Problem 107 - Chemistry The Central Science

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?

Step-by-Step Solution

Answer

The molar mass of the unknown gas is approximately 100.98 g/mol. We assumed that the same volume and pressure were used for both argon and the unknown gas, the unknown gas behaves as an ideal gas, and the temperature remains constant throughout the experiment.

Step 1: Gather the given information

We are given the following information: - Mass increase when flask is filled with argon gas: 3.224 g - Mass increase when flask is filled with unknown gas: 8.102 g - Molar mass of argon: 39.95 g/mol Assume that the same volume and pressure of gas were used in both cases. As we don't have those values, we'll use V and P as placeholders in the calculations.

Step 2: Calculate the moles of argon gas

First, we'll calculate the moles of argon gas using the mass increase value and the molar mass of argon. The formula for calculating moles (n) is: n = mass / molar_mass Plugging in the given values, we get the moles of argon gas: n(Ar) = 3.224 g / 39.95 g/mol ≈ 0.0807 mol

Step 3: Apply the ideal gas law to determine volume and pressure

The ideal gas law is given by: PV = nRT where: - P is the pressure - V is the volume - n is the number of moles - R is the ideal gas constant (8.314 J/mol·K) - T is the temperature in kelvins. We are not given the temperature, but since the conditions are the same for both argon and the unknown gas, we can assume that the temperature is constant. Converting the ideal gas law into a relationship between moles (n) and volume and pressure (V and P), we get: V and P = n R T For argon, we have: V and P(Ar) = 0.0807 mol * R * T

Step 4: Calculate the moles of unknown gas

Since, according to ideal gas law, PV = nRT, and we're assuming the pressure and temperature are the same for both argon and the unknown gas, we can infer that: m(Ar)*M(unknown) = m(unknown)*M(Ar) Where m is mass and M is molar mass. Rearranging, we find the molar mass of the unknown gas: M(unknown) = (m(unknown) * M(Ar)) / m(Ar) Plugging in the given values: M(unknown) = (8.102 g * 39.95 g/mol) / 3.224 g ≈ 100.98 g/mol

Step 5: State assumptions and final answer

The molar mass of the unknown gas is approximately 100.98 g/mol. The assumptions we made in the process of solving the problem are: 1. The same volume and pressure were used for both argon and the unknown gas. 2. The unknown gas also behaves as an ideal gas. 3. The temperature remains constant throughout the experiment.

Key Concepts

Molar Mass

Molar mass plays a critical role in understanding the amount of a substance present. It is defined as the mass of one mole of a given substance and is expressed in grams per mole (g/mol). In this exercise, we used the molar mass of argon gas, which is given as 39.95 g/mol, to relate it to the mass increase observed when the flask is filled. By knowing the molar mass of one gas, we can estimate the molar mass of another via calculation based on mass and moles, which follows the formula:

• n = mass / molar mass

Understanding this concept allows us to interchangeably use different elements and compounds by comparing their masses, allowing for calculations like the one shown in this exercise, where we opted to find the unknown gas's molar mass.

Evacuated Flask

An evacuated flask serves as a useful tool in chemistry for collecting gas without the interference of other substances. It is a container from which all air and other gases have been removed, making it a vacuum environment. This ensures that when we introduce another gas, like argon or an unknown gas, any change in mass is solely due to the gas being introduced.
In our exercise, the flask's initial empty state provides a baseline. When filled first with argon and then with an unknown gas, the resultant mass increases give us important details about the amount and mass of gas added. This step of ensuring the flask is evacuated each time guarantees accuracy in weighing the actual gas inside, removing errors from otherwise present air mass.

Argon Gas

Argon gas is a noble gas and is known for being chemically inert and colorless. In the periodic table, it sits between neon and krypton and is widely used in environments where chemical reactions are harmful or unintended, such as in gas-filled electric lightbulbs.
In the exercise, argon's known properties and molar mass (39.95 g/mol) are utilized to benchmark the measurements against another unknown gas. Because it behaves ideally under many conditions and doesn't react, it provides a stable measure for conducting gas volume and mass comparison, rendering it an ideal candidate for preliminary calculations in experiments.

Assumptions in Chemistry

Assumptions are a fundamental part of conducting chemical calculations, particularly when detailed experimental data aren't all available. Here, we made several assumptions to simplify our calculations:

• The same volume and pressure applied to both the argon and unknown gases. This allows us to comparably calculate molar masses since the equation under equal conditions becomes simpler.
• The unknown gas behaves like an ideal gas, meaning it follows the ideal gas law closely (PV = nRT). This is generally reasonable under standard conditions.
• Constant temperature is assumed to make sure that any changes in the behavior of gases are not due to temperature fluctuations.

Such assumptions streamline complex calculations, but it is crucial to recognize when these might lead to slight inaccuracies, especially when a gas doesn't behave ideally or other conditions are variable.