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Graphical Representations of Boyle's Law
Consider an experiment in which a known amount of hydrogen gas in a syringe has a volume of 23 mL at atmospheric pressure (760 mm Hg or 1 atm or 101.3 kPa).
You then apply an external pressure of 912 mm Hg (1.2 atmospheres or 121.6 kPa) by pressing down on the plunger in the syringe.
The volume of hydrogen gas is then recorded as 19.2 mL.
You continue to apply external pressure by pushing the plunger down further, recording the volume of hydrogen gas as shown in the table below:
Pressure (mm Hg)* 
Volume (mL) 
Trend 
760 
23 
Increasing the pressure applied to the plunger causes a reduction in the gas volume.
Decreasing the applied pressure increases the volume of the gas.

912 
19.2 
1064 
16.4 
1216 
14.4 
1368 
12.8 
1520 
11.5 
* A pressure of 760 mm Hg is equal to 1 atmosphere (atm) or 101.3 kilopascals (kPa)

If we plot these points on a graph, the graph looks like the one below:
volume (mL)

Gas Volume versus Pressure
Pressure (mm Hg)

Note that this is not a linear relationship, the line in the graph is curved, it is not a straight line.
But look what happens if we multiply volume and pressure (P × V):
Pressure (mm Hg) 
Volume (mL) 
P × V 
Trend 
760 
23 
1.75 × 10^{4} 
P × V is a constant!
For this amount of gas at this temperature:
P × V = 1.75 × 10^{4}

912 
19.2 
1.75 × 10^{4} 
1064 
16.4 
1.75 × 10^{4} 
1216 
14.4 
1.75 × 10^{4} 
1368 
12.8 
1.75 × 10^{4} 
1520 
11.5 
1.75 × 10^{4} 
For a given amount of gas at constant temperature we now we can write the equation:
P × V = constant
If we divide both sides of the equation by P, we get:
Recall that the equation for a straight line that runs through the point (0,0) is
y = mx
where m is the slope (or gradient) of the line
Then a graph of V against ^{1}/_{P}, should be a straight line with a slope (or gradient) equal to the value of the constant.
The table below shows what happens if we calculate ^{1}/_{P} for each volume, V, in the experiment above and then graph the results:
Volume (mL) 
Pressure (mm Hg) 
1/Pressure (1/mm Hg)* 
Comments 
11.5 
1520 
6.6 × 10^{4} 
As gas volume (V) increases, the value of ^{1}/_{P} increases.
As gas volume (V) decreases, the value of ^{1}/_{P} decreases.

12.8 
1368 
7.3 × 10^{4} 
14.4 
1216 
8.2 × 10^{4} 
16.4 
1064 
9.4 × 10^{4} 
19.2 
912 
1.1 × 10^{3} 
23 
760 
1.3 × 10^{3} 
By plotting these points on a graph, we can see that the relationship is linear:
volume (mL)

Gas Volume versus 1/Pressure
1/Pressure (1/mm Hg)

We now have a simple method for determining the value of the constant:
Recall that we can calculate the slope (gradient, m) of a straight line using two points on the line
m = 
(y_{2}  y_{1}) (x_{2}  x_{1}) 
Choosing the points (0.00094,16.4) and (0.0013,23)
m 
= 
(23  16.4) (0.0013  0.00094) 

= 
(6.6) (0.00036) 

= 
1.8 × 10^{4} 
and the equation for this straight line is
This equation then allows us to calculate the volume of the gas at any pressure, as long as we use the same amount of gas and keep the temperature the same.
Let us say we have a specific amount of gas and keep the temperature constant, then initially at pressure P_{i} the gas has a volume of V_{i} and we know that:
P_{i}V_{i} = constant
If we maintain the same temperature and the same amount of gas, but change the pressure to P_{f}, then the new gas volume will be V_{f}, and
P_{f}V_{f} = the same constant
So, as we long as we use the same amount of gas at the same temperature:
P_{i}V_{i} = constant = P_{f}V_{f}
that is:
P_{i}V_{i} = P_{f}V_{f}
This means that if we know the initial conditions (P_{i} and V_{i}), and, we know the final pressure (P_{f}), we can calculate the final volume (V_{f}):
V_{f} = 
P_{i} × V_{i} P_{f} 
or we can calculate the final pressure (P_{f}) if we know the final volume (V_{f}):
P_{f} = 
P_{i} × V_{i} V_{f} 
Similarly, if we know the final conditions (P_{f} and V_{f}), and, we know the initial pressure (P_{i}), we can calculate the initial volume (V_{i}):
V_{i} = 
P_{f} × V_{f} P_{i} 
or we can calculate the initial pressure (P_{i}) if we know the initial volume (V_{i}):
P_{i} = 
P_{f} × V_{f} V_{i} 