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# How do you graph Boyle's law?

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

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

Let’s assume you get the following data.

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

At this point we (theoretically) don’t know the relationship, so we plot ##V## vs. ##P## 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, as in the figure, looks like a hyperbola.

The equation for a hyperbola is ##y = k/x##. It looks as if we should try a plot of ##V## vs. ##1/P##.

We calculate the values of ##1/P## and then plot ##V## vs. ##1/P##. We get the straight line plot shown above.

If you plot the data manually, you extend the line backwards until it reaches ##V ≈ "0 cm"^3##.

The graph of ##V## against ##1/P## is a straight line through the origin.

This means that the measured volume is INVERSELY PROPORTIONAL to its pressure — .

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 = 6185.8/P - 0.0827##

This says that when ##1/P = 0##, ##V = "-0.08 cm"^3##.

Since we measured the volumes only to the nearest cubic centimetre, ##"0.08 cm"^3## is negligible. Therefore, within experimental uncertainty,

##V = 6185.8/P##

This is Boyle’s Law, ##V = k/P##.

If ##V = k/P##, then ##PV = k##.

The graph of ##PV## against ##P## should be a straight line parallel to the ##P## axis. In other words, the product ##PV## is a constant at a fixed temperature.

We can test this by plotting ##PV## vs. ##P## as shown below.

The graph looks like a horizontal straight line.

The computer gives the equation for the best fit line as

##PV= -0.149P + 6192.8##

That means that the line starts at ##PV = 6183.8## at the left hand end and finishes at ##6163.0## at the right hand end.

However, since we are justified in using only two for the PV product, it starts and ends at ##6200##.

Therefore, within the limits of experimental uncertainty,

##PV = "constant"## → BOYLE'S LAW