Question

\(\mathrm{X} \mathrm{mL}\) of \(\mathrm{H}_{2}\) has effused through a hole in a container in 5 seconds. The time taken for the effusion of the same volume of the gas specified below under identical conditions is: (a) 10 seconds:He (b) 20 seconds: \(\mathrm{O}_{2}\) (c) 25 seconds:CO (d) 55 seconds: \(\mathrm{CO}_{2}\)

Short answer

The time taken for O₂ effusion is 20 seconds, corresponding to option (b).

Step-by-step solution

Understanding Graham's Law of Effusion

Graham's Law of Effusion states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. The formula is given by: \( \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \) where \( r_1 \) and \( r_2 \) are the rates of effusion of two gases, and \( M_1 \) and \( M_2 \) are their molar masses.

Calculate Rate of Effusion for H₂

First, we need to determine the rate of effusion for \( \mathrm{H}_{2} \) given that it effuses in 5 seconds. The rate is \( r_1 = \frac{X}{5} \).

Compare Each Gas to H₂ Using Effusion Rates

For each gas, use the relation \( \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \). Since H₂ is our basis, \( M_1 = 2 \). Let's calculate \( r_2 \) for each gas.

Effusion Calculation for Helium

Helium (He) has a molar mass \( M_2 = 4 \). \( \frac{r_1}{r_2} = \sqrt{\frac{4}{2}} = \sqrt{2} \). Hence, \( r_2 = \frac{r_1}{\sqrt{2}} \). Time \( t_2 = \frac{5}{\sqrt{2}} \).

Effusion Calculation for Oxygen

Oxygen (O₂) has a molar mass \( M_2 = 32 \). \( \frac{r_1}{r_2} = \sqrt{\frac{32}{2}} = 4 \). Hence, \( r_2 = \frac{r_1}{4} \), so \( t_2 = 5 \times 4 = 20 \) seconds.

Effusion Calculation for Carbon Monoxide

Carbon Monoxide (CO) has a molar mass \( M_2 = 28 \). \( \frac{r_1}{r_2} = \sqrt{\frac{28}{2}} = \sqrt{14} \). Hence, \( r_2 = \frac{r_1}{\sqrt{14}} \), and \( t_2 \approx 5 \times \sqrt{14} \approx 18.7 \) seconds.

Effusion Calculation for Carbon Dioxide

Carbon Dioxide (CO₂) has a molar mass \( M_2 = 44 \). \( \frac{r_1}{r_2} = \sqrt{\frac{44}{2}} = \approx 4.69 \). Thus, \( t_2 = 5 \times 4.69 \approx 23.45 \) seconds.

Key concepts

Rate of Effusion

The rate of effusion is a key concept in understanding how gases escape through small openings from an area of higher concentration to one of lower concentration. This rate is influenced by several factors: the type of gas, the size of the opening, and importantly, the conditions under which the effusion occurs. According to Graham's Law of Effusion, the rate of effusion of a gas is inversely proportional to the square root of its molar mass. The equation \[ \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \]helps us compare the effusion rates of two gases under identical conditions. Here, \( r_1 \) and \( r_2 \) are the rates of effusion, while \( M_1 \) and \( M_2 \) represent their respective molar masses. With a higher effusion rate, a gas will escape more rapidly. This is why lighter gases like hydrogen effuse faster than heavier gases like carbon dioxide. Understanding and calculating the rate of effusion can help you predict how long it will take different gases to effuse under similar conditions.

Molar Mass

Molar mass is a fundamental concept in chemistry that affects several properties of a gas, including its rate of effusion. Molar mass is defined as the mass of one mole of a given substance, usually measured in grams per mole (g/mol). The molar mass of a gas determines its density and affects its speed when moving through an opening – a key factor in its effusion rate. For instance, helium, with a molar mass of 4 g/mol, will effuse more quickly than oxygen, which has a molar mass of 32 g/mol. This is because as molar mass increases, the velocity and speed of effusion decrease, resulting in a longer time for the gas to effuse. In problems involving effusion, always compare the molar masses of different gases to predict and calculate differences in their effusion rates, as shown in calculations using Graham's Law of Effusion.

Kinetic Molecular Theory

Kinetic Molecular Theory provides essential insights into the behavior of gases, particularly in terms of their movement and energy. This theory reveals that gas molecules are in constant, random motion and that they frequently collide with each other and with the walls of their container. These collisions and the speed at which molecules move are largely determined by the temperature and the mass of the molecules, relating closely to the concept of molar mass. According to this theory, lighter gas molecules move quicker than heavier ones, which accounts for the differences in effusion rates. In terms of energy, all gases at the same temperature have the same average kinetic energy. However, lighter molecules will move faster to maintain this energy equivalence. The kinetic molecular theory thus provides the underlying explanation for Graham’s Law of Effusion, explaining why gases of different masses behave differently when effusing through a small opening.