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 \(\left.23^{\circ} \mathrm{C}\right)\)

Step by step solution

01

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.

02

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.

03

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.

04

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.

05

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 \( \frac{Mass_{Al}}{Mass_{alloy}} * 100% \).

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