## The Empirical Gas Laws: Pressure and Boyle's Law

James Richard Fromm

Since the days of Aristotle, all substances have been classified into one of three physical states. A substance having a fixed volume and shape is a solid. A substance which has a fixed volume but not a fixed shape is a liquid; liquids assume the shape of their container but do not necessarily fill it. A substance having neither a fixed shape nor a fixed volume is a gas; gases assume both the shape and the volume of their container. The structures of gases, and their behavior, are simpler than the structures and behavior of the two condensed phases, the solids and the liquids. In this section we begin to study the behavior of substances in the gaseous state.

Quantitative measurements on gases were first made in a systematic manner by the English chemist Robert Boyle (1627 - 1691). The instruments used by Boyle to measure pressure were two: the manometer, which measures differences in pressure, and the barometer, which measures the total pressure of the atmosphere.

A manometer is simply a bent piece of tubing, preferably glass, with one end closed. When the liquid level in both arms is the same, the pressure of the sample of gas inside the closed end must equal the pressure of the external atmosphere since the downward force on the two columns of liquid is then equal. When the liquid levels are unequal, the pressures must differ. The difference in pressure can be measured in units of length of the vertical column of liquid. The mmHg, or its modern version the torr, originated in this use of the manometer. Mercury is particularly convenient for use in manometers (and barometers) because at room temperature it has low vapor pressure, does not wet glass, and has a high density. Other liquids such as linseed oil or water have also been used in manometers.

The barometer was invented by Evangelista Torricelli (1608-1647), one of Galileo's students. It is a device for measuring the total pressure of the atmosphere. A Torricellian barometer can easily be constructed by taking a glass tube about a meter long, sealing one end, filling the tube completely with mercury, placing your thumb firmly over the open end, and carefully inverting the tube into an open dish filled with mercury. The mercury will fall to a height independent of the diameter of the tube and a vacuum will be created above it. The height of the mercury column will be the height which the atmospheric pressure can support. The standard atmospheric pressure, one atmosphere (atm), is 760 mmHg but the actual atmospheric pressure varies depending upon altitude and local weather conditions. For this reason barometers can be used to help predict the weather. A falling barometer indicates the arrival of a low pressure air system, which often means stormy weather. A rising barometer indicates the arrival of a high pressure air system, and that often means clear weather.

While mercury is again the most convenient liquid for use in barometers it is by no means the only liquid which can be used. Preparation of a water barometer, and many of the early barometers did use water, requires use of a vacuum pump (or arms 13 meters long).

With the manometer and barometer used together, the actual pressure of a sample of gas can be measured. Combining the barometer reading of atmospheric pressure with the manometer reading of pressure difference gives the actual pressure. If the manometer is as shown on the left-hand side of the Figure below, then p2 = p(atmospheric) + p1, while if the manometer is as shown on the left-hand side of the Figure below, then p2 = p(atmospheric) - p1.

#### Units of Pressure

Units of pressure were originally all based on the length of the column of liquid, usually mercury, supported in a manometer or barometer. By far the most common of these units was the mmHg, although inches of mercury were also used in English-speaking countries. However, the modern SI unit of pressure is derived from the fundamental units of the SI. Pressure is force per unit area, and force is the product of mass times acceleration, so the SI unit of pressure is the kg m s-2/m2 or newton/m2, which is called the pascal (Pa).

All of the older units of pressure have now been redefined in terms of the pascal. One standard atmosphere or atm, the pressure of the atmosphere at sea level, is by definition exactly 101325 Pa. The torr, named in honor of Torricelli, is defined as 1/760 of a standard atmosphere or as 101325/760 Pa. The mmHg, which is almost but not quite identical to the torr, is defined as (13.5951 x 9.80665) Pa, using a fixed density of mercury and a standard force of terrestrial gravitation. The term bar is used for 100000 Pa, which is slightly below one standard atmosphere.

#### Boyle's Law

Boyle used the manometer and barometer to study the pressures and volumes of different samples of different gases. The results of his studies can be summarized in a simple statement which has come to be known as the law of Boyle or Boyle's law:

At any constant temperature, the product of the pressure and the volume of any size sample of any gas is a constant.

For a particular sample of any gas, Boyle's law can be shown graphically as is done in the Figure below. It is more common to express it mathematically as p1V1 = p2V2 or as pV = k, where k is a constant which depends upon the particular sample. The pressure and the volume vary inversely; as the pressure of the sample increases the volume of the sample of gas must decrease. The law as formulated by Boyle does not suggest any particular scale of volume or of pressure. The units of volume are simply the cube of any convenient unit of length; the volume is actually measured in a separate experiment in which the tube is filled to the same mark with a liquid, just as Archimedes once measured the volume of the crown of Hiero of Syracuse.

Example. A sample of gas occupies a volume of 47.3 mL at 20oC when the pressure is 30 cm of mercury. If the pressure is increased to 75 cm of mercury, the sample will occupy a volume of 47.3 mL (30 cm Hg/75 cm Hg) = 18.9 mL.

Copyright 1997 James R. Fromm