Problem 104 - Chemistry: The Central Science in SI Units, Global Edition

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. \((\mathbf{b})\) What assumptions did you make in arriving at your answer?

Step-by-Step Solution

Answer

The estimated molar mass of the unknown gas is approximately \(102.89 \frac{\mathrm{g}}{\mathrm{mol}}\). In deriving this answer, we assumed the volume of the flask remained constant, the gases follow the ideal gas law, and the temperature and pressure remained constant during the filling process.

Step 1: Identify the Given Information

We're given the following information: 1. The mass increase when the flask is filled with argon gas is 3.224 g. 2. The mass increase when the flask is filled with the unknown gas is 8.102 g. 3. The molar mass of argon is \(39.95 \frac{\mathrm{g}}{\mathrm{mol}}\). We want to estimate the molar mass of the unknown gas.

Step 2: Establish a Relationship

Since we know that the volume of the flask is constant, we can relate the mass increase directly to the number of moles of gas filled in each case. For both argon and the unknown gas, we can write the relationship: \[\frac{\text{mass increase}}{\text{molar mass}} = \text{number of moles}\] For both argon and the unknown gas: \[\frac{\text{mass increase (Argon)}}{\text{molar mass (Argon)}} = \frac{\text{mass increase (Unknown)}}{\text{molar mass (Unknown)}}\]

Step 3: Substitute the Given Values

We will now substitute the values given in the exercise to find the molar mass of the unknown gas. Let \(M_{\text{unknown}}\) denote the molar mass of the unknown gas. \[\frac{3.224 \,\mathrm{g}}{39.95\frac{\mathrm{g}}{\mathrm{mol}}} = \frac{8.102 \,\mathrm{g}}{M_{\text{unknown}}}\]

Step 4: Solve for the Molar Mass

Now, we need to solve for the molar mass of the unknown gas, \(M_{\text{unknown}}\). \[M_{\text{unknown}} = \frac{8.102 \mathrm{~g} \cdot 39.95\frac{\mathrm{g}}{\mathrm{mol}}}{3.224 \,\mathrm{g}} \approx 102.89\frac{\mathrm{g}}{\mathrm{mol}}\]

Step 5: State the Results

The estimated molar mass of the unknown gas is approximately \(102.89 \frac{\mathrm{g}}{\mathrm{mol}}\).

Step 6: Assumptions Made

In arriving at this answer, we made the following assumptions: 1. The volume of the flask is constant for both argon and the unknown gas. 2. The gases follow the ideal gas law, so the relationship between mass, molar mass, and the number of moles holds true. 3. The temperature and pressure during the process of filling the flask with argon and the unknown gas were constant.

Key Concepts

Argon Gas

Argon gas is one of the noble gases and is chemically very stable. It makes up about 0.93% of the earth’s atmosphere. Being inert, it doesn't react with most substances, which makes it safe and versatile for various uses. In this context, argon gas is used to fill the flask in order to determine its mass properties. This gives us a basis for comparison when another unknown gas is used.

When argon gas fills a container, like a flask, we can measure the increase in mass. This is due to the added weight of the gas itself. For argon, this mass increase is 3.224 grams. Knowing the molar mass of argon is 39.95 g/mol allows us to find the number of moles of argon that has filled the flask using the formula:

• Mass = moles x molar mass

Understanding these factors is crucial for determining the molar mass of an unknown gas, by comparing to a known quantity like argon.

Ideal Gas Law

The Ideal Gas Law is an important concept when studying gases and their behavior. It is expressed in the formula: \[ PV = nRT \]where:

• \( P \) is the pressure
• \( V \) is the volume
• \( n \) is the number of moles
• \( R \) is the ideal gas constant
• \( T \) is the temperature

In this exercise, we assume the gases behave ideally. This means the gas molecules do not interact with each other and occupy no volume themselves. The relationship \( \frac{\text{mass}}{\text{molar mass}} = \text{number of moles} \) stems from the Ideal Gas Law, when conditions of temperature and pressure are constant.

In practice, we used argon gas, which is known, to determine the behavior of the unknown gas, assuming both gases behave ideally under the same conditions.

Unknown Gas

In this problem, the unknown gas is the target for estimation of its molar mass. Once the flask is filled with this gas, its mass increases by 8.102 grams. To find the molar mass, we can use the comparison with the known argon gas. By setting up proportions based on the masses and solving for the unknown, we obtain a value.

The following equation helps find the molar mass of the unknown gas:\[\frac{\text{mass increase (Argon)}}{\text{molar mass (Argon)}} = \frac{\text{mass increase (Unknown)}}{M_{\text{unknown}}}\]By substituting the known variables and solving this equation, we find \( M_{\text{unknown}} \approx 102.89 \frac{\mathrm{g}}{\mathrm{mol}} \).

• This result assumes variables like temperature and pressure are constant
• And that the flask volume remains unchanged when different gases are filled