Problem 104 - Chemistry The Central Science In Si Units

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 \(8.102 \mathrm{~g}\). (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?

Step-by-Step Solution

Answer

The molar mass of the unknown gas is approximately \(100.4\,\mathrm{g/mol}\). Assumptions include same flask volume and consistent conditions.

Step 1: Understanding the problem

We are given a flask that is initially evacuated and then filled with argon. The mass increases by \(3.224 \mathrm{~g}\). When filled with an unknown gas, the mass increases by \(8.102 \mathrm{~g}\). We need to calculate the molar mass of the unknown gas.

Step 2: Determining molar mass of argon

The molar mass of argon (\(\mathrm{Ar}\)) is \(39.95\,\mathrm{g/mol}\). This value will be used as a reference to estimate the molar mass of the unknown gas.

Step 3: Calculating mass ratio

To find the relationship between the argon and the unknown gas, calculate the ratio of their masses: \( \frac{8.102\,\mathrm{g}}{3.224\,\mathrm{g}} = 2.513. \)

Step 4: Estimating molar mass of unknown gas

Since the mass of the unknown gas is \(2.513\) times that of argon, apply this ratio to the molar mass of argon to estimate the molar mass of the unknown gas: \(\mathrm{Molar\,mass\,of\,unknown\,gas}=2.513 \times 39.95\,\mathrm{g/mol} = 100.4\,\mathrm{g/mol}.\)

Step 5: Assumptions made

The main assumption is that both argon and the unknown gas completely fill the flask to the same volume. Additionally, we assume that the temperature and pressure conditions are consistent for both experiments.

Key Concepts

Gas Laws

Gas laws are foundational to understanding how gases behave under different conditions. They describe relationships between pressure, volume, temperature, and the amount of gas. In the given exercise, we are indirectly applying the gas laws by comparing how argon and an unknown gas fill a flask.

• Ideal Gas Law: This is often represented by the formula \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature. This formula helps us understand that for a fixed amount of gas, the volume will change with shifts in pressure and temperature.
• Application in the Exercise: The task assumed that the gases behave ideally, meaning their behavior fits the ideal gas law. The calculation uses the idea that under consistent conditions of temperature and pressure, different gases fill the same volume in proportion to their molar mass.

It's important to be aware that real gases might show slight deviations from ideal behavior, especially under high-pressure or low-temperature conditions.

Mass Ratio

Mass ratio is a key concept in determining the relative sizes or amounts of substances. It's particularly useful when comparing different substances by mass under similar conditions.
In the exercise, we used the mass ratio to find the molar mass of an unknown gas. Here's how:

• The flask was first filled with argon, which increased the mass by \(3.224 \, \text{g} \).
• When the same flask was filled with an unknown gas, the mass increased by \(8.102 \, \text{g} \).
• By calculating the mass ratio \( \frac{8.102}{3.224} = 2.513 \), we determined the unknown gas's relative mass compared to argon.
• This ratio was then multiplied by argon's known molar mass \(39.95 \, \text{g/mol} \) to estimate the molar mass of the unknown gas, giving \(100.4 \, \text{g/mol} \).

This concept is useful in many chemical calculations, where direct measurement might not be possible.

Assumptions in Chemistry

In chemical calculations and experiments, making assumptions is a common practice to simplify complex systems. It helps us focus on main variables and ignore inconsequential factors.

• Assumption of Constant Conditions: In the exercise, an assumption was made that both gases were under the same pressure and temperature, ensuring valid comparison.
• Ideal Gas Behavior: It was presumed that gases behave ideally, which means they occupy the same volume under similar conditions, regardless of their identity.
• Full Filling of Volume: The experiment assumed both gases completely filled the flask. This is crucial to ensure the weight measurement accurately reflects the molar differences.

While assumptions can simplify calculations, it's important to recognize their limitations. In real-world scenarios, conditions might not always align with these assumptions leading to slight variances from expected results.