For an ideal gas where the pressure and temperature are held constant, which of the following is the correct proportionality relation between the volume 𝑉 of the gas and the number of moles 𝑛? (A) 𝑉 is proportional to 𝑛 squared. (B) 𝑉 is proportional to 𝑛. (C) 𝑉 is proportional to the square root of 𝑛. (D) 𝑉 is proportional to one divided by 𝑛. Or (E) 𝑉 is proportional to one divided by 𝑛 squared.
Here, we’re considering an ideal gas, which means we’re approximating that the molecules of this gas don’t interact with each other and that they have negligible size. Let’s recall that any ideal gas will obey the formula 𝑃𝑉 equals 𝑛𝑅𝑇, where 𝑃 is the pressure of the gas, 𝑉 is its volume, 𝑛 is the number of moles, 𝑅 is the molar gas constant, and 𝑇 is its absolute temperature. This formula is known as the ideal gas law. And in this question we’ll use it to devise a proportionality relation between the volume and the number of moles of the gas. To do this, we can first replace the equal sign with this symbol, which means “is proportional to.”
Remember that a statement of proportionality helps us understand how variables change with respect to each other. So, when we write it out, it should not include any constants because we know those values don’t change. For this reason, we’ll basically ignore any constant term by setting it equal to one, since a factor of one has no impact on overall value.
Now, we know that 𝑅 represents the molar gas constant, so it’ll be excluded from this relationship. And although 𝑃 and 𝑇 do appear as variables in the ideal gas law itself, remember that we’ve been told that this gas has a constant pressure and temperature. So we’ll leave those terms out as well. Thus, the only two true variables that we’re concerned with are 𝑉 and 𝑛. So the relationship reads 𝑉 is proportional to 𝑛. And we can see right away that this agrees with answer choice (B).
Another way to say this is 𝑉 is directly proportional to 𝑛 because a change in one value corresponds to the same magnitude change for the other value. For instance, if we were to increase the volume of the gas by a factor of three, with pressure and temperature held constant, the number of moles must also increase by a factor of three. But we could not say the same thing if any of these other relationships were true. Notice that in all the other answer choices, 𝑛 has a different exponent. In option (A), 𝑛 is raised to the second power. In (C), we can say 𝑛 is raised to the one-half. (D) shows 𝑛 to the negative one, and (E) is 𝑛 to the negative two. Thus, for any of these relationships, a change in 𝑉 would not correspond to the same magnitude of change in 𝑛.
But we know this isn’t the case, since in the actual ideal gas law, 𝑉 and 𝑛 are simply 𝑉 and 𝑛. Neither term has some other exponent. Therefore, we’ve seen that the ideal gas law has a direct proportionality between volume and the number of moles. So we know that answer choice (B) is correct. For an ideal gas with constant pressure and temperature, 𝑉 is proportional to 𝑛.