Question

Calculate the ratio of effusion rates for Ar and Kr.

Short answer

The ratio of effusion rates for Argon (Ar) to Krypton (Kr) is approximately 1.448.

Step-by-step solution

Write down the effusion rate formula

The effusion rate for a gas can be calculated using Graham's law of effusion, which states that the rate of effusion is inversely proportional to the square root of its molar mass (M). The formula for effusion rate (r) is given by: \[ r \propto \frac{1}{\sqrt{M}} \]

Identify the molar masses of Argon (Ar) and Krypton (Kr)

From the periodic table, we find the molar masses of the gases: Argon (Ar) has a molar mass of approximately 39.95 g/mol, and Krypton (Kr) has a molar mass of approximately 83.80 g/mol.

Establish the ratio of the effusion rates using Graham's law

Now apply Graham's law to find the ratio of the effusion rates of Argon to Krypton: \[ \frac{r_{\text{Ar}}}{r_{\text{Kr}}} = \sqrt{\frac{M_{\text{Kr}}}{M_{\text{Ar}}}} \] Substitute the molar masses into the equation: \[ \frac{r_{\text{Ar}}}{r_{\text{Kr}}} = \sqrt{\frac{83.80}{39.95}} \]

Calculate the effusion rate ratio

Perform the calculations: \[ \frac{r_{\text{Ar}}}{r_{\text{Kr}}} = \sqrt{\frac{83.80}{39.95}} = \sqrt{2.099} \approx 1.448 \] Thus, the ratio of effusion rates for Argon to Krypton is approximately 1.448.

Key concepts

Effusion Rate Calculation

Understanding the concept of effusion rate calculation is crucial for students studying gas behaviors. Effusion is a process where gas molecules escape through a tiny opening into a vacuum. According to Graham's law of effusion, the rate at which a gas effuses is inversely proportional to the square root of its molar mass.

The effusion rate can be expressed mathematically as:\[\begin{equation} r \bigg(\frac{\text{volume}}{\text{time}}\bigg) \propto \frac{1}{\bigg(\sqrt{M}\bigg)}\end{equation}\]In practice, this means that lighter gases effuse faster than heavier gases. When comparing two different gases, the rate of effusion of one gas (gas 1) to another (gas 2) can be determined by using the following formula based on Graham's law:\[\begin{equation} \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}\end{equation}\]This ratio shows how much quicker one gas effuses compared to another. To use this formula effectively, it is important to know the molar masses of the gases being compared, as this will directly affect the calculation of effusion rates.

Molar Mass

The molar mass of a chemical element or compound is a fundamental concept in chemistry, representing the mass of one mole of that substance. It is typically expressed in grams per mole (g/mol) and can be found by summing the atomic masses of all atoms in a molecule for compounds or by looking at the atomic weight for elements on the periodic table.

For gases, the molar mass is essential when calculating effusion rates according to Graham's law. As heavier molecules have a larger molar mass, they effuse more slowly than lighter molecules with a smaller molar mass. In the given example, we see that Argon (Ar) has a molar mass of approximately 39.95 g/mol, whereas Krypton (Kr) has a higher molar mass of approximately 83.80 g/mol. This difference in molar mass explains why Argon effuses more quickly than Krypton, as shown in the effusion rate ratio calculation.

Gas Properties

Gases have unique properties that differentiate them from solids and liquids. They have no fixed shape or volume, filling any container they are placed in. These properties depend on a variety of factors such as pressure, temperature, volume, and the amount of gas (typically measured in moles).When dealing with the effusion of gases, the kinetic molecular theory of gases provides insights into the behavior of gas molecules in motion. It explains that gas molecules are in constant random motion and that these molecules move faster at higher temperatures due to increased kinetic energy. This behavior directly ties into the concept of effusion, where faster-moving molecules will escape through an opening more quickly than slower ones.Moreover, the properties of gas also include the ability to diffuse, which is similar to effusion but occurs when a gas spreads out through another gas, and this process is also influenced by the molar mass of the gases involved. All in all, a deep understanding of gas properties and how they interact under different conditions is fundamental to master concepts like effusion and diffusion in chemistry.