Question

A 1.65 g sample of \(\mathrm{Al}\) reacts with excess \(\mathrm{HCl}\), and the liberated \(\mathrm{H}_{2}\) is collected over water at \(25^{\circ} \mathrm{C}\) at a barometric pressure of \(744 \mathrm{mmHg} .\) What volume of gaseous mixture, in liters, is collected? $$2 \mathrm{Al}(\mathrm{s})+6 \mathrm{HCl}(\mathrm{aq}) \longrightarrow 2 \mathrm{AlCl}_{3}(\mathrm{aq})+3 \mathrm{H}_{2}(\mathrm{g})$$

Short answer

The corrected volume of the gaseous mixture collected is 2.42 L.

Step-by-step solution

Calculate moles of Al

First, compute for the moles of Al using the given mass and the molar mass of Al, which is \( 26.982 \, g/mol\). Moles of Al \( = \frac{1.65 \, g}{26.982 \, g/mol} = 0.0611 \, mol\)

Calculate moles of H2

Determine the moles of \( H_{2} \) using the stoichiometric ratio in the balanced chemical reaction, \( 2Al:3H_{2} = 0.0611 \, mol \, Al:x \, mol \, H_{2}\). Solve for \( x \), getting \( x = \frac{3}{2} \times 0.0611 \, mol \, Al = 0.0916 \, mol \, H_{2}\).

Determine the gas volume using the Ideal Gas Law

Use the Ideal Gas Law \(PV = nRT\) where \(P\) is the pressure in atm \(= 744 \, mmHg \times \frac{1 \, atm}{760 \, mmHg} = 0.9789 \, atm\), \(n\) is the number of moles \(= 0.0916 \, mol\), \(R\) is the gas constant \(= 0.0821 \, L.atm/(mol.K)\), and \(T\) is the temperature in Kelvin \(= 25^{\circ} C + 273.15 = 298.15 \, K\). Solving for \(V\), \(V = \frac{nRT}{P}\) gets \(V = \frac{0.0916 \, mol \times 0.0821 \, L.atm/(mol.K) \times 298.15 \, K }{0.9789 \, atm} = 2.30 \, L\).

Correct the collected volume for vapor pressure

Lastly, correct for the presence of water vapor in the collected gas. The partial pressure of hydrogen is given by subtracting the water vapor pressure from the total pressure. The vapor pressure of water at \(25^{\circ}C\) is 23.76 mmHg or 0.0313 atm. Hence, the corrected volume of \(H_{2}\) gas is \(V' = nRT/P' = \frac{0.0916 \, mol \times 0.0821 \, L.atm/(mol.K) \times 298.15 \, K }{0.9789 \, atm - 0.0313 \, atm} = 2.42 \, L\).

Key concepts

Stoichiometry

Stoichiometry is the study of quantitative relationships in chemical reactions. It involves using the balanced chemical equation to determine the proportions of reactants and products. In this exercise, stoichiometry helps us convert the amount of aluminum (Al) into hydrogen gas (\(H_{2}\)) produced. The balanced equation tells us that 2 moles of Al produce 3 moles of \(H_{2}\):

• Start with the given mass of Al.
• Convert mass to moles using Al's molar mass.
• Use the mole ratio from the balanced equation to find moles of \(H_{2}\).

This process bridges the gap between the amount of reactants and products, allowing precise predictions in chemical reactions.

Ideal Gas Law

The Ideal Gas Law links the physical properties of gases using the formula \(PV = nRT\). It helps determine the behavior of gases under different conditions of pressure (\(P\)), volume (\(V\)), temperature (\(T\)), and moles (\(n\)). In this problem, this law helps calculate the volume of hydrogen gas collected.

• First, convert all units to match: pressure to atm, and temperature to Kelvin.
• Insert the number of moles of hydrogen gas calculated from stoichiometry.
• Use the gas constant \(R = 0.0821\, \text{L.atm/(mol.K)}\).

Solving for the unknown volume \(V\), we find the initial gas volume before adjusting for water vapor.This calculation gives us insight into how gasses behave and how the physical conditions affect this behavior.

Chemical Reactions

Chemical reactions describe how atoms are reorganized, resulting in products with new properties. They are represented by balanced equations that maintain the conservation of mass and atoms. In this exercise, we examine the reaction between aluminum (\(2Al\)) and hydrochloric acid (\(6HCl\)), producing aluminum chloride (\(2AlCl_{3}\)) and hydrogen gas (\(3H_{2}\)).

• Balanced equations ensure the same number of each type of atom on both sides.
• Determine which compounds are reactants and products.
• Predict the reaction outcomes using the balanced equation.

This reaction particularly involves single replacement, where hydrogen in hydrochloric acid is displaced by aluminum, showcasing the rearrangement of atoms into different substances. Understanding reactions further helps us control and predict outcomes in various chemical processes.