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

Dec 16, 2015

Here's what's going on here.

#### Explanation:

First, make sure that you have a clear understanding of what directly proportional actually means.

In order for two quantities to be directly proportional, you need one to increase as the other increases or decrease as the other decreases, in both cases at the same rate.

Starting from the ideal gas law equation, isolate $P$ on one side of the equation first

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

Since $R$ represents a constant, you can write it separately

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

Now focus on the ratio that exists between number of moles, $n$, and temperature, $T$, on one side, and volume, $V$, on the other.

Let's start with the number of moles. In order to establish direct or inverse proportionality, you need to keep the other two variables constant.

$P = n \cdot {\overbrace{\frac{T}{V} \cdot R}}^{\textcolor{b l u e}{\text{constant}}}$

So, under these circumstance, what would happen to $P$ if $n$ were to increase? Well, in order for the equal sign to remain valid, $P$ would have to Increase as well.

Likewise, if $n$ were to decrease, $P$ would have to decrease as well. Therefore, you can say that

Pressure is directly proportional with number of moles when temperature and volume are kept constant

The same can be said for temperature, $T$. Keeping the other two variables constant will result in

$P = T \cdot {\overbrace{\frac{n}{V} \cdot R}}^{\textcolor{b l u e}{\text{constant}}}$

Once again, an increase in temperature will result in an Increase in pressure, and a decrease in temperature will result in a decrease in pressure. Therefore, you can say that

Pressure is directly proportional with temperature when number of moles and volume are kept constant - Gay Lussac's Law

Finally, keep number of moles and temperature constant and check to see what happens to pressure when the volume varies.

$P = \frac{1}{V} \cdot {\overbrace{n \cdot T \cdot R}}^{\textcolor{b l u e}{\text{constant}}}$

This time, an increase in volume would result in a decrease in pressure, since volume is now in the denominator

$V \uparrow \implies \frac{1}{V} \downarrow$

Likewise, a decrease in volume would result in an increase in pressure. Therefore, pressure is not directly proportional to volume when number of moles and temperature are kept constant.

You can say, however, that

Pressure is inversely proportional with volume when number of moles and temperature are kept constant - Boyle's Law