Problem 54 - 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, where a simple, compact means of generating \(\mathrm{H}_{2}\) is desired. How many grams of \(\mathrm{CaH}_{2}\) are needed to generate \(53.5 \mathrm{~L}\) of \(\mathrm{H}_{2}\) gas if the pressure of \(\mathrm{H}_{2}\) is 814 torr at \(21{ }^{\circ} \mathrm{C}\) ?

Step-by-Step Solution

Answer

About 53.25 grams of CaH2 are needed to generate 53.5 L of H2 gas at the given conditions of 814 torr pressure and 21°C temperature.

Step 1: Convert the given conditions to appropriate units

To use the Ideal Gas Law, we need to convert the given conditions into the appropriate units. Convert the temperature from Celsius to Kelvin and the pressure from torr to atmospheres. Temperature in Kelvin: \(T = 21 + 273.15 = 294.15 \ \mathrm{K}\) Pressure in atmospheres: \(P = \frac{814 \ \mathrm{torr}}{760 \ \mathrm{torr/atm}} = 1.0711 \ \mathrm{atm}\)

Step 2: Determine the number of moles of H2 gas

Use the Ideal Gas Law (PV = nRT) to calculate the number of moles of H2 gas. Rearrange the formula to solve for n (moles of the gas): \(n_\mathrm{H_2} = \frac{PV}{RT}\) Plug in the given values and the gas constant (R = 0.0821 L atm/mol K) for the calculation: \(n_\mathrm{H_2} = \frac{(1.0711 \ \mathrm{atm})(53.5 \ \mathrm{L})}{(0.0821 \ \mathrm{L \ atm/mol\ K})(294.15 \ \mathrm{K})} \approx 2.529 \ \mathrm{moles \ of \ H_2}\)

Step 3: Use stoichiometry to determine the moles of CaH2 needed

From the balanced chemical equation, we can see that 1 mole of CaH2 produces 2 moles of H2. To calculate the moles of CaH2 needed, we can use the stoichiometry ratio: Moles of CaH2 needed = \(\frac{1 \ \mathrm{mole \ of \ CaH_2}}{2 \ \mathrm{moles \ of \ H_2}} \times 2.529 \ \mathrm{moles \ of \ H_2}\) Moles of CaH2 needed = \(1.2645 \ \mathrm{moles}\)

Step 4: Convert the moles of CaH2 to grams

Finally, we will convert the moles of CaH2 to grams using its molar mass. The molar mass of CaH2 is approximately 42.094 g/mol (40.08 g/mol for Ca and 2.014 g/mol for 2H). Grams of CaH2 needed = \(1.2645 \ \mathrm{moles \ of \ CaH_2} \times 42.094 \ \mathrm{g/mol} \approx 53.25 \ \mathrm{g}\) Hence, about 53.25 grams of CaH2 are needed to generate 53.5 L of H2 gas at the given conditions.

Key Concepts

Understanding Stoichiometry

Stoichiometry is a branch of chemistry that deals with the quantitative relationships between the reactants and products in a chemical reaction. It's like the math behind chemical equations, helping you figure out how much of each substance you'll need or get.
In our case, we used stoichiometry to determine how much calcium hydride (\( \mathrm{CaH}_{2} \)) is required to produce a specific volume of hydrogen gas (\( \mathrm{H}_{2} \)).
Here's what makes stoichiometry so useful:

• It allows you to convert from moles of one substance in a reaction to moles of another using a balanced equation.
• From the balanced equation \( \mathrm{CaH}_{2}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \rightarrow \mathrm{Ca}(\mathrm{OH})_{2}(aq)+2 \mathrm{H}_{2}(g) \), you can calculate that 1 mole of \( \mathrm{CaH}_{2} \) produces 2 moles of \( \mathrm{H}_{2} \).
• By knowing the moles of \( \mathrm{H}_{2} \) produced, we can work backwards to find the moles of \( \mathrm{CaH}_{2} \) needed.

This basic concept is powerful and is used in everything from simple classroom experiments to industrial-scale chemical production.
You can think of stoichiometry as your chemical recipe book, telling you the exact proportions needed to "cook" up a chemical reaction.

Grasping Molar Mass

Molar mass is key to connecting the microscopic world of atoms to the macroscopic quantities we can measure. It tells you the weight of a mole of a given substance and is expressed in grams per mole (g/mol).
In our problem, knowing the molar mass of calcium hydride (\( \mathrm{CaH}_{2} \)) was essential to figure out how many grams are needed.

• We found the molar mass by adding up the atomic masses of calcium (approximately 40.08 g/mol) and hydrogen (approximately 1.008 g/mol each, but consider 2 hydrogen atoms, summing to about 2.016 g/mol).
• Thus, the molar mass of \( \mathrm{CaH}_{2} \) is about 42.094 g/mol.

Once we had this number, converting moles of \( \mathrm{CaH}_{2} \) to grams was straightforward: simply multiply the number of moles by the molar mass.
This concept is essential in chemistry, as molar mass allows us to translate between the amount of a substance in moles (a countable quantity) and its mass (a measurable quantity), making it a bridge between chemistry and real-world applications.

Basics of Chemical Reactions

Chemical reactions are processes in which substances interact to form new products. Understanding the basics of these reactions is crucial, especially when you need to predict the outcomes or quantities involved.
Our example reaction involves calcium hydride (\( \mathrm{CaH}_{2} \)) reacting with water to produce hydrogen gas (\( \mathrm{H}_{2} \)) and calcium hydroxide (\( \mathrm{Ca(OH)}_{2} \)).

• A balanced chemical equation is key. Each atom on the reactant side should match those on the product side, maintaining mass and charge balance.
• This specific reaction is written as: \( \mathrm{CaH}_{2}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \rightarrow \mathrm{Ca}(\mathrm{OH})_{2}(aq)+2 \mathrm{H}_{2}(g) \) which shows the stoichiometric coefficients needed for balance.
• Reactions may also involve energy changes, either absorbing (endothermic) or releasing energy (exothermic).

Understanding chemical reactions helps in various applications, from generating gases for inflating balloons to powering vehicles with reactions inside engines.
Knowing how to read and balance these equations forms the foundation for further learning in chemistry.