# Considering the ideal gas law PV=nRT, what is P directly proportional to what variable?

Jul 27, 2015

Pressure is directly proportional to temperature and number of moles.

#### Explanation:

The ideal gas law equation looks like this

$P V = n R T$, where

$P$ - the pressure of the gas;
$V$ - its volume;
$n$ - the number of moles of gas;
$R$ - the gas constant, usually given as $0.082 \left(\text{atm" * "L")/("mol" * "K}\right)$
$T$ - the temperature of the gas.

In order for two variables to have a direct relationship, they must increase or decrease at the same rate.

In other words, two variables are directly proportional if increasing one will cause the other to increase at the same rate. LIkewsie, decreasing one will cause the other to decrease at the same rate.

To see which variables are directly proportional to pressure, isolate $P$ on one side of the equation

$\frac{P \cdot \textcolor{red}{\cancel{\textcolor{b l a c k}{V}}}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{V}}}} = \frac{n R T}{V}$

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

Since $R$ is a constant, you can say that $P$ is proportional to

$P \propto \frac{n T}{V}$

So, $P$ is directly proportional to the product between $n$ and $T$ and inversely proportional to $V$.

In other words, if $n$ increases while $T$ and $V$ remain unchanged, then $P$ will increase as well.

Likewise, if $T$ increases while $n$ and $V$ remain constant, the $P$ will Increase as well.

On the other hand, if $n$ and $T$ are unchanged, then any increase in $V$ will cause $P$ to decrease.

As a conclusion, $P$ is directly proportional to $n$ and $T$ and inversely proportional to $V$.