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Chapter 10: Problem 111

A 1.42-g sample of helium and an unknown mass of \(\mathrm{O}_{2}\) are mixed in a flask at room temperature. The partial pressure of the helium is 42.5 torr, and that of the oxygen is 158 torr. What is the mass of the oxygen?

Short Answer

Expert verified
To find the mass of oxygen, first, calculate the moles of helium using the Ideal Gas Law: \(n_{He} = \frac{P_{He}V}{RT}\). Then, find the moles of oxygen using the ratio of partial pressures: \(\frac{n_{He}}{n_{O_2}} = \frac{P_{He}}{P_{O_2}}\). Finally, calculate the mass of oxygen: \(m_{O_2} = n_{O_2}\times M_{O_2}\) where \(M_{O_2}\) is the molar mass of oxygen (32 g/mol).

Step by step solution

01

Calculate the moles of Helium

The Ideal Gas Law can be expressed as: \(PV = nRT\) Where: - P is the pressure (in atm), - V is the volume (in L), - n is the moles of gas, - R is the gas constant (0.08206 L atm/(mol K)), - T is the temperature (in K). We know the pressure, mass, and molar mass of helium. First, we need to convert the given pressure from torr to atm. Then, we can calculate the moles by rearranging the Ideal Gas Law formula. 1 atm = 760 torr \(P_{He} (atm) = \frac{P_{He}(torr)}{760} = \frac{42.5}{760} \) Now, rearrange the Ideal Gas Law to get moles (n): \(n_{He} = \frac{PV}{RT}\)
02

Calculate the moles of Oxygen

We are given the partial pressures of helium and oxygen. Then the total pressure in the flask can be calculated as: \(P_{total} = P_{He} + P_{O_2}\) Convert \(P_{O_2}\) to atm: \(P_{O_2} (atm) = \frac{P_{O_2}(torr)}{760} = \frac{158}{760}\) Now, use the Ideal Gas Law to calculate the number of moles of oxygen: \(n_{O_2} = \frac{P_{O_2}V}{RT}\) Since helium and oxygen are in the same flask, the volume V and temperature T are the same for both gases. Therefore, the ratio of the moles of helium to the moles of oxygen can be calculated as: \(\frac{n_{He}}{n_{O_2}} = \frac{P_{He}}{P_{O_2}}\) From which we can find the moles of oxygen: \(n_{O_2} = n_{He}\times \frac{P_{O_2}}{P_{He}}\)
03

Find the mass of Oxygen

Now that we have the moles of oxygen, we can find the mass of the oxygen gas using the molar mass. The molar mass of \(\mathrm{O}_2\) is 32 g/mol. Then the mass of oxygen (m) can be calculated as: \(m_{O_2} = n_{O_2}\times M_{O_2}\) Where: - \(n_{O_2}\) is the moles of oxygen, - \(M_{O_2}\) is the molar mass of oxygen (32 g/mol). Substitute the values and calculate the mass of the oxygen.

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