Question

Aluminum reacts with excess hydrochloric acid to form aqueous aluminum chloride and \(35.8 \mathrm{~mL}\) of hydrogen gas over water at \(27^{\circ} \mathrm{C}\) and \(751 \mathrm{mmHg} .\) How many grams of aluminum react?

Short answer

0.0250 g of aluminum.

Step-by-step solution

Write the Balanced Chemical Equation

First, write the balanced equation for the reaction between aluminum and hydrochloric acid: \[\text{2 Al + 6 HCl → 2 AlCl}_3 \text{ + 3 H}_2\]

Calculate the Partial Pressure of Hydrogen Gas

The total pressure is the sum of the pressure of hydrogen gas and the vapor pressure of water at 27°C (which is 26.74 mmHg). \[\text{P}_H₂ = 751 \text{ mmHg} - 26.74 \text{ mmHg} = 724.26 \text{ mmHg}\]

Convert the Volume of Gas to Liters

Convert the given volume from mL to liters: \[35.8 \text{ mL} \times \frac{1 \text{ L}}{1000 \text{ mL}} = 0.0358 \text{ L}\]

Use the Ideal Gas Law to Find Moles of Hydrogen

Use the ideal gas law \[PV = nRT\] and convert the pressure to atm. Also, use the value of the gas constant R = 0.0821 L·atm/(K·mol) and temperature T = 27 + 273 = 300 K: \[724.26 \text{ mmHg} \times \frac{1 \text{ atm}}{760 \text{ mmHg}} = 0.9532 \text{ atm}\] \[n = \frac{PV}{RT} = \frac{(0.9532 \text{ atm})(0.0358 \text{ L})}{(0.0821 \text{ L·atm/(K·mol)})(300 \text{ K})} = 0.00139 \text{ mol of H}_2\]

Use Stoichiometry to Find Moles of Aluminum

From the balanced chemical equation, 3 moles of \[H_2\] are produced per 2 moles of \[Al\], hence: \[ \text{Moles of Al} = 0.00139 \text{ mol of H}_2 \times \frac{2 \text{ mol Al}}{3 \text{ mol H}_2} = 0.000926 \text{ mol Al} \]

Calculate Mass of Aluminum

Finally, convert moles of \[Al\] to grams using the molar mass of aluminum (26.98 g/mol): \[ \text{Mass of Al} = 0.000926 \text{ mol Al} \times 26.98 \text{ g/mol} = 0.0250 \text{ g} \]

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry, expressed as \( PV = nRT \). It describes the state of an ideal gas by relating pressure (P), volume (V), amount of substance (n), and temperature (T) with the gas constant (R). In this exercise, we used the Ideal Gas Law to determine the number of moles of hydrogen gas produced in the reaction.
The steps involve:

• Converting pressure from mmHg to atm.
• Calculating the number of moles of hydrogen (n), using the given volume (V), temperature (T), and the gas constant (R).

Remember, the gas constant (R) has a value of 0.0821 L·atm/(K·mol). It is vital to use consistent units when applying the Ideal Gas Law.

Stoichiometry

Stoichiometry involves the calculation of reactants and products in chemical reactions. It is based on the balanced chemical equation. For our reaction, the balanced equation is:
\[ \text{2 Al + 6 HCl} \rightarrow \text{2 AlCl}_3 \text{ + 3 H}_2 \]
This tells us the molar ratio of aluminum (Al) to hydrogen gas (H2). For every 2 moles of Al, 3 moles of H2 are produced. Thus, by knowing the moles of H2, we can calculate the moles of Al.
In this exercise:

• We found the moles of hydrogen gas (H2).
• Using the ratio from the balanced equation, we calculated the corresponding moles of aluminum (Al).

Stoichiometry is essential for converting between masses and moles using molar ratios.

Partial Pressure

Partial pressure is the pressure exerted by a single component of a gas mixture. It is significant when collecting gases over water, as the total pressure includes the vapor pressure of water.
For instance, the total pressure given is the sum of the hydrogen gas pressure and the vapor pressure of water at the reaction temperature. To find the partial pressure of hydrogen gas:
\[ \text{P}_\text{H2} = \text{Total Pressure} - \text{Vapor Pressure of Water} \]In our exercise:
\[ \text{P}_\text{H2} = 751 \text{ mmHg} - 26.74 \text{ mmHg} = 724.26 \text{ mmHg} \]This partial pressure is then converted to atm for use in the Ideal Gas Law, ensuring accurate calculations.

Gas Volumes

Gas volumes often need conversion to appropriate units for calculations. In this exercise, the volume of hydrogen gas is given in milliliters but must be converted to liters to use in the Ideal Gas Law:
\[ 35.8 \text{ mL} \times \frac{1 \text{ L}}{1000 \text{ mL}} = 0.0358 \text{ L} \]Understanding how to convert between units is crucial. Always ensure volume is in liters (L) when using the gas constant in the Ideal Gas Law.
Correct unit conversion allows for accurate calculations of moles or masses in chemical reactions.

Chemical Equations

A balanced chemical equation is fundamental to stoichiometric calculations. The equation provides the necessary mole ratios of reactants and products. In this exercise, the balanced equation:
\[ \text{2 Al + 6 HCl} \rightarrow \text{2 AlCl}_3 \text{ + 3 H}_2 \]shows the reaction of aluminum (Al) with hydrochloric acid (HCl) to produce aluminum chloride (AlCl3) and hydrogen gas (H2).
Balance the equation by adjusting coefficients to obey the law of conservation of mass. This ensures the number of atoms of each element is the same on both sides of the equation.
Balanced equations are essential for determining the proportion of each substance involved, enabling the calculation of quantities needed or produced in a reaction.