# How do you graph Boyle's law?

Jan 2, 2014

WARNING! Long answer! Boyle’s Law examines the relationship between the volume of a gas and its pressure.

#### Explanation:

So, you would do an experiment in which you measure the volume of a gas at various pressures.

A common lab experiment uses a pop can with various amounts of water to compress the air in a syringe.

Let’s assume you get the following data.

$\underline{\boldsymbol{\text{Mass/g"color(white)(m) "Volume/mL}}}$
$\textcolor{w h i t e}{m l} 100 \textcolor{w h i t e}{m m m m m} 34$
$\textcolor{w h i t e}{m l} 150 \textcolor{w h i t e}{m m m m m} 22$
$\textcolor{w h i t e}{m l} 200 \textcolor{w h i t e}{m m m m m} 17$
$\textcolor{w h i t e}{m l} 250 \textcolor{w h i t e}{m m m m m} 14$
$\textcolor{w h i t e}{m l} 300 \textcolor{w h i t e}{m m m m m} 11$
$\textcolor{w h i t e}{m l} 350 \textcolor{w h i t e}{m m m m m} 10$
$\textcolor{w h i t e}{m l} 400 \textcolor{w h i t e}{m m m m m l l} 8$
$\textcolor{w h i t e}{m l} 450 \textcolor{w h i t e}{m m m m m l l} 8$
$\textcolor{w h i t e}{m l} 500 \textcolor{w h i t e}{m m m m m l l} 7$
$\textcolor{w h i t e}{m l} 550 \textcolor{w h i t e}{m m m m m l l} 6$

The best way to determine a relationship is to plot a graph that gives a straight line.

Since the pressure is directly proportional to the mass of the can plus water, we can use the mass instead of pressure.

At this point we (theoretically) don’t know the relationship, so we plot $V$ vs. $\text{Mass}$ to see what the plot looks like.

We plot pressure as the independent variable (along the horizontal or $x$ axis) and volume as the dependent variable (along the vertical or $y$ axis).

You could do this by hand, but it is more convenient to use a computer to do the job for you.

Your graph looks like a hyperbola, as in the diagram above.

The equation for a hyperbola is $y = \frac{k}{x}$.

It looks as if we should try a plot of $V$ vs. $\frac{1}{p}$.

We calculate the values of $\frac{1}{p}$ and then plot $V$ vs. $\frac{1}{\text{Mass}}$.

We get the straight line plot shown below.

You can extend the line backwards until it reaches V ≈ "0 cm"^3.

If you use a computer or a calculator, you can tell it to calculate the equation for the line that best fits all the points (the regression line).

My computer tells me that the equation is

$V = 3365.9 x$ or $V = \frac{3365.9}{p}$ or $p V = 3365.9$.

The graph of $V$ against $\frac{1}{p}$ is a straight line through the origin.

This means that the measured volume is inversely proportional to the pressure — Boyle's Law.