Problem 60 - Chemistry The Central Science

Question

Both Jacques Charles and Joseph Louis Guy-Lussac were avid balloonists. In his original flight in 1783 , Jacques Charles used a balloon that contained approximately \(31,150 \mathrm{~L}\) of \(\mathrm{H}_{2}\). He generated the \(\mathrm{H}_{2}\) using the reaction between iron and hydrochloric acid: $$ \mathrm{Fe}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{FeCl}_{2}(a q)+\mathrm{H}_{2}(g) $$ How many kilograms of iron were needed to produce this volume of \(\mathrm{H}\), if the temperature was \(22{ }^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

To find out how many kilograms of iron are needed to produce $31150 \mathrm{~L}$ of $\mathrm{H}_{2}$ gas at $22{ }^{\circ} \mathrm{C}$, we need to follow these steps: 1. Calculate moles of hydrogen gas using the Ideal Gas Law equation. 2. Use stoichiometry to find the required moles of iron based on the balanced chemical equation. 3. Convert moles of iron to kilograms using the molar mass of iron. After performing these calculations, we find that approximately \(24.82 \mathrm{~kg}\) of iron is needed to produce the given volume of hydrogen gas at the given temperature.

Step 1: Calculate the moles of hydrogen gas

To calculate the moles of hydrogen gas, we will use the Ideal Gas Law equation, which is: $$ PV = nRT $$ Where: - P: Pressure of the gas - V: Volume of the gas - n: Moles of the gas - R: Ideal gas constant (8.314 J/(mol·K) or 0.0821 L·atm/(mol·K)) - T: Temperature of the gas in Kelvin We are given the volume, temperature, and we can assume the pressure is 1 atm (standard conditions). Thus, we can rearrange the equation to solve for the moles (n) of hydrogen gas: $$ n = \frac{PV}{RT} $$ Given, V = 31150 L, T = 22°C = 295.15 K and R = 0.0821 L·atm/(mol·K), assuming the pressure is 1 atm. Now calculate the moles of hydrogen gas: $$ n_{H_{2}} = \frac{31150 × 1}{0.0821 × 295.15} $$

Step 2: Find the moles of iron

Using stoichiometry, we can find out how many moles of iron are required to produce the above-calculated moles of hydrogen gas. According to the balanced chemical equation: $$ \mathrm{Fe}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{FeCl}_{2}(a q)+\mathrm{H}_{2}(g) $$ 1 mole of iron reacts with 2 moles of hydrochloric acid to produce 1 mole of hydrogen gas. Therefore, the moles of iron required are equal to the moles of hydrogen gas produced in the reaction. $$ n_{Fe} = n_{H_{2}} $$

Step 3: Convert moles of iron to kilograms

Now that we have the moles of iron, we can convert it to kilograms using the molar mass of iron: - Molar mass of iron, \(M_{Fe}\) = 55.845 g/mol Now calculate the mass of iron (in grams) by multiplying the moles of iron with its molar mass and convert it to kilograms: $$ m_{Fe} = n_{Fe} × M_{Fe} $$ $$ m_{Fe} (kg) = \frac{m_{Fe} (g)}{1000} $$

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a powerful tool in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. By combining Avogadro's Law, Boyle's Law, and Charles's Law, the Ideal Gas Law provides a comprehensive equation to describe the behavior of gases under various conditions. In its simplest form, the equation is written as:
\[ PV = nRT \]
where P is pressure in atmospheres (atm), V is volume in liters (L), n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin (K).

• Pressure (P): The force per unit area exerted by gas particles as they collide with the walls of their container.
• Volume (V): The space that the gas occupies.
• Moles (n): Represents the quantity of gas particles in terms of Avogadro's number, which is approximately \(6.022 \times 10^{23}\) particles per mole.
• Temperature (T): A measure of the average kinetic energy of the gas particles. It must be in Kelvin for the Ideal Gas Law to work correctly.
• Ideal Gas Constant (R): A proportionality constant that is the same for all ideal gases. Its value depends on the units used for pressure and volume. Common values are \(8.314 \frac{J}{mol\cdot K}\) or \(0.0821 \frac{L\cdot atm}{mol\cdot K}\).

To solve problems involving the Ideal Gas Law, one must first make sure that all the units are consistent and that the temperature is in Kelvin. It is critical for students to understand how each variable in the equation affects the others, and how changes in one can lead to changes in the others. For example, increasing the temperature of a gas will increase its volume if the pressure and number of moles remain constant.

Chemical Reaction

A chemical reaction is a process where substances, known as reactants, transform into new substances, termed products. These reactions involve the breaking and forming of chemical bonds, and each reaction is governed by a balanced chemical equation that follows the conservation of mass. The equation for the formation of hydrogen gas from a reaction of iron with hydrochloric acid can be written as:
\[ \mathrm{Fe}(s) + 2 \mathrm{HCl}(aq) \longrightarrow \mathrm{FeCl}_2(aq) + \mathrm{H}_2(g) \]
This equation tells us that solid iron reacts with hydrochloric acid in aqueous solution to produce iron(II) chloride, also in aqueous solution, and hydrogen gas. The reaction is balanced, indicating that for every one mole of iron that reacts, two moles of hydrochloric acid are consumed, and one mole of hydrogen gas is produced.
In the context of the example provided, stoichiometry, which is the quantitative relationship between reactants and products in a chemical reaction, allows us to calculate the amount of reactant needed (iron in this case) to produce a certain amount of product (hydrogen gas).

• Reactants: The starting substances in a chemical reaction (iron and hydrochloric acid).
• Products: The substances formed from a chemical reaction (iron(II) chloride and hydrogen gas).
• Stoichiometry: The calculation of the quantitative aspects of a chemical reaction, ensuring that the quantities of reactants and products follow the stoichiometric coefficients in the balanced equation.

Comprehending the ratios of reactants and products as well as the conservation of mass is crucial in solving stoichiometric problems. Students should focus on the coefficients in the chemical equation, as they dictate the mole-to-mole ratios that are fundamental to these calculations.

Molar Mass

Molar mass is an essential concept in chemistry, referring to the mass of one mole of a substance. One mole is defined as the amount of a substance containing as many entities (atoms, molecules, or ions) as there are atoms in 12 grams of pure carbon-12. The molar mass is typically expressed in grams per mole (g/mol) and is calculated by summing the atomic masses of all atoms in a molecule.

• Molar Mass: The mass of one mole of a substance, typically in units of g/mol.
• Atomic Mass: The mass of an individual atom, often found on the periodic table.
• Mole: A unit of measurement for amount of substance, exactly \(6.022 \times 10^{23}\) entities of that substance.

In the provided example, the molar mass of iron (Fe) is \(55.845 g/mol\). This value is derived from the weighted average of the isotopes of iron as they occur naturally. Once the moles of iron needed for the reaction are known, as calculated through stoichiometry, one can determine the mass of iron required by multiplying the number of moles by the molar mass of iron.
Knowing how to convert between mass, moles, and number of atoms using the molar mass is fundamental in chemistry, especially in measuring and predicting the outcomes of reactions. This is because the number of atoms or molecules, rather than their mass, determines the proportions in which substances react. The ability to convert mass to moles and vice versa is a critical skill for students when analyzing chemical reactions and performing laboratory experiments.