in joules. Additionally, when comparing this energy to another form, such as the energy obtained from burning oil, you need to perform a conversion from joules to kilojoules, then proceed with the problem.

Mole Calculation

Mole calculations are fundamental to chemistry and are frequently used in problems involving chemical reactions and stoichiometry. In this exercise, calculating the number of moles of hydrogen gas using the Ideal Gas Law is the first step. Once you have the moles of H₂, you can use them in subsequent calculations such as Faraday's Law for determining the required charge and converting that to energy. Understanding how to manipulate and convert between different units, such as moles, volume, and pressure, is crucial to solving complex chemical problems efficiently.