# Keeping n and T constant, how do I prove that P prop 1/V? What about keeping n and P constant and showing that V prop T?

Oct 23, 2016

You can check this just from the ideal gas law.

$P V = n R T$

1)

If you try to double the pressure, then you are clearly increasing it. As a result, what do you have to do to get $P V = n R T$ back to how it was, assuming $n$ and $T$ remain constant?

$2 P V = n R T$

$\implies 2 P \cdot \frac{1}{2} V = n R T$

Therefore, the volume must have halved to cancel out the pressure doubling. In general, it means $P \propto \frac{1}{V}$, or that they are inversely proportional.

You can also prove it further:

$P = n R T \cdot \frac{1}{V}$

Therefore, $P \propto \frac{1}{V}$.

2) Keeping $n$ and $P$ constant, then by now in this answer, you should realize that if $V$ changes, $T$ must change. It can't stay the same.

Can you show that $V \propto T$ for constant $P$ and $n$? To do that, simply identify the constant in the equation:

$V = \frac{n R}{P} \cdot T$