It is an elementary observation that air has a
"spring" to it: if you squeeze a balloon, the balloon rebounds to
its original shape. As you pump air into a bicycle tire, the air
pushes back against the piston of the pump. Furthermore, this
resistance of the air against the piston clearly increases as the
piston is pushed farther in. The "spring" of the air is
measured as a pressure, where the pressure
PP is
defined

FF is the
force exerted by the air on the surface of the piston head and
AA is the
surface area of the piston head.

For our purposes, a simple pressure gauge is
sufficient. We trap a small quantity of air in a syringe (a piston
inside a cylinder) connected to the pressure gauge, and measure
both the volume of air trapped inside the syringe and the pressure
reading on the gauge. In one such sample measurement, we might find
that, at atmospheric pressure (760 torr), the volume of gas trapped
inside the syringe is 29.0 ml. We then compress the syringe
slightly, so that the volume is now 23.0 ml. We feel the increased
spring of the air, and this is registered on the gauge as an
increase in pressure to 960 torr. It is simple to make many
measurements in this manner. A sample set of data appears in
Table 1. We note that, in
agreement with our experience with gases, the pressure increases as
the volume decreases. These data are plotted here.

Table 1: **Sample Data from Pressure-Volume Measurement**
Pressure (torr) |
Volume (ml) |

760 |
29.0 |

960 |
23.0 |

1160 |
19.0 |

1360 |
16.2 |

1500 |
14.7 |

1650 |
13.3 |

An initial question is whether there is a
quantitative relationship between the pressure measurements and the
volume measurements. To explore this possibility, we try to plot
the data in such a way that both quantities increase together. This
can be accomplished by plotting the pressure versus the inverse of
the volume, rather than versus the volume. The data are given in
Table 2 and plotted here.

Table 2: **Analysis of Sample Data**
Pressure (torr) |
Volume (ml) |
1/Volume (1/ml) |
Pressure × Volume |

760 |
29.0 |
0.0345 |
22040 |

960 |
23.0 |
0.0435 |
22080 |

1160 |
19.0 |
0.0526 |
22040 |

1360 |
16.2 |
0.0617 |
22032 |

1500 |
14.7 |
0.0680 |
22050 |

1650 |
13.3 |
0.0752 |
21945 |

Notice also that, with elegant simplicity, the
data points form a straight line. Furthermore, the straight line
seems to connect to the origin
0000.
This means that the pressure must simply be a constant multiplied
by
1V1V:

If we multiply both sides of this equation by
VV, then we
notice that

In other words, if we go back and multiply the
pressure and the volume together for each experiment, we should get
the same number each time. These results are shown in the last
column of Table 2, and we see
that, within the error of our data, all of the data points give the
same value of the product of pressure and volume. (The volume
measurements are given to three decimal places and hence are
accurate to a little better than 1%. The values of
Pressure×Volume×PressureVolume
are all within 1% of each other, so the fluctuations are not
meaningful.)

We should wonder what significance, if any,
can be assigned to the number
22040(torrml)22040torrml
we have observed. It is easy to demonstrate that this
"constant" is not so constant. We can easily trap any
amount of air in the syringe at atmospheric pressure. This will
give us any volume of air we wish at 760 torr pressure. Hence, the
value
22040(torrml)22040torrml
is only observed for the particular amount of air we happened to
choose in our sample measurement. Furthermore, if we heat the
syringe with a fixed amount of air, we observe that the volume
increases, thus changing the value of the
22040(torrml)22040torrml.
Thus, we should be careful to note that the *product
of pressure and volume is a constant for a given amount of air at a
fixed temperature*. This observation is referred to
as Boyle's Law, dating to
1662.

The data given in Table 1 assumed that we used air for the
gas sample. (That, of course, was the only gas with which Boyle was
familiar.) We now experiment with varying the composition of the
gas sample. For example, we can put oxygen, hydrogen, nitrogen,
helium, argon, carbon dioxide, water vapor, nitrogen dioxide, or
methane into the cylinder. In each case we start with 29.0 ml of
gas at 760 torr and 25°C. We then vary the volumes as in
Table 1 and measure the
pressures. Remarkably, we find that the pressure of each gas is
exactly the same as every other gas at each volume given. For
example, if we press the syringe to a volume of 16.2 ml, we observe
a pressure of 1360 torr, no matter which gas is in the
cylinder. This result also applies equally well to mixtures of
different gases, the most familiar example being air, of
course.

We conclude that the pressure of a gas sample
depends on the volume of the gas and the temperature, but not on
the composition of the gas sample. We now add to this result a
conclusion from a previous study. Specifically, we
recall the Law of Combining
Volumes, which states that, when gases combine during
a chemical reaction at a fixed pressure and temperature, the ratios
of their volumes are simple whole number ratios. We further recall
that this result can be explained in the context of the atomic
molecular theory by hypothesizing that equal volumes of gas contain
equal numbers of gas particles, independent of the type of gas, a
conclusion we call Avogadro's
Hypothesis. Combining this result with Boyle's
law reveals that the *pressure* of a
gas depends on the *number* of gas
particles, the *volume* in which
they are contained, and the
*temperature* of the sample. The
pressure does *not* depend on the
type of gas particles in the sample or whether they are even all
the same.

We can express this result in terms of
Boyle's law by noting that, in the equation
PV=kPVk,
the "constant"
kk is actually
a function which varies with both number of gas particles in the
sample and the temperature of the sample. Thus, we can more
accurately write

explicitly showing that the product of
pressure and volume depends on
NN, the number
of particles in the gas sample, and
tt,the
temperature.

It is interesting to note that, in 1738,
Bernoulli showed that the inverse relationship between pressure and
volume could be proven by assuming that a gas consists of
individual particles colliding with the walls of the container.
However, this early evidence for the existence of atoms was ignored
for roughly 120 years, and the atomic molecular theory was not to
be developed for another 70 years, based on mass measurements
rather than pressure measurements.

Comments:"Reviewer's Comments: 'I recommend this book. It is suitable as a primary text for first-year community college students. It is a very well-written introductory general chemistry textbook. This […]"