Gas transformation for different molar number

Question

When a gas at normal conditions (1atm, 273K, 22.4L) we say like the $$\frac{PV}{T} = 0.0820...$$ And by the know equation: $$PV = nRT$$ Where R is equals to $0.08205...$ and $n$ is the number of mols of the substance. When we want to compare a previous state of a gas with its next state, we use the formula: $$\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$$ But the $R$ is just the first state of the gas, already in a constant term ($\frac{P_1 V_1}{T_1}$ is $\frac{1 \cdot 22.4}{273}$ that is $R=0.082...$) So, instead of doing: $$\frac{1 \cdot 22.4}{273} = \frac{P_2 V_2}{T_2}$$ We do: $$R = \frac{P_2 V_2}{T_2}\tag{assuming the same n of mols}$$ But let's say we're doing this equality for a gas that has a given number $n$ mols of the substance. By what I know it is $$nR = \frac{P_2 V_2}{T_2}$$ I have $2$ questions:

$(1)$ - How do I know that for a $1/2$ number of mols (for example), the quotient $\frac{PV}{T}$ is gonna be exactly $1/2$ (in other words, how do I know that they are linear)

$(2)$ - Why do I have to multiply the $n$ always for the $\frac{1 \cdot 22.4}{273}$ and not for the $ \frac{P_2 V_2}{T_2}$ in the equation? (in other words, if I'm just equating two states of the gas, I guess there should be no problem with which side I multiply by $n$). I don't get what "multiplying by $n$" means. Like, the two states of the gas that I'm equating have the same amount of molecules, doesn't mean if they are $1/2$ mol or anything, they're the same quantity.

Thanks!

Answer 1

(1) The equation $PV=nRT$ is a "proportionality" type of equation, which necessarily has a linear dependence when only one variable is independent. The class of problems your are dealing with now has only one independent (you control the change) and one dependent (the equation tells you the change) variable, while the others are held constant. And from a pure math perspective, for it to be a non-linear dependence with only one independent variable there would need to be operators other than multiplication/division (e.g. exponent, logarithm, sin, cos, etc.).

(2) When you are equating two gas states you are really solving the equation:

$$ \frac{P_1 V_1}{nRT_1}=\frac{P_2V_2}{nRT_2} $$ And for the case you are looking at $n_1=n_2 $ the variable $n$ is on both sides and cancels. So in this case you aren't multiplying either side by $n$.
When you are only looking at one gas state, you are not setting equations equal to each other and there is no cancellation, so you still have the variable $n$.