Hydrogen diffuses 4 times as rapidly as a certain
gas. What is the density and molecular weight of the
gas when that of Hydrogen is unity?

We start with Graham’s law of diffusion which tells us that the rate at which a gas diffuses (or effuses) is inversely proportional to the square root of its molecular weight. In mathematical form, if we have two gases (gas 1 and gas 2), then

where is the rate of diffusion and is the molecular weight.

In this problem, hydrogen (which we will call gas ) diffuses 4 times as rapidly as the unknown gas (which we will call gas ). That is,

Also, hydrogen is given a reference density of unity and, by the ideal gas behavior at constant temperature and pressure, its density is proportional to its molecular weight. When we “take hydrogen as unity”, we can also treat its molecular weight as 1 for the purpose of comparing relative quantities.

Using Graham’s law, we have

Squaring both sides gives

Since is taken as unity,

Because the density of a gas (under the same conditions) is proportional to its molecular weight, if hydrogen’s density is unity then the density of the unknown gas is also 16 times as great. Therefore, the density of gas must also be 16.

Thus, the answers are: