Pressure-Volume Measurements on Gases

Author: John Hutchinson

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 P is defined

P=F/A

[1]

F is the force exerted by the air on the surface of the piston head and A 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.

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

Figure 1: Measurements on Spring of the Air.

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.

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

Figure 2: Analysis of Measurements on Spring of the Air.

Notice also that, with elegant simplicity, the data points form a straight line. Furthermore, the straight line seems to connect to the origin {0,0}. This means that the pressure must simply be a constant multiplied by 1/V:

P=k/V

[2]

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

PV = k

[3]

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) 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) 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) 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). 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 = k, the "constant" k 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

PV = k(N,T)

[4]

explicitly showing that the product of pressure and volume depends on N, the number of particles in the gas sample, and T, 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.