# 7.8 Gas Laws

Try this trick. Fill a glass with water to its brim and place a smooth playing card over the top. Hold the card in place with your finger and invert the glass. Now remove your finger. The card does not move and the water stays in the glass! The trick works because air exerts pressure that counters the weight of the water; it would not work on the Moon where there is no atmospheric pressure. Scientists realized in the 17th century that we live under an ocean of air. Pressure is one of the attributes of air or any other gas.

von Guericke's vacuum experiment. Click here for original source URL.

Otto von Guericke. Click here for original source URL.

Otto von Guericke was the mayor of the German town of Magdeburg and a talented amateur scientist. He invented a vacuum pump and showed that feathers and stones fall at the same rate in the absence of air resistance — confirming a prediction of Galileo. He also showed that air is essential to life by pumping the air out of a jar containing small animals. von Guericke demonstrated the power of air pressure with a spectacular experiment. He built a copper sphere half a meter across composed of two hemispheres. After he removed the air from the sphere with a pump, two teams of eight horses could not pull the hemispheres apart. Yet when he opened the valve and let the air back in, the hemispheres fell apart.

We define pressure as force acting over an area. Mathematically, P = F/A. So the same force acting over a smaller area makes for higher pressure. Distributing force (or weight) over a larger area is the reason why you sink into snow when wearing shoes but not when wearing snowshoes. It also explains why it is possible to lie safely on a bed of nails when sitting on any one nail would cause it to penetrate the skin. The air pressure exerted on a large window in your house is equal to about ten times your weight! However, the air inside exerts the same pressure outward so the two forces are balanced.

Robert Boyle. Click here for original source URL.

The first systematic studies of the properties of gases were carried out by Robert Boyle in the middle of the 17th century, while Newton was still alive. Boyle was the youngest of fourteen children of the Earl of Cork, one of the richest men in Ireland. He compressed and expanded a fixed amount of air, which was the only known gas at the time, and measured its pressure. Under conditions of constant temperature, he observed a simple inverse relation between pressure and volume

P ∝ 1 / V

If we increase the pressure on a gas, its volume will shrink. You can observe this easily with a bicycle pump. If you cover the valve and press on the plunger, the volume of the column of air in the pump decreases. Boyle concluded that "there is a spring or elastic power to the air we live in" and you can feel it too as you press up and down on the pump. As you apply larger force and therefore more pressure, the air compresses even more.

In the late 18th century, several French scientists made another important discovery about gases. If a gas is heated (and the pressure is fixed), it expands by 0.3% of its volume for every degree Celsius it is heated. If we use the Kelvin temperature scale, the simple proportionality is

V ∝ T

Gas expands as it gets hotter. Double the temperature and the volume will double. You can see this behavior with a bicycle pump also. Press as hard as you can on the plunger and hold it down. After a minute or so the barrel of the pump will be noticeably warmer. The relationship holds for any gas. Combining the two linear relationships for a gas, between pressure and volume and between volume and temperature, we have

P V ∝ T

This equation is a powerful tool for understanding and predicting the behavior of gases. It is often called the ideal gas law because it makes a number of assumptions about the molecules that make up a gas. These assumptions are: (1) the number of molecules is very large, (2) the molecules occupy a negligible fraction of the gas volume, (3) the molecules are in constant random motion, (4) there are no forces between molecules, and (5) the molecules bounce off each other without losing energy. Under normal terrestrial condition, these assumptions are all valid.

The ideal gas law describes the temperature and density variation of the Earth’s atmosphere with altitude. It also describes the variation of temperature and density of any planetary atmosphere. This in turn allows us to predict the conditions under which a gas might be compressed into a liquid or a solid. The exotic structure of giant planet atmospheres is a manifestation of a well-tested law of physics.

Global gas properties like pressure and temperature are related to the microscopic properties of atoms and molecules. Daniel Bernoulli was the first to get this insight, in the mid-18th century. Bernoulli was the head of an extraordinary family — no less than eight of his sons and grandsons became noted scientists and mathematicians! Bernoulli realized that gas pressure was caused by the continual microscopic collisions of gas molecules with their container. Compressing a gas increases the number of collision each second, hence the pressure. Compressing a gas also increases the energy of each collision, hence the temperature. Bernoulli’s calculation showed that the product of pressure and volume is related to the number of molecules (N) and the average kinetic energy of each one

Daniel Bernoulli. Click here for original source URL.

P V = 2/3 N (1/2 mv^{2})

The kinetic energy of a particle is related to temperature by 1/2 mv^{2} = 3/2 kT. Substituting in the equation above, we get

P V = N k T

The constant of proportionality is given by the Boltzmann constant (k) and the number of molecules (N). This elegant little law applies for any gas and for the physical conditions on any planet.