Question

Which among the following has rate of effusion less than the moist air? (a) \(\mathrm{He}\) (b) Dry air (c) \(\mathrm{NH}_{3}\) (d) heavy hydrogen

Short answer

None of the given gases has a rate of effusion less than moist air since all the listed gases have a lower molar mass compared to moist air.

Step-by-step solution

Understanding Effusion and Graham's Law

Graham's law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. The formula is expressed as \( \frac{\text{Rate of effusion of gas 1}}{\text{Rate of effusion of gas 2}} = \sqrt{\frac{\text{Molar mass of gas 2}}{\text{Molar mass of gas 1}}} \). Use this law to compare the rates of effusion of the given gases with moist air.

Comparing Molar Masses

Identify the molar mass of each gas and compare it with the molar mass of moist air, which is approximately \( \text{29 g/mol} \) (average due to mixture of nitrogen, oxygen and water vapor). Helium (\text{He}) has a molar mass of 4 g/mol, dry air has a molar mass close to that of moist air but slightly less because of the absence of water vapor, Ammonia (\text{NH}_3) has a molar mass of 17 g/mol, and heavy hydrogen (deuterium, \text{D}_2) has a molar mass of 4 g/mol per deuterium atom or 8 g/mol for \text{D}_2.

Determining the Gas with Lesser Rate of Effusion

Apply Graham's law to find that the gas with the larger molar mass compared to moist air will effuse less rapidly. Among the given options, dry air has the closest molar mass to moist air, but since it is slightly less, which indicates a higher effusion rate for dry air; helium and heavy hydrogen have significantly lower molar masses, which means they will effuse more rapidly; ammonia has a lower molar mass as well, which also leads to a higher rate of effusion. Therefore, none of the gases listed have a rate of effusion less than moist air based on their molar masses.

Key concepts

Effusion Rate

Effusion is when gas molecules pass through a tiny opening into a vacuum or an area of lower pressure. It's one of those phenomena in physical chemistry that we can witness happening, even if we don't see it at the molecular level. To describe how fast this happens for different gases, we use the 'effusion rate'.

The effusion rate essentially tells us how quickly a gas can escape through a small hole. It's affected by things like the size of the molecules and the temperature. But mainly, Graham's Law shows us that it's all about molar mass—the lighter the gas, the higher the effusion rate. A higher effusion rate means a gas will spread out faster, which is why a popped balloon deflates so quickly.

Molar Mass

In the world of atoms and molecules, weight is discussed in terms of 'molar mass'. It's like the batch weight of a mole of molecules, where a mole is Avogadro's number of units—kind of like a baker's dozen but much, much bigger, at approximately 6.022 × 10^{23} particles. This molar mass is what Graham's Law uses to compare different gases.

The molar mass is measured in grams per mole (g/mol), and it is essential in predicting how a gas will behave when it effuses. Because gases with a higher molar mass move slower (they're like the heavy trucks on a molecular highway), they will have a lower rate of effusion according to Graham's law, which creates a predictable relationship between molar mass and effusion rate.

Physical Chemistry Problems

Now, how does all this talk about effusion rate and molar mass fit into physical chemistry problems? In this field, we often deal with how molecules behave and interact, and how that behavior is affected by physical laws, like Graham's Law.

The problems you'll encounter usually involve calculating these rates, understanding the implications of molecular motion, or predicting the outcome of a gas-related process. It's like being a detective but for molecules. For example, knowing that heavier gases effuse more slowly could explain why some smells linger longer than others. This is practical stuff—beyond textbook exercises, effusion affects the design of gas masks, industrial processes, and even how we store and use gases in medicine.

Comparative Effusion Analysis

Comparative effusion analysis is quite the skill to have in your chemistry toolkit. It's not just about seeing which gas zips through a hole the fastest; it's a way to understand the behavior of gases under various conditions. By using Graham's Law, we compare the effusion rates of gases to interpret or predict how they'll act in the real world.

For instance, in our exercise, knowing that moist air has an average molar mass allows us to predict how other gases will behave in comparison when they effuse. This analysis is essential for safety protocols in labs and industry because it can predict how quickly a hazardous gas might spread. Learning to perform these comparative effusion analyses comes in handy for scientists and engineers who need to control or harness gas properties for their projects.