Problem 56 - Chemistry: The Central Science

Question

Calcium hydride, \(\mathrm{CaH}_{2}\), reacts with water to form hydrogen gas: $$ \mathrm{CaH}_{2}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{Ca}(\mathrm{OH})_{2}(a q)+2 \mathrm{H}_{2}(g) $$ This reaction is sometimes used to inflate life rafts, weather balloons, and the like, when a simple, compact means of generating \(\mathrm{H}_{2}\) is desired. How many grams of \(\mathrm{CaH}_{2}\) are needed to generate \(145 \mathrm{~L}\) of \(\mathrm{H}_{2}\) gas if the pressure of \(\mathrm{H}_{2}\) is 825 torr at \(21^{\circ} \mathrm{C}\) ?

Step-by-Step Solution

Answer

To generate 145 L of hydrogen gas at a pressure of 825 torr and a temperature of 21°C, it is required to use 142.04 grams of calcium hydride (\(\mathrm{CaH}_{2}\)).

Step 1: Convert the given Data to Required Units

First, let's convert the given pressure and temperature to standard units. Pressure in atm and temperature in Kelvin. Pressure, P (in atm) = 825 torr × (1 atm / 760 torr) = 1.0855 atm Temperature, T (in K) = 21°C + 273.15 = 294.15 K

Step 2: Apply the Ideal Gas Law Equation

The ideal gas law equation is as follows: PV = nRT Where: P = pressure (in atm) V = volume (in L) n = number of moles of gas R = ideal gas constant (0.0821 L atm / mol K) T = temperature (in K) Given volume, V = 145 L, Pressure, P = 1.0855 atm, and Temperature, T = 294.15 K, we can now solve for the number of moles of hydrogen gas (H2) as shown below: n = PV / RT

Step 3: Calculate Number of Moles of Hydrogen Gas

Plug the given data into the equation: n = (1.0855 atm × 145 L) / (0.0821 L atm / mol K × 294.15 K) n = 6.7463 mol We have 6.7463 moles of hydrogen gas (H2).

Step 4: Stoichiometric Calculations

From the balanced chemical equation, \(\mathrm{CaH}_{2}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{Ca}(\mathrm{OH})_{2}(a q)+2 \mathrm{H}_{2}(g)\), we can infer the stoichiometric ratio between \(\mathrm{CaH}_{2}\) and \(\mathrm{H}_{2}\) as 1:2. Now, we can calculate the moles of \(\mathrm{CaH}_{2}\) required to produce 6.7463 moles of \(\mathrm{H}_{2}\) using the stoichiometric ratio: moles of \(\mathrm{CaH}_{2}\) = (moles of \(\mathrm{H}_{2}\)) × (1 mol \(\mathrm{CaH}_{2}\) / 2 mol \(\mathrm{H}_{2}\)) moles of \(\mathrm{CaH}_{2}\) = 6.7463 mol × (1/2) moles of \(\mathrm{CaH}_{2}\) = 3.37315 mol

Step 5: Convert Moles of \(\mathrm{CaH}_{2}\) to Grams

Finally, convert the moles of calcium hydride (CaH2) to grams using its molar mass. The molar mass of \(\mathrm{CaH}_{2}\) is 42.094 g/mol. grams of \(\mathrm{CaH}_{2}\) = (moles of \(\mathrm{CaH}_{2}\)) × (molar mass of \(\mathrm{CaH}_{2}\)) grams of \(\mathrm{CaH}_{2}\) = 3.37315 mol × 42.094 g/mol grams of \(\mathrm{CaH}_{2}\) = 142.04 g Hence, 142.04 grams of calcium hydride (\(\mathrm{CaH}_{2}\)) are needed to generate 145 L of hydrogen gas at a pressure of 825 torr and a temperature of 21°C.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation that describes the behavior of gases, allowing us to interrelate the pressure, volume, temperature, and quantity (in moles) of a gas. It is expressed as:
\[ PV = nRT \]
where:

• \( P \): Pressure in atmospheres (atm)
• \( V \): Volume in liters (L)
• \( n \): Number of moles of gas
• \( R \): Ideal gas constant, approximately 0.0821 L atm/mol K
• \( T \): Temperature in Kelvin (K)

This equation helps us calculate one of these properties if the others are known.
In the context of our problem, we use the Ideal Gas Law to determine the number of moles of hydrogen gas produced in a reaction. By rearranging the equation to solve for \( n \) (moles), we substitute the known values of pressure, volume, and temperature into the equation.
This straightforward approach provides a clear pathway to relate physical conditions of a gas with its chemical representation.

Chemical Reactions

Chemical reactions involve the transformation of reactants into products. The essence of a balanced chemical reaction is to convey the conservation of mass during the reaction process, with atoms of each element accounted for on both sides of the equation.
In our example, calcium hydride \((\mathrm{CaH}_2)\) reacts with water to produce calcium hydroxide and hydrogen gas. This reaction is represented by the equation:
\[\mathrm{CaH}_{2}(s) + 2 \mathrm{H}_{2}\mathrm{O}(l) \rightarrow \mathrm{Ca}(\mathrm{OH})_{2}(aq)+2 \mathrm{H}_{2}(g)\]
From this balanced equation, we derive the stoichiometric ratios: one mole of \(\mathrm{CaH}_2\) reacts with two moles of \(\mathrm{H}_2\mathrm{O}\), producing two moles of \(\mathrm{H}_2\) gas.
Understanding these ratios is crucial as they serve as the bridge between chemistry and calculations.
It enables us to determine how much reactant is needed to produce a certain amount of product, reinforcing the practical application of stoichiometry.

Molar Mass

Molar mass is the weight of one mole of a substance, expressed in grams per mole (g/mol). It is determined by summing the atomic masses of all atoms in a molecular formula.
For example, the molar mass of calcium hydride (\(\mathrm{CaH}_2\)) is calculated by adding the atomic masses of calcium (approximately 40.08 g/mol) and two hydrogen atoms (approximately 1.01 g/mol each):

• Molar mass of \(\mathrm{CaH}_2\) = 40.08 + (2 \times 1.01) = 42.10 g/mol

Molar mass serves as a conversion factor between the mass of a compound and moles, enabling chemists to measure quantities needed for reactions.
In our problem, understanding molar mass allows us to convert moles of \(\mathrm{CaH}_2\) calculated from the stoichiometric relationships into grams, which is more practical for actual laboratory use.
Using this conversion effectively bridges theoretical calculations and real-world applications.

Gas Laws

The Ideal Gas Law is just one component of the broader group known as gas laws. These laws describe the behavior and interactions of gases under varying conditions.
Some of the key gas laws include:

• Boyle's Law: Relates pressure and volume \((P \times V = \, \text{constant})\) indicating that pressure increases as volume decreases when temperature is held constant.
• Charles's Law: Shows the relationship between volume and temperature \((V/T = \, \text{constant})\) indicating that volume increases with temperature when pressure is constant.
• Avogadro's Law: Expresses the relationship between volume and moles \((V/n = \, \text{constant})\), suggesting that volume increases with the number of gas molecules at constant pressure and temperature.

Collectively, these laws, along with the Ideal Gas Law, describe how gases will react to changes in environmental conditions, providing a foundational understanding of gas behavior.
For students, these laws illustrate the dynamic nature of gases and are essential in solving problems involving chemical reactions in gaseous states.