Question

Suppose you are given two \(1-\) flasks and told that one contains a gas of molar mass 30 , the other a gas of molar mass 60 , both at the same temperature. The pressure in flask \(A\) is \(x\) atm, and the mass of gas in the flask is 1.2 \(\mathrm{g}\) . The pressure in flask \(\mathrm{B}\) is 0.5\(x\) atm, and the mass of gas in that flask is 1.2 \(\mathrm{g}\) . Which flask contains the gas of molar mass \(30,\) and which contains the gas of molar mass 60\(?\)

Short answer

Flask A contains the gas with molar mass 30 and Flask B contains the gas with molar mass 60. This is because Flask A has twice the volume of Flask B, and we find that the number of moles in Flask A is greater than the number of moles in Flask B (\(0.04 > 0.02\)).

Step-by-step solution

List given information and find the number of moles in each flask

We have the following information: - Flask A: Molar mass of either 30 or 60, pressure x atm, mass 1.2 g - Flask B: Molar mass of either 30 or 60, pressure 0.5x atm, mass 1.2 g Using the Ideal Gas Law, we can also write the equation for both flasks: \(V_A = \frac{n_A RT}{Px}\), where V_A represents the volume of flask A, and n_A the number of moles. \(V_B = \frac{n_B RT}{0.5Px}\), where V_B represents the volume of flask B, and n_B the number of moles.

Determine the volume ratio of flask A to flask B

To compare the volumes of flask A and flask B, we can divide the equation for V_A by the equation for V_B: \(\frac{V_A}{V_B} = \frac{n_A RT}{Px}\times \frac{0.5Px}{n_B RT}\) Since the same gas constant R and temperature T are given for both flasks, these values will cancel out: \(\frac{V_A}{V_B} = \frac{n_A}{0.5n_B}\) It means that flask A has a volume twice the volume of Flask B.

Use molar mass to find which flask contains the gas of molar mass 30 and 60

Using the information provided for each flask, we know that: In flask A: Molar Mass A = \(\frac{1.2}{n_A}\) In flask B: Molar Mass B = \(\frac{1.2}{n_B}\) Since flask A has twice the volume of flask B, if flask A has the lighter gas (molar mass 30), the number of moles in flask A would be greater than the number of moles in flask B due to the Ideal Gas Law. Therefore, we can assume: - Flask A contains the gas with molar mass 30 - Flask B contains the gas with molar mass 60 We can now verify this by calculating the number of moles for each flask: For flask A (molar mass 30): \(n_A = \frac{1.2}{30} = 0.04\, moles\) For flask B (molar mass 60): \(n_B = \frac{1.2}{60} = 0.02\, moles\) Since \(n_A > n_B\), and flask A has twice the volume of flask B, our assumption is correct, and we can conclude that flask A contains the gas with molar mass 30 and flask B contains the gas with molar mass 60.

Key concepts

Molar Mass

When studying gases, the concept of molar mass is critical in understanding how much of a particular gas is present. Molar mass, typically expressed in units of grams per mole (g/mol), refers to the mass of one mole of a substance. It is the bridge between the microscopic world of atoms and molecules and the macroscopic world where we can measure mass in grams.

In the exercise, by knowing the mass of gas and its molar mass, we can calculate the number of moles present in the flask using the relationship:
\[ \text{{number of moles (n)}} = \frac{{\text{{mass of gas}}}}{{\text{{molar mass}}}}. \]
This allows us to compare the amount of substance present in two flasks containing different gases. By calculating and comparing the number of moles of gas in each flask, we can deduce which gas has which molar mass based on the given conditions - in this case, the gases that occupy the flask with double the volume should have half the molar mass. Molar mass helps us to understand the ratio of mass to moles, providing the basis for further calculations related to gas behavior.

Gas Pressure

The pressure exerted by a gas is a result of gas particles colliding with the walls of their container. Gas pressure is a measure of the force applied by the gas particles per unit area and is influenced by several factors, including the amount of gas (number of moles), temperature, and volume of the container.

In our exercise, pressure is a variable that we considered in determining the volumes of the gases. Given that the two flasks have different pressures, it tells us about the force per unit area that the gas particles in flask A and flask B are applying to the walls of their respective containers. We used the concept of pressure in the Ideal Gas Law equation to further understand the relationship between the pressure and volume of the gases:
\[ PV = nRT, \]
where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature. This relationship is vital for comparing the conditions of the two gases and ultimately aids in identifying which flask contains the lighter or heavier gas based on the differing pressures.

Number of Moles

Number of moles is a fundamental quantity in chemistry that represents a count of the number of particles in a substance. One mole of any substance contains the same number of particles – Avogadro's number, which is approximately \(6.022 \times 10^{23}\) particles.

In the context of the exercise, understanding the concept of moles is essential for interpreting the amount of gas within each flask. By using the molar mass and the mass of the gas, we're able to calculate the number of moles. This information is important because the Ideal Gas Law assumes that the same number of moles of any gas will occupy the same volume at a given temperature and pressure - a cornerstone of comparing different gases under various conditions.
To solve the problem, we calculated the number of moles in each flask with the equation provided earlier. This step is key in determining which gas corresponds to which molar mass, completing the puzzle that combines all concepts - the molar mass, gas pressure, and number of moles. By linking these core concepts, students can better approach and solve exercises on gas properties and behavior.