Problem 48 - Chemistry and Chemical Reactivity

Question

Argon gas is ten times denser than helium gas at the same temperature and pressure. Which gas is predicted to effuse faster? How much faster?

Step-by-Step Solution

Answer

Helium effuses 3.16 times faster than Argon.

Step 1: Understand the relationship between density and molecular mass

The density of a gas is directly proportional to the molecular mass when the temperature and pressure are constant. Since Argon is ten times denser than Helium, we infer that Argon's molecular mass is ten times that of Helium's molecular mass.

Step 2: Write down the effusion equation

Graham's law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. The relationship can be written as: \[ \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \] where \(r_1\) and \(r_2\) are the effusion rates of Helium and Argon, and \(M_1\) and \(M_2\) are their respective molar masses.

Step 3: Substitute the known values

Since Argon's molar mass is ten times that of Helium, \(M_2 = 10M_1\). Substitute this into the effusion equation to get: \[ \frac{r_1}{r_2} = \sqrt{\frac{10M_1}{M_1}} = \sqrt{10} \]

Step 4: Calculate the rate of effusion

Calculate \(\sqrt{10}\) to find the ratio of the effusion rates. \(\sqrt{10} \approx 3.16\). Hence, Helium effuses 3.16 times faster than Argon.

Key Concepts

Density and Molecular Mass

The density of a gas is a measure of how much mass it has compared to the volume it occupies. In gases, density is directly related to molecular mass, especially under the same conditions of temperature and pressure. This means that, at constant temperature and pressure, a gas with a higher molecular mass will be denser than a gas with a lower molecular mass. This relationship is crucial in understanding how different gases behave in varying situations. For instance, in our exercise, Argon gas is ten times denser than Helium gas, indicating that Argon has a molecular mass ten times greater than that of Helium.

Effusion Rates

Effusion is the process by which gas particles pass through a tiny opening from one container to another. The rate at which a gas effuses is called its effusion rate. Effusion rates can tell us how fast different gases escape through such an opening in controlled conditions. Knowing how quickly a gas effuses is important in applications ranging from industrial separation processes to scientific experiments.
Graham's Law of Effusion provides a mathematical relationship between the effusion rates and molar masses of gases. It indicates that lighter gases effuse faster than heavier ones. For example, in our scenario, Helium effuses faster than Argon, due to its lower molecular mass.

Molecular Mass and Effusion

Molecular mass, also known as molar mass, refers to the mass of a particular molecule. It is a critical factor in determining the effusion rate of a gas, according to Graham's Law. The mathematical aspiration of this relationship is seen through the inverse proportionality in effusion rates.The equation \[ \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \]expresses this idea by reminding us that as the molecular mass increases, the effusion rate decreases. Therefore, understanding molecular mass helps predict how quickly a particular gas will effuse relative to another gas.

Inverse Proportionality in Effusion

Inverse proportionality in effusion, as governed by Graham's Law, is a key principle for predicting and analyzing gas behavior. This principle indicates that the effusion rate of a gas is inversely proportional to the square root of its molar mass. This means simply: gases with smaller molar masses will effuse more rapidly than those with larger molar masses. When dealing with two gases at the same temperature and pressure, as in our exercise with Helium and Argon, understanding this relationship allows us to deduce that Helium effuses more rapidly. The faster effusion rate of Helium by a factor of \(\sqrt{10} \approx 3.16\) highlights its lower molar mass compared to Argon.