Problem 32 - Chemistry The Central Science In Si Units

Question

Suppose you are given two flasks at the same temperature, one of volume \(2 \mathrm{~L}\) and the other of volume \(3 \mathrm{~L}\). The 2 -L flask contains \(4.8 \mathrm{~g}\) of gas, and the gas pressure is \(x \mathrm{kPa}\). The 3 -L flask contains \(0.36 \mathrm{~g}\) of gas, and the gas pressure is \(0.1 x\). Do the two gases have the same molar mass? If not, which contains the gas of higher molar mass?

Step-by-Step Solution

Answer

No, the gas in the 2-L flask has a higher molar mass.

Step 1: Calculate Molar Mass for 2-L Flask

First, we need to find the molar mass of the gas in the 2-L flask using the ideal gas law. Rearrange the formula for molar mass \( M = \frac{m}{PV/RT} \), where \( m = 4.8 \, \text{g} \), \( P = x \, \text{kPa} \), \( V = 2 \, \text{L} \), and \( T \) is constant.Let \( R \) = 8.3145 J/mol K, and since PV = nRT, find the number of moles: \( n = \frac{PV}{RT} \).Thus, \( M = \frac{4.8 \, \text{g}}{\frac{2x \cdot 10^3}{8.3145 \times T}} \), which simplifies to get \( M_1 = \frac{4.8 \times 8.3145T}{2x \cdot 1000} \).

Step 2: Calculate Molar Mass for 3-L Flask

For the 3-L flask, the pressure is \(0.1x\). Again, using the ideal gas law in the form \( M = \frac{m}{PV/RT} \), where \( m = 0.36 \, \text{g} \), \( P = 0.1x \, \text{kPa} \), and \( V = 3 \, \text{L} \).Find the number of moles: \( n = \frac{PV}{RT} \).Thus, \( M = \frac{0.36 \, \text{g}}{\frac{0.3x \cdot 10^3}{8.3145 \times T}} \), which simplifies to get \( M_2 = \frac{0.36 \times 8.3145T}{0.3x \cdot 1000} \).

Step 3: Compare Molar Masses

Now, compare \( M_1 \) and \( M_2 \) to determine which gas has a higher molar mass.- Simplified expressions: \( M_1 = \frac{4.8 \times 8.3145T}{2x \times 1000} \) and \( M_2 = \frac{0.36 \times 8.3145T}{0.3x \times 1000} \).- Plug in similar terms and constants will be canceled due to same \( T \) and \( R \). Compare the ratios of the weights and volume-pressure terms: \( \frac{4.8}{2} \) vs. \( \frac{0.36}{0.3} \). - 2-L Calculations: \( 4.8/2 = 2.4 \) - 3-L Calculations: \( 0.36/0.3 = 1.2 \)Thus, \( M_1 > M_2 \).

Step 4: Conclusion: Determine which Gas has a Higher Molar Mass

Since \( M_1 > M_2 \), the gas in the 2-L flask has a higher molar mass. Therefore, the molecules in the 2-L flask are heavier compared to the 3-L flask.

Key Concepts

Ideal Gas Law

The ideal gas law is an essential tool in understanding the behavior of gases under various conditions. It combines several fundamental gas laws into one powerful equation: \[ PV = nRT \] where \( P \) represents the pressure of the gas, \( V \) is its volume, \( n \) denotes the number of moles, \( R \) is the ideal gas constant, and \( T \) stands for temperature in Kelvin. This law provides a relationship between these parameters and allows us to predict the behavior of a gas by knowing some of the variables. The idea is that if you have a certain gas and you know how it behaves at known conditions, you can predict its behavior at new conditions when the temperature, pressure, or volume changes. By rearranging the ideal gas law into different forms, we can solve for any unknown when the other parameters are given. For instance, solving for the molar mass \( M \) can be done through a rearrangement involving density and the ideal gas law's components, which is crucial for problems where determining the identity of a gas is necessary.

Pressure-Volume Relationship

Gases behave in a way that their pressure and volume are related, particularly if the temperature and number of moles remain consistent. This relationship is often explained through Boyle's Law, part of the ideal gas law's foundation. Boyle's Law states that, for a given amount of gas held at a constant temperature, the pressure of the gas is inversely proportional to its volume:\[ P_1V_1 = P_2V_2 \] This means that as the volume of the gas increases, its pressure decreases if no other factors change. Conversely, reducing the volume will increase the pressure. This principle was showcased in the exercise by comparing the gases in a 2-L and a 3-L flask. Understanding this principle allows us to predict changes in pressure when a gas expands or contracts, thereby inferring something about its density and molar mass.

Gas Flask Comparison

When comparing gases within containers, especially under identical conditions other than volume, significant insights into their properties can be gained. In the context of the original problem, the task was to compare two gases distributed in flasks of different sizes at the same temperature. The objective was to determine if the gases had the same molar mass using the rearranged ideal gas law equation. By calculating the molar mass for each gas based on given conditions—such as pressure, volume, and mass—we were able to establish a mathematical expression for each flask scenario. In the step-by-step solution given, both gas masses and the volume-pressure relationships were used to derive the molar masses. A comparison of these masses revealed the 2-L flask gas had a higher molar mass. This conclusion shows the application of the ideal gas law and supports understanding the relationship between pressure, volume, and mass in determining a gas’s molar mass.