Problem 31 - Chemistry: The Central Science in SI Units, Global Edition

Question

Suppose you are given two 2 -L flasks and told that one contains a gas of molar mass 28 , the other a gas of molar mass 56 , both at the same temperature and pressure. The mass of gas in the flask \(A\) is \(1.0 \mathrm{~g}\) and the mass of gas in the flask \(\mathrm{B}\) is \(2.0 \mathrm{~g}\). Which flask contains the gas of molar mass 28 , and which contains the gas of molar mass 56 ?

Step-by-Step Solution

Answer

Flask A contains the gas of molar mass 56, and Flask B contains the gas of molar mass 28.

Step 1: Write down the given information

We are given the following information: - Flask A contains 1.0 g of gas - Flask B contains 2.0 g of gas - Molar mass of gas 1 = 28 g/mol - Molar mass of gas 2 = 56 g/mol - Both flasks have the same temperature and pressure

Step 2: Calculate the number of moles for each flask

Using the given mass and molar mass, we can calculate the number of moles for each flask. Let's denote gas 1 as the gas with molar mass 28 and gas 2 as the gas with molar mass 56. For Flask A, let the number of moles be \(n_A\), we have: \(n_A = \frac{mass_A}{molar \: mass \: of \: gas}\) For Flask B, let the number of moles be \(n_B\), we have: \(n_B = \frac{mass_B}{molar \: mass \: of \: gas}\) We will calculate these values assuming both possibilities, i.e., Flask A contains gas 1 and Flask B contains gas 2 and vice versa.

Step 3: Compare the number of moles for each possibility

Now, we will calculate and compare the number of moles in both possibilities: Case 1 - Flask A contains gas 1 and Flask B contains gas 2: - For Flask A: \(n_A = \frac{1.0 g}{28\: g/mol} = 0.0357\: mol\) - For Flask B: \(n_B = \frac{2.0 g}{56\: g/mol} = 0.0357\: mol\) Case 2 - Flask A contains gas 2 and Flask B contains gas 1: - For Flask A: \(n_A = \frac{1.0 g}{56\: g/mol} = 0.0179\: mol\) - For Flask B: \(n_B = \frac{2.0 g}{28\: g/mol} = 0.0714\: mol\)

Step 4: Find the correct possibility

From the ideal gas law, we know that the number of moles of gas present in a container depends on the pressure, volume, and temperature of the system. Since both flasks have the same temperature and pressure, the flask containing more gas molecules (higher number of moles) will have lower molar mass. From the calculations in Step 3, we can infer the following: - In Case 1, Flask A and Flask B have the same number of moles (0.0357 mol), which is not possible since one flask contains gas with a molar mass of 28, and the other contains gas with a molar mass of 56. - In Case 2, Flask B contains more gas molecules (0.0714 mol) than Flask A (0.0179 mol). Given the higher number of moles in Flask B, it must contain the gas with the lower molar mass, making Flask B the flask that contains the gas of molar mass 28, and Flask A the flask containing the gas of molar mass 56.

Step 5: Conclusion

Flask A contains the gas of molar mass 56, and Flask B contains the gas of molar mass 28.

Key Concepts

Molar Mass

Molar mass is an essential concept when dealing with chemical reactions and compounds. It represents the mass of one mole of a substance, measured in grams per mole (g/mol). Think of it as a bridge between the microscopic world of atoms and molecules and the macroscopic amounts we can measure in the lab. Calculating the molar mass of a compound involves summing the atomic masses of the elements in the compound's formula.
To find the molar mass:

• Identify each element in the compound.
• Find the atomic mass of each element, usually found on the periodic table.
• Add the atomic masses, considering the number of atoms of each element present in the compound.

In our original exercise, the molar mass of the gases is critical to solving which gas is in which flask. For example, a gas with a molar mass of 28 g/mol is lighter (per mole) than a gas with a molar mass of 56 g/mol, influencing other calculations and conclusions.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that connects several gas properties, providing a useful tool for understanding gas behavior. The law is expressed as:\[ PV = nRT \]Where:

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

This formula allows scientists to predict how a gas will respond to changes in conditions such as temperature, pressure, and volume. In the original exercise, the ideal gas law highlights that, at constant temperature and pressure, gases in the same volume must have the same number of moles if they were of the same molar mass. Because we observe different masses, comparing moles shows us the differences in molar mass between the gases in the flasks.

Moles Calculation

Calculating the number of moles is crucial in stoichiometry for relating mass to the amount of substance. Moles provide a way to count particles by relating their mass to the quantity of substance through Avogadro’s number. The mole formula is:\[ n = \frac{mass}{molar\ mass} \]Where:

• \( n \) is the number of moles
• \( mass \) is the mass of the substance
• \( molar\ mass \) is the molar mass of the substance

In the exercise context, by calculating the number of moles of gas in each flask, we could distinguish which gas corresponded to which flask despite having different masses, since the moles could ultimately indicate the gas's molar mass. Therefore, by observing the moles, we can solve the exercise by assigning lighter and heavier molar masses to the correct flasks.