Problem 109 - Chemistry: The Central Science

Question

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?

Step-by-Step Solution

Answer

The mass of oxygen in the flask is 42.21 g. To find this value, we used the Ideal Gas Law for partial pressures, calculated the moles of helium and oxygen, and then converted the moles of oxygen to mass, considering their molar masses.

Step 1: Write the Ideal Gas Law equation for partial pressures.

We can write the Ideal Gas Law in terms of partial pressures as: \(P_{total} V = n_{total} RT\) , where \(P_{total}\) = partial pressure of Helium + partial pressure of Oxygen. We also have the individual Ideal Gas Law for each gas in the mixture: \(P_{He} V = n_{He} RT\) \(P_{O_2} V = n_{O_2} RT\)

Step 2: Calculate the moles of helium.

We can use the Ideal Gas Law to find the moles of helium. First, find the molar mass of helium: Molar mass of He = 4.00 g/mol. Now, divide the mass of helium by its molar mass to find the moles of helium: \(n_{He} = \frac{mass}{molar\;mass} = \frac{1.42}{4.00} = 0.355\;mol\)

Step 3: Calculate the moles of oxygen.

We know the partial pressures of helium and oxygen. Divide the partial pressure of oxygen by the partial pressure of helium to find the ratio of moles of oxygen to moles of helium: \(\frac{n_{O_{2}}}{n_{He}} = \frac{P_{O_{2}}}{P_{He}} = \frac{158\;torr}{42.5\;torr} = 3.7176\) Now, use the ratio to find the moles of oxygen: \(n_{O_{2}} = 3.7176 \times n_{He} = 3.7176 \times 0.355 = 1.319\;mol\)

Step 4: Calculate the mass of oxygen.

Now that we have the moles of oxygen, we can find the mass of oxygen. First, find the molar mass of oxygen: Molar mass of \(O_2\) = 32.00 g/mol. Now, multiply the moles of oxygen by its molar mass to find the mass of oxygen: \(mass_{O_{2}} = moles \times molar\;mass = 1.319 \times 32.00 = 42.21\;g\) Thus, the mass of oxygen in the flask is 42.21 g.

Key Concepts

Partial Pressures Simplified

When dealing with gases, partial pressures represent the pressure contribution of an individual gas in a mixture. This is an important concept in the study of gas behavior, especially when using the Ideal Gas Law. In a mixture, each gas exerts its own pressure, known as its partial pressure. This pressure depends on the number of moles of the gas, the temperature, and the volume of its container.

The total pressure of a gas mixture is simply the sum of these partial pressures. For the problem you are working on, the partial pressures of helium and oxygen sum up to provide the total pressure of the mix:

• The partial pressure of helium is given as 42.5 torr.
• The partial pressure of oxygen is given as 158 torr.

Knowing how to add these pressures is key to understanding the system's behavior under these conditions.

Moles Calculation Explained

Calculating the number of moles is a fundamental part of applying the Ideal Gas Law. Moles are a way to express amounts of a substance in chemistry. They help us connect macroscopic and microscopic events through the use of Avogadro's number.

To convert from mass to moles, we need to use the formula: \[ n = \frac{\text{mass of the gas}}{\text{molar mass}} \] Let’s see how this works:

• For helium, with a molar mass of 4.00 g/mol, if we have 1.42 grams, we find the moles by dividing the mass by the molar mass: \[ n_{He} = \frac{1.42}{4.00} = 0.355 \text{ mol} \]
• Understanding this conversion is crucial as it allows us to apply the ratio of partial pressures to find moles of the other gas, in this case, oxygen, within the mixture.

By using the partial pressure data, you can find a relation between different gases in the mixture without knowing individual temperatures and volumes.

Molar Mass: A Key Tool

Molar mass is an essential element when working with the Ideal Gas Law. It allows you to convert between grams and moles, which is a critical skill when solving chemistry problems.

Each element has a specific molar mass which tells you how much one mole of that substance weighs in grams. For oxygen in its diatomic form, which is common in problems like yours, the molar mass is 32.00 g/mol.

• You calculate the molar mass of a compound by adding up the molar masses of its individual elements.
• For example, oxygen gas \((O_2)\) is made of two oxygen atoms, each with a molar mass of 16.00 g/mol, leading to a molar mass of 32.00 g/mol for \(O_2\).
• Once you find the moles of a substance using the partial pressure and moles calculation, like for oxygen, multiply by its molar mass: \[ \text{mass of } O_2 = 1.319 \text{ mol} \times 32.00 \frac{\text{g}}{\text{mol}} = 42.21 \text{ g} \]

This conversion from moles to mass helps you quantify how much of a substance you actually have in a container, tying together the concepts of partial pressures, moles, and molar mass.