Question

If water is added to magnesium nitride, ammonia gas is produced when the mixture is heated. $$ \mathrm{Mg}_{3} \mathrm{~N}_{2}(s)+3 \mathrm{H}_{2} \mathrm{O}(l) \rightarrow 3 \mathrm{MgO}(s)+2 \mathrm{NH}_{3}(g) $$ If \(10.3 \mathrm{~g}\) of magnesium nitride is treated with water, what volume of ammonia gas would be collected at 24 ' \(\mathrm{C}\) and \(752 \mathrm{~mm} \mathrm{Hg}\) ?

Short answer

The volume of ammonia gas produced at 24°C and 752 mm Hg is approximately 5.12 Liters.

Step-by-step solution

Convert mass of Magnesium Nitride to moles

First, we need to find the molar mass of Magnesium Nitride (Mg₃N₂). The molar mass of magnesium (Mg) is 24.31 g/mol and of nitrogen (N) is 14.01 g/mol. Molar mass of Mg₃N₂ = (3 × 24.31 g/mol) + (2 × 14.01 g/mol) = 100.95 g/mol Now, we will convert the given mass of Magnesium Nitride (10.3 g) to moles: Moles of Mg₃N₂ = Mass / Molar Mass = 10.3 g / 100.95 g/mol ≈ 0.102 moles

Find moles of ammonia produced using stoichiometry

Using the balanced chemical equation, we can see the mole ratio between Magnesium Nitride (Mg₃N₂) and ammonia (NH₃) is 1:2. Therefore, we can find the moles of ammonia produced by doubling the moles of Magnesium Nitride: Moles of NH₃ = 2 × Moles of Mg₃N₂ = 2 × 0.102 moles ≈ 0.204 moles

Calculate the volume of ammonia gas using the ideal gas law

Now, we can use the ideal gas law to find the volume of ammonia gas produced. The ideal gas law formula is: \(PV = nRT\) where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. First, we need to convert the given pressure to atm. 1 atm = 760 mm Hg. So the given pressure (752 mm Hg) in atm is: P = 752 mm Hg × (1 atm / 760 mm Hg) ≈ 0.9895 atm Next, we need to convert the given temperature from Celsius to Kelvin: T = 24°C + 273.15 ≈ 297.15 K The gas constant R = 0.08206 L atm/(mol K). Now, we can rearrange the ideal gas law equation to find the volume V: V = nRT / P Plugging in the values: V = (0.204 moles)(0.08206 L atm/(mol K))(297.15 K) / 0.9895 atm ≈ 5.12 L The volume of ammonia gas produced at 24°C and 752 mm Hg is approximately 5.12 Liters.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. It is expressed mathematically as \(PV = nRT\), where:

• \(P\) is the pressure of the gas,
• \(V\) is the volume of the gas,
• \(n\) is the number of moles of the gas,
• \(R\) is the ideal gas constant (0.08206 L atm/mol K),
• \(T\) is the temperature in Kelvin.

To use the ideal gas law effectively, you need to ensure that all units are consistent. For example, pressure needs to be in atmospheres, volume in liters, and temperature in Kelvin.
When given pressure in mm Hg, you can convert it to atmospheres using the conversion factor 1 atm = 760 mm Hg. Similarly, to convert temperature from Celsius to Kelvin, add 273.15 to the Celsius temperature. This consistency in units is crucial for accurate calculations and understanding of gas behavior.

Molar Mass

Molar mass is a key concept in stoichiometry, providing a bridge between the mass of a substance and the amount in moles.
The molar mass of a compound is the sum of the masses of all the atoms in its molecular formula, expressed in grams per mole (g/mol).

• For instance, for magnesium nitride (\(\text{Mg}_3\text{N}_2\)), calculate the molar mass by summing the contributions of magnesium and nitrogen.
• The molar mass of magnesium is 24.31 g/mol, and for nitrogen, it is 14.01 g/mol.
• Thus, the molar mass of \(\text{Mg}_3\text{N}_2\) is \((3 \times 24.31) + (2 \times 14.01) = 100.95 \text{ g/mol}\).

Once you have the molar mass, you can convert between the mass of a sample and the number of moles, which is fundamental in chemical calculations and reactions.

Chemical Reactions

Chemical reactions involve transforming reactants into products, adhering to the principle of conservation of mass. Each chemical reaction has a balanced chemical equation that outlines the relationship and quantities of reactants and products.

• In our example, the reaction \(\mathrm{Mg}_3\mathrm{N}_2 + 3 \mathrm{H}_2\mathrm{O} \rightarrow 3 \mathrm{MgO} + 2 \mathrm{NH}_3\) describes how magnesium nitride and water react to form magnesium oxide and ammonia.
• Stoichiometry is used to calculate the amount of products formed from given amounts of reactants. It leverages the mole ratio derived from the balanced equation to relate the quantities.
• This mole ratio dictates that 1 mole of \(\text{Mg}_3\text{N}_2\) yields 2 moles of \(\text{NH}_3\), which is crucial for converting reactant quantities into product quantities.

Understanding these relationships is essential in predicting the outcomes of reactions and in optimizing chemical processes.

Gas Volume Calculations

Calculating the volume of a gas produced in a chemical reaction requires understanding of both stoichiometry and the ideal gas law.
Once the number of moles of a gas is determined from a reaction using stoichiometry, the ideal gas law \(PV = nRT\) is used to find the volume.

• Using stoichiometry, convert the mass of reactants to moles of product gas. In this problem, 0.102 moles of \(\text{Mg}_3\text{N}_2\) yields approximately 0.204 moles of \(\text{NH}_3\).
• These moles are then used in the ideal gas law to calculate the volume of gas formed under specific conditions of temperature and pressure.
• Proper unit conversion, especially of pressure and temperature to atm and Kelvin respectively, is essential to ensure accurate calculations.

These calculations demonstrate how known physical properties and mathematical relationships can predict volumes of gases involved in or produced by chemical reactions.