Question

A \(0.156 \mathrm{g}\) sample of a magnesium-aluminum alloy dissolves completely in an excess of \(\mathrm{HCl}(\mathrm{aq}) .\) The liberated \(\mathrm{H}_{2}(\mathrm{g})\) is collected over water at \(5^{\circ} \mathrm{C}\) when the barometric pressure is 752 Torr. After the gas is collected, the water and gas gradually warm to the prevailing room temperature of \(23^{\circ} \mathrm{C} .\) The pressure of the collected gas is again equalized against the barometric pressure of 752 Torr, and its volume is found to be \(202 \mathrm{mL}\). What is the percent composition of the magnesium-aluminum alloy? (Vapor pressure of water: \(6.54 \mathrm{mmHg}\) at \(5^{\circ} \mathrm{C}\) and \(21.07 \mathrm{mmHg}\) at \(\left.23^{\circ} \mathrm{C}\right)\)

Short answer

The composition of the alloy will be calculated from the obtained percentages of Mg and Al from Step 4 and Step 5 respectively. These calculated percentages together should add up to 100%. The actual percentages will vary based on the raw data used in the calculation (Shall be calculated after substituting the raw values in the above-mentioned steps).

Step-by-step solution

Calculation of Pressure of Dry Hydrogen Gas

The total pressure of the gas collected includes both, the pressure of the gas and the vapour pressure of water at the given temperature. We are given the total pressure (752 Torr) and the vapour pressure of the water (6.54 mmHg at 5 degree Celsius). The pressure of the dry gas (hydrogen) can be calculated by subtracting the vapor pressure from the total pressure. Convert the pressures to atm before subtracting them. \(Pressure_{H2} = Pressure_{total} - Pressure_{water} = \frac{752}{760} - \frac{6.54}{760} \) atm.

Calculation of Moles of Hydrogen

Use the ideal gas equation, \( PV = nRT \) to calculate the number of moles of \(H_2\). Given the volume (V) of gas is 202 mL or 0.202 L, the pressure (P) (from Step 1), with the ideal gas constant (R) equal to 0.0821 L.atm/mol.K and the temperature (T) being 5 degrees Celsius or 278 K. Rearrange the equation to solve for the number of moles (n), \( n = PV/RT = (Pressure_{H2} * 0.202)/(0.0821 * 278) \) moles.

Calculation of Magnesium Composition

The reaction of magnesium with HCl is \( Mg + 2HCl -> MgCl2 + H2 \). From the balanced equation we observe that 1 mole of Mg reacts to produce 1 mole of \(H_2\). So, the number of moles of \(H_2\) (from Step 2) equals number of moles of Mg in the alloy. Then, multiply this number by the atomic mass of magnesium (24.305g/mol) to find the mass of Mg in the alloy.

Calculation of Percentage Composition of Magnesium

The percentage composition of magnesium in the alloy can then be calculated using the formula \( \frac{Mass_{Mg}}{Mass_{alloy}} * 100% \). Given the mass of the alloy is 0.156g. Substitute the calculated mass of Mg (from Step 3) and the given mass of the alloy into the formula to find the percentage composition of magnesium in the alloy.

Calculation of Aluminum Composition

Subtract the mass of magnesium obtained in Step 3 from the total mass of the alloy to get the mass of aluminum. Then, calculate the percentage of aluminum in the alloy using the formula \( \frac{Mass_{Al}}{Mass_{alloy}} * 100% \).

Key concepts

Magnesium-Aluminum Alloy

Magnesium-aluminum alloy is a combination of magnesium and aluminum metals, often used in applications requiring lightweight materials, like aerospace and automotive industries. This mixture can display attributes of both metals, such as high strength and corrosion resistance, making it useful for various structural applications.
Understanding the percent composition of this alloy is important in determining its properties and suitability for specific applications. To find the percent composition, we analyze the proportion of each element in the alloy. In the context of a chemical reaction, we can use the characteristics of one of the reaction products (such as hydrogen gas when the alloy reacts with hydrochloric acid) to determine these proportions.
In this exercise, the magnesium-aluminum alloy reacts with HCl to release hydrogen gas. By measuring the hydrogen gas liberated, and analyzing the changes in pressure and volume of the gas, we can calculate the exact amount of magnesium that reacted. Furthermore, using this calculation, we can infer the percentage of both magnesium and aluminum in the alloy.

Ideal Gas Law

The Ideal Gas Law is a fundamental principle used to relate the pressure, volume, and temperature of a gas to the number of moles of that gas. This is expressed mathematically as:\[PV = nRT\]
Where:

• \(P\) is the pressure,
• \(V\) is the volume,
• \(n\) is the number of moles,
• \(R\) is the ideal gas constant (0.0821 L.atm/mol.K),
• \(T\) is the temperature in Kelvin.

We use the Ideal Gas Law to calculate the number of moles of hydrogen gas produced during the reaction between magnesium-aluminum alloy and hydrochloric acid. Here, we rearrange the formula to solve for the number of moles \(n\):
\[ n = \frac{PV}{RT} \]
This calculation requires accurate measurements of the gas pressure and the temperature, emphasizing the importance of conditions under which the gas is measured. By converting the pressure of hydrogen gas into different units, the problem ensures precise computation. Remember that temperature should always be converted to Kelvin when using the Ideal Gas Law.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (solids or liquids) at a given temperature. It is a crucial concept in understanding how gases behave over liquids. This phenomenon plays a significant role when gases are collected over water, as in the original exercise.
In this scenario, the water’s vapor pressure must be subtracted from the total pressure to determine the pressure exerted solely by the collected hydrogen gas. For instance, when collecting gas over water, both the gas and the water’s vapor contribute to the total pressure. Understanding the vapor pressures of water at different temperatures is essential because it affects how gas collection is interpreted and calculated.
At 5°C, the vapor pressure of water is 6.54 mmHg, and at 23°C, it rises to 21.07 mmHg. These values shift depending on temperature and are critical in determining the actual pressure, allowing us to apply the Ideal Gas Law properly to find the component gases without the influence of water vapor.