Question

When a large evacuated flask is filled with argon gas, its mass increases by 3.224 \(\mathrm{g}\) . When the same flask is again evacuated and then filled with a gas of unknown molar mass, the mass increase is 8.102 g. (a) Based on the molar mass of argon, estimate the molar mass of the unknown gas. (b) What assumptions did you make in arriving at your answer?

Short answer

The molar mass of the unknown gas is estimated to be 100.4 g/mol. We made the following assumptions: the volume, temperature, and pressure of the flask remain the same for both gases; the unknown gas behaves as an ideal gas; and Avogadro's hypothesis holds true for both gases.

Step-by-step solution

Identify the given information and the variables

We are given the mass increase for the argon gas and the unknown gas. We can denote the mass increase of the argon gas as m₁=3.224 g and the mass increase of the unknown gas as m₂=8.102 g. We also know the molar mass of argon gas M₁=39.95 g/mol. We need to calculate the molar mass of the unknown gas, which we can denote as M₂.

Calculate number of moles of argon gas

We can calculate the number of moles of argon gas using the mass of argon gas and its molar mass. The formula to find the number of moles (n) is: n = mass / molar mass For argon gas, we can write: n₁ = m₁ / M₁ n₁ = 3.224 g/ 39.95 g/mol n₁ = 0.0807 mol

Apply Avogadro's hypothesis

Avogadro's hypothesis states that equal volumes of gases at the same temperature and pressure have the same number of moles. As we are assuming that the volume, temperature, and pressure of the flask remain the same for both gases, we can apply Avogadro's hypothesis. Since the number of moles is the same for both gases, we can write: n₁ = n₂ 0.0807 mol = n₂

Calculate the molar mass of the unknown gas

Now that we know the number of moles of the unknown gas, we can find its molar mass using the mass of the unknown gas. We can use the formula: M₂ = m₂ / n₂ M₂= 8.102 g / 0.0807 mol M₂ = 100.4 g/mol The estimated molar mass of the unknown gas is 100.4 g/mol.

Identify assumptions made in the calculations

The following assumptions were made during the calculations: 1. The volume, temperature, and pressure of the flask remain the same for both gases. 2. The unknown gas behaves as an ideal gas. 3. Avogadro's hypothesis holds true for both gases. Finally, the molar mass of the unknown gas is estimated to be 100.4 g/mol.

Key concepts

Avogadro's Hypothesis

Avogadro's Hypothesis is an important concept in gas chemistry. It states that equal volumes of gases at the same temperature and pressure contain the same number of particles, or moles. This is true regardless of the kind of gas we're dealing with.

In our exercise, we dealt with two gases in a flask. Since we assume the conditions of temperature, pressure, and volume remain constant for both gases, Avogadro's Hypothesis allows us to equate the number of moles of argon to that of the unknown gas.

Therefore, when we calculated the number of moles of argon, we could directly set it equal to the moles of the unknown gas. This simplifies the solution significantly, since without this principle, we would need additional information about the unknown gas, which we don’t have.

Ideal Gas Law

The Ideal Gas Law is a fundamental tool in understanding gas behavior. It’s expressed as PV = nRT, where:

• P is pressure.
• V is volume.
• n is the number of moles.
• R is the gas constant.
• T is temperature.

Even though we did not directly use the Ideal Gas Law formula in solving our exercise, the steps rely on its principles.

For both gases in the flask, the conditions—pressure, volume, and temperature—are assumed to be constant. This aligns with the assumptions derived from the Ideal Gas Law. By maintaining these conditions, it allows us to assert that both gases behave ideally.

This assumption simplifies calculations and helps us use the molar masses to determine the properties of the gases, such as mass change, similar to how we substitute in the gas law.

Moles Calculation

Calculating the number of moles is crucial for solving gas-related problems. Moles represent the quantity of particles in a given mass of substance. The formula used is:

• n = \( \frac{\text{mass}}{\text{molar mass}} \)

In our case, this formula was used to find the number of moles of argon gas in the flask.

Given that argon has a known molar mass of 39.95 g/mol, and the mass increase observed was 3.224 g, we simply divided these values to get the moles of argon. This step is essential because it then allows us to determine the moles of the unknown gas using Avogadro's Hypothesis.

Ultimately, understanding moles calculations helps bridge the gap between mass and molar mass, giving us a complete understanding of the gas properties inside the flask.