Calculate the relative rate of diffusion of \(^{1} \mathrm{H}_{2}\) (molar mass \(2.0 \mathrm{g} / \mathrm{mol}\) ) compared with \(^{2} \mathrm{H}_{2}\) (molar mass 4.0 g/ mol) and the relative rate of diffusion of \(\mathrm{O}_{2}\) (molar mass \(32 \mathrm{g} / \mathrm{mol}\) ) compared with \(\mathrm{O}_{3}\) (molar mass \(48 \mathrm{g} / \mathrm{mol}\) ).

The rate of diffusion is a key concept in understanding how gases mix and spread out over time. It refers to the speed at which gas molecules disperse or spread out in a medium, such as air or another gas. Imagine you open a bottle of perfume in a room; the rate at which the scent spreads throughout the room is an example of the rate of diffusion. In scientific terms, the rate of diffusion is influenced by several factors, including the size of the molecules, the temperature of the environment, and the difference in concentration of the gas.

In the context of Graham's Law, when dealing with gases under the same conditions, the lighter a gas molecule is (meaning it has a lower molar mass), the faster it will diffuse. This law helps us predict how quickly one gas will diffuse in relation to another. It can be quite useful in industrial applications where separating gases is necessary or in predicting the spread of pollutants in the atmosphere.

Moving forward from the basic rate of diffusion, let's delve into the relative rate of diffusion. This describes the ratio of the diffusion rates between two different gases. Since diffusion rate is affected by molar mass, as explained in Graham's Law, comparing two gases gives us a 'relative' sense of how much quicker one gas will mix with context in contrast to another. The relative rate of diffusion becomes particularly important in practical scenarios, such as the administration of medical gases or in the design of chemical synthesis processes.

For example, our textbook exercise highlighted the comparison between hydrogen isotopes (differing forms of hydrogen with varying molar masses) and between different oxygen molecules, \( \text{O}_2 \) and \( \text{O}_3 \). By applying Graham's Law, we can calculate the relative rate of diffusion, which tells us how much faster \( ^1\text{H}_2 \) diffuses compared to \( ^2\text{H}_2 \) and how \( \text{O}_2 \) diffuses in relation to \( \text{O}_3 \). These comparisons can be critical when the subtle differences in diffusion rates matter, such as in the isotopic separation processes or environmental assessments of ozone dispersion.

Finally, the concept of molar mass is the cornerstone of understanding how different gases will diffuse according to Graham's Law. The molar mass is the weight of one mole (approximately \( 6.02 \times 10^{23} \)) of particles for a particular substance and is typically expressed in grams per mole (g/mol). It's an intrinsic property of each substance that directly affects its rate of diffusion.

Knowing the molar mass of a gas, using the periodic table of elements or from given data, allows us to use Graham's Law to compare the diffusion rates of gases. Lighter gases, with smaller molar masses, will diffuse faster than heavier gases. In the exercise, for instance, \( ^1\text{H}_2 \) with a molar mass of 2.0 g/mol will diffuse faster than \( ^2\text{H}_2 \) with a molar mass of 4.0 g/mol, as indicated by the calculated relative rate of diffusion using Graham's Law. Knowing such details is essential not only for academic exercises but also for applications ranging from chemical analysis to the manufacturing of synthetic materials.