Problem 37 - Chemistry and Chemical Reactivity

Question

Aluminum dissolves readily in hot aqueous NaOH to give the aluminate ion, \(\mathrm{Al}(\mathrm{OH})_{4}^{-},\) and \(\mathrm{H}_{2} .\) Write a balanced equation for this reaction. If you begin with \(13.2 \mathrm{g}\) of \(\mathrm{Al}\), what volume (in milliliters) of \(\mathrm{H}_{2}\) gas is produced when the gas is measured at \(735 \mathrm{mm}\) Hg and \(22.5^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

18500 mL of hydrogen gas is produced.

Step 1: Write the Chemical Equation

The chemical reaction between aluminum and aqueous sodium hydroxide can be represented as follows:\[\mathrm{2Al (s) + 2NaOH (aq) + 6H_2O (l) \rightarrow 2Na[Al(OH)_4] (aq) + 3H_2 (g)}\]This equation shows that aluminum reacts with sodium hydroxide and water to produce sodium aluminate and hydrogen gas.

Step 2: Calculate Moles of Aluminum

First, we need to find the moles of aluminum used in the reaction. The molar mass of aluminum, \(\mathrm{Al}\), is approximately \(26.98 \, \mathrm{g/mol}\).\[\text{Moles of Al} = \frac{13.2 \, \mathrm{g}}{26.98 \, \mathrm{g/mol}} \approx 0.489 \text{ moles}\]

Step 3: Determine Moles of Hydrogen Gas Produced

From the balanced equation, \(2\) moles of aluminum produce \(3\) moles of hydrogen gas. Using this stoichiometric relationship:\[\text{Moles of } H_2 = 0.489 \text{ moles Al} \times \frac{3 \text{ moles } H_2}{2 \text{ moles Al}} \approx 0.7335 \text{ moles } H_2\]

Step 4: Apply Ideal Gas Law

To find the volume of the hydrogen gas produced, we use the ideal gas law equation \(PV = nRT\).First, convert \(735 \text{ mm Hg}\) to atmospheres since the gas constant \(R = 0.0821 \text{ L atm/mol K}\):\[P = \frac{735 \text{ mm Hg}}{760 \text{ mm Hg/atm}} \approx 0.967 \text{ atm}\]Convert temperature from Celsius to Kelvin:\[T = 22.5 + 273.15 = 295.65 \text{ K}\]Rearrange the ideal gas law to find the volume \(V\):\[V = \frac{nRT}{P}\]\[V = \frac{0.7335 \text{ moles} \times 0.0821 \text{ L atm/mol K} \times 295.65 \text{ K}}{0.967 \text{ atm}} \approx 18.5 \text{ L}\]

Step 5: Convert Volume to Milliliters

Finally, convert the volume from liters to milliliters:\[18.5 \text{ L} = 18500 \text{ mL}\]

Key Concepts

Balancing Chemical Equations

Balancing chemical equations is an essential step in understanding chemical reactions. In the reaction between aluminum and sodium hydroxide in water, all chemical species must obey the conservation of mass. This means the number of each type of atom on the reactant side must be equal to the number on the product side.

For this specific reaction, aluminum (Al) reacts with sodium hydroxide (NaOH) and water (H₂O) to produce sodium aluminate \[\text{(Na[Al(OH)}_4)]\] and hydrogen gas (H₂). The balanced chemical equation is:

• 2 Al (s) + 2 NaOH (aq) + 6 H₂O (l) \rightarrow 2 Na[Al(OH)_4] (aq) + 3 H₂ (g)

This equation tells us that two moles of solid aluminum react with two moles of aqueous sodium hydroxide and six moles of water to yield two moles of aqueous sodium aluminate and three moles of hydrogen gas. Balancing this equation ensures that both mass and charge are conserved in the chemical reaction.

Ideal Gas Law Calculations

The ideal gas law is a useful equation for finding the properties of a gas. The formula is expressed as \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

In this problem, we need to find the volume of hydrogen gas produced. First, the pressure of hydrogen gas is given as \(735\) mm Hg, which we convert to atmospheres by dividing by \(760\) mm Hg/atm:

• \(P = \frac{735}{760} \approx 0.967\) atm

The temperature is \(22.5^{\circ}C\), which needs converting to Kelvin by adding \(273.15\).

• \(T = 22.5 + 273.15 = 295.65\) K

By plugging the known values into the rearranged ideal gas law formula \(V = \frac{nRT}{P}\) and using \(R = 0.0821\) L atm/mol K, we find the volume of hydrogen gas under these conditions.

• \(V = \frac{0.7335 \text{ moles} \times 0.0821 \text{ L atm/mol K} \times 295.65 \text{ K}}{0.967 \text{ atm}} \approx 18.5 \text{ L}\)

Finally, convert the volume from liters to milliliters for clarity:

• \(18.5 \text{ L} = 18500 \text{ mL}\)

Stoichiometry

Stoichiometry involves calculating the amounts of reactants or products in a chemical reaction based on the balanced equation. This concept is crucial for determining how much of a substance will react or be produced.

In the reaction of aluminum with sodium hydroxide, stoichiometry helps us find the amount of hydrogen gas produced from a given mass of aluminum. First, we convert the mass of aluminum to moles using its molar mass.

• Molar mass of Al = 26.98 g/mol
• \(\text{Moles of Al} = \frac{13.2 \text{ g}}{26.98 \text{ g/mol}} \approx 0.489\) moles

Then, using the stoichiometric ratio from the balanced equation \(2 \text{ moles Al} \rightarrow 3 \text{ moles } H_2\), we can determine the moles of hydrogen gas:

• \(\text{Moles of } H_2 = 0.489 \text{ moles Al} \times \frac{3 \text{ moles } H_2}{2 \text{ moles Al}} \approx 0.7335 \text{ moles } H_2\)

Understanding stoichiometry allows us to predict the outcomes of chemical reactions and make informed decisions about the necessary quantities of different reactants.