How does the volume of a mole of ideal gas relate to pressure and temperature?

Question

The volume of a mole of gas is:

1. $\ce{22.4 l}$
2. directly proportional to pressure and Kelvin temperature.
3. directly proportional to pressure, inversely proportional to Kelvin temperature.
4. inversely proportional to pressure, directly proportional to Kelvin temperature.
5. inversely proportional to pressure and to Kelvin temperature.

I said the answer is 3 because the ideal gas law is $PV=nRT$ and "proportional to" is the same as saying which ones are on opposite sides. So $n$ and $P$ are on opposite sides and temp is on the same side as $n$. But I'm wrong and the answer is not 3, but I don't know why.

Answer 1

Your starting point of reasoning is good, but you misread the equation. The question is about the volume of 1 mole of gas so let's rewrite the ideal gas law to give this relation: $$P V = n R T \rightarrow \frac{V}{n}=\frac{R T}{P} $$ Now we have the volume of a mole of gas $\frac{V}{n}$ on the left hand side so we can see the proportionalities on the right hand side.

The volume of a mole of gas is inversely proportional to the pressure and directly proportional to the temperature, i.e. 4 in your list of answers

Answer 2

According to Boyle's law, at constant temperature, pressure is inversely proportional to volume.

According to Charles law, at constant pressure, volume is directly proportional to temperature.

Combining the above and the Avagadro's law, we get the ideal gas equation. Hence, the correct answer is 4.