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Chapter 20: Problem 36

A \(0.500-\mathrm{g}\) sample of zinc-copper alloy was treated with dilute hydrochloric acid. The hydrogen gas evolved was collected by water displacement at \(27^{\circ} \mathrm{C}\) and a total pressure of \(755 \mathrm{~mm} \mathrm{Hg}\). The volume of the water displaced by the gas is \(105.7 \mathrm{~mL}\). What is the percent composition, by mass, of the alloy? (Vapor pressure of \(\mathrm{H}_{2} \mathrm{O}\) at \(27^{\circ} \mathrm{C}\) is \(26.74 \mathrm{~mm} \mathrm{Hg}\).) Assume only the zinc reacts.

Short Answer

Expert verified
The percent composition of the zinc-copper alloy is approximately 47.58% zinc and 52.42% copper.

Step by step solution

01

Calculate the pressure of hydrogen gas

First, we need to find the pressure of hydrogen gas in the system. The total pressure measured is of both hydrogen gas and water vapor. We are given the vapor pressure of water at \(27^{\circ} \mathrm{C}\), which is \(26.74 \mathrm{~mm} \mathrm{Hg}\). So, the pressure of hydrogen gas can be calculated as: Pressure of hydrogen gas = Total pressure - Pressure of water vapor Pressure of hydrogen gas = \(755 \mathrm{~mm} \mathrm{Hg} - 26.74 \mathrm{~mm} \mathrm{Hg} = 728.26 \mathrm{~mm} \mathrm{Hg}\)
02

Convert pressure to atm

We need to convert the pressure of hydrogen gas from mmHg to atm to make it compatible with other units when using the ideal gas law: Pressure of hydrogen gas in atm = Pressure of hydrogen gas in mmHg / 760 Pressure of hydrogen gas in atm = \(728.26 \mathrm{~mm} \mathrm{Hg} / 760 = 0.958 \mathrm{~atm}\)
03

Convert temperature to Kelvin

To use the ideal gas law, we also need to convert the temperature in Celsius to Kelvin: Temperature in Kelvin = Temperature in Celsius + 273.15 Temperature in Kelvin = \(27^{\circ} \mathrm{C} + 273.15 = 300.15 \mathrm{K}\)
04

Find moles of hydrogen gas

Now, we can use the ideal gas law (\(PV=nRT\)) to determine the amount (in moles) of hydrogen gas produced. We know the volume of gas (\(V=105.7 \mathrm{~mL}\)), which we can convert to liters: \(V = 105.7 \mathrm{~mL} \times \frac{1 \mathrm{L}}{1000 \mathrm{~mL}} = 0.1057 \mathrm{L}\) Next, we can rearrange the ideal gas law to solve for moles (n): \(n = \frac{PV}{RT}\) \(n = \frac{(0.958 \mathrm{~atm})(0.1057 \mathrm{L})}{(0.0821 \mathrm{~\frac{L \cdot atm}{mol \cdot K}})(300.15 \mathrm{K})}\) \(n = 0.00364 \mathrm{~mol}\)
05

Find moles of zinc

As only zinc reacts with hydrochloric acid in the alloy, the moles of hydrogen produced correspond to the moles of zinc. Therefore, moles of zinc = moles of hydrogen: \(0.00364 \mathrm{~mol}\).
06

Find mass of zinc

Now, we can determine the mass of zinc in the alloy by multiplying moles of zinc by its molar mass (\(65.38 \mathrm{~g/mol}\)): Mass of zinc = Moles of zinc × Molar mass of zinc Mass of zinc = \(0.00364 \mathrm{~mol} \times 65.38 \mathrm{~g/mol} = 0.2379 \mathrm{~g}\)
07

Calculate the percent composition of zinc

Finally, we can calculate the percent composition of zinc in the alloy by dividing the mass of zinc by the total mass of the alloy, and then multiplying by 100: Percent composition of zinc = \(\frac{0.2379 \mathrm{~g}}{0.500 \mathrm{~g}} \times 100 \% = 47.58 \%\)
08

Calculate the percent composition of copper

Since there are only two elements in the alloy, the percent composition of copper can be calculated by subtracting the percent composition of zinc from 100%: Percent composition of copper = \(100 \% - 47.58 \% = 52.42 \%\) Thus, the percent composition of the zinc-copper alloy is approximately 47.58% zinc and 52.42% copper.

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