Problem 110 - General Chemistry: Principles and Modern Applications

Question

Aluminum (Al) and iron (Fe) each react with hydrochloric acid solution (HCl) to produce a chloride salt and hydrogen gas, \(\mathrm{H}_{2}(\mathrm{g}) .\) A \(0.1924 \mathrm{g}\) sample of a mixture of \(\mathrm{Al}\) and \(\mathrm{Fe}\) is treated with excess \(\mathrm{HCl}\) solution. A volume of \(159 \mathrm{mL}\) of \(\mathrm{H}_{2}\) gas is collected over water at \(19.0^{\circ} \mathrm{C}\) and 841 Torr. What is the percent (by mass) of Fe in the mixture? The vapor pressure of water at \(19.0^{\circ} \mathrm{C}\) is 16.5 Torr.

Step-by-Step Solution

Answer

The percentage (by mass) of Fe in the mixture is 34.0 %.

Step 1: Calculate the pressure of Hydrogen Gas

The total pressure given is the sum of the pressure of hydrogen gas and the vapor pressure of water. So, the pressure of the hydrogen gas will be total pressure subtracted by the vapor pressure of water. Hence, \(P_{H_{2}} = P_{total} - P_{H_{2}O} = 841 Torr - 16.5 Torr = 824.5 Torr \)

Step 2: Convert pressure to atm and volume to L

To apply ideal gas law, convert the pressure from Torr to atm by dividing by 760 (1 atm = 760 Torr), and the volume from mL to L by multiplying by 0.001 (1L = 1000mL). Thus, \[ P_{H_{2}} = \frac{824.5 Torr}{760 Torr/atm} = 1.085 atm \] and \[ V_{H_{2}} = 159 mL*0.001 L/mL = 0.159 L\]

Step 3: Calculate the amount of Hydrogen Gas

Use the ideal gas law \( PV = nRT \) where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, T is the temperature. Solve for n (number of moles of hydrogen gas). \[ n_{H_{2}} = \frac{P_{H_{2}}V_{H_{2}}}{RT} = \frac{(1.085 atm)*(0.159 L)}{(0.0821 L*atm/mol*K)*(292 K)} = 0.00708 mol \]

Step 4: Calculate the mass of Aluminum

From the balanced chemical reaction of Aluminum with hydrochloric acid, 3 moles of H2 are produced for each 2 moles of Aluminum. So, the moles of Aluminum reacting is \[ n_{Al} = \frac{2}{3}* n_{H_{2}} = \frac{2}{3}*0.00708 mol = 0.00472 mol \] Given Aluminum's molar mass is 26.98 g/mol, the mass of Aluminium can be calculated as \[ mass_{Al} = n_{Al} * molar mass_{Al} = 0.00472 mol * 26.98 g/mol = 0.127 g \]

Step 5: Calculate the mass and percent of Iron

The mass of iron in the sample will be the total sample mass subtracted by the mass of aluminum. So, \[ mass_{Fe} = mass_{total} - mass_{Al} = 0.1924 g - 0.127 g = 0.0654 g \] The percentage of iron in the sample is calculated by \[ \% Fe = \frac{mass_{Fe}}{mass_{total}}* 100\% = \frac{0.0654 g}{0.1924 g}* 100\% = 34.0\% \]

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a cornerstone concept in chemistry and is used to relate the four important properties of gases: pressure (P), volume (V), number of moles (n), and temperature (T). The equation is expressed as \( PV = nRT \), where \( R \) is the ideal gas constant, typically \( 0.0821 \text{ L atm/mol K} \). This law helps predict how a gas will behave under different conditions.

In the exercise at hand, the Ideal Gas Law is utilized to determine the number of moles of hydrogen gas produced from the reaction of metals with hydrochloric acid. By converting the measurements to suitable units (pressure in atm, volume in liters, and temperature in Kelvin) and substituting these into the Ideal Gas Law, you can solve for the number of moles \( n \).

As a tip, always convert your units properly:

• Pressure: 1 atm = 760 Torr
• Volume: 1 L = 1000 mL
• Temperature: Kelvin = Celsius + 273.15

These conversions ensure accuracy when applying the law to real-world scenarios and calculations.

Chemical Reactions

Chemical reactions are processes where substances, known as reactants, transform into products. In this exercise, we have the reaction of aluminum and iron with hydrochloric acid (HCl). These reactions produce hydrogen gas and metal chlorides:

\[2 \text{Al} + 6 \text{HCl} \rightarrow 2 \text{AlCl}_3 + 3 \text{H}_2 \]\[\text{Fe} + 2 \text{HCl} \rightarrow \text{FeCl}_2 + \text{H}_2\]\
Understanding the stoichiometry of these reactions is crucial. Stoichiometry involves using balanced chemical equations to calculate the relative quantities of reactants and products involved in a reaction.

In our problem, the stoichiometry helps us determine how much aluminum reacts by comparing the moles of hydrogen generated. Always make sure to balance your chemical equations so that you can correctly calculate the amount of reactants needed or the amount of products formed.

Molar Mass Calculation

Molar mass is the mass of a given substance divided by its amount of substance, typically expressed in \( \text{g/mol} \). It is used to convert between the mass of a chemical substance and the number of moles in that substance, a crucial step in stoichiometric calculations.

For example, the molar mass of aluminum (Al) is \( 26.98 \text{ g/mol} \). In this exercise, you are required to find how much aluminum reacts by calculating the mass of aluminum from the moles derived through stoichiometry.

Here’s how it’s done:

• First, find the number of moles from the Ideal Gas Law.
• Next, use the stoichiometric ratios from the balanced reaction equations to find moles of aluminum.
• Finally, multiply the number of moles of aluminum by its molar mass to find the mass.

Correct calculations ensure you can determine the mass and percentage of each element in the sample mixture accurately.