Gas Laws and Kinetic Theory of Gases: Formulas and PYQs

A gas has no shape and no fixed volume, so its behaviour is described by only four quantities: pressure, volume, temperature and the amount of substance. The gas laws are the experimental relations between them, and the kinetic theory is the molecular picture that explains why those relations hold. Together they are among the most profitable chapters in general science for competitive examinations, because the questions are short, formula-based and repeat almost unchanged from year to year.

The experimental gas laws

Charles's law also fixes the absolute scale of temperature. If the volume of a gas falls by one two hundred seventy-third of its volume at zero degrees Celsius for every degree of cooling, the volume would vanish at minus 273.15 degrees Celsius, and that limit is absolute zero, or zero kelvin. This is why every gas-law calculation must use the kelvin scale: a question that gives temperatures in Celsius is testing whether the candidate converts.

The ideal gas equation

Combining the three laws with Avogadro's law gives the equation of state of an ideal gas, PV equals nRT, where n is the number of moles and R is the universal gas constant. The value of R is 8.314 joule per mole per kelvin, which is the same as 0.0821 litre atmosphere per mole per kelvin or about 1.987 calorie per mole per kelvin. Writing the equation for a single molecule gives PV equal to NkT, where N is the number of molecules and k is the Boltzmann constant, 1.38 into ten to the power minus twenty-three joule per kelvin; R divided by the Avogadro number, 6.022 into ten to the power twenty-three, gives k. One mole of any ideal gas occupies 22.4 litres at 273.15 kelvin and one atmosphere, the figure every examination uses, while the modern standard pressure of one bar gives 22.7 litres.

The postulates of the kinetic theory

• A gas is made of a very large number of identical molecules in constant random motion.
• The volume actually occupied by the molecules is negligible compared with the volume of the container.
• There are no attractive or repulsive forces between molecules except during a collision.
• Collisions between molecules, and with the walls, are perfectly elastic, so kinetic energy is conserved.
• The time spent in a collision is negligible compared with the time between collisions, and between collisions a molecule moves in a straight line obeying Newton's laws.
• Pressure arises from the continual bombardment of the walls by the molecules, and no direction of motion is preferred.

What the theory gives us

From these postulates follows the kinetic gas equation, which states that the pressure of a gas equals one third of the product of its density and the mean square speed of its molecules. Comparing it with the ideal gas equation gives the most important single result of the chapter: the average kinetic energy of a molecule is three halves of the product of the Boltzmann constant and the absolute temperature, and for one mole it is three halves of RT. Temperature is therefore nothing but a measure of the average kinetic energy of molecular motion, and at the same temperature a light molecule and a heavy one have the same average kinetic energy though not the same speed. The root mean square speed equals the square root of three RT divided by the molar mass, so hydrogen molecules move about four times as fast as oxygen molecules at the same temperature. The three molecular speeds always stand in the order most probable speed, then average speed, then root mean square speed, in the ratio of the square root of two, the square root of eight divided by pi, and the square root of three. The mean free path, the average distance a molecule travels between collisions, grows as the pressure falls, which is why a vacuum tube works.

Degrees of freedom and specific heats

The law of equipartition of energy gives each degree of freedom an energy of half kT per molecule. A monatomic gas such as helium or argon has three translational degrees of freedom, so its molar specific heat at constant volume is three halves of R and the ratio of its specific heats, gamma, is 1.67. A diatomic gas such as oxygen or nitrogen has three translational and two rotational degrees of freedom at ordinary temperature, giving a specific heat of five halves of R and a gamma of 1.40. For a non-linear triatomic gas gamma is about 1.33. Mayer's relation connects the two specific heats of any ideal gas: the molar specific heat at constant pressure minus that at constant volume equals R.

Real gases and everyday examples

Real gases obey these laws only at low pressure and high temperature. Molecules do occupy space and do attract one another, and van der Waals corrected the ideal equation for both effects, adding a pressure term for attraction and subtracting a volume term for the size of the molecules. Below its critical temperature a gas can be liquefied by pressure alone, a result Andrews established with carbon dioxide, and the cooling produced when a compressed gas expands through a valve, the Joule-Thomson effect, is used in refrigeration and in liquefying air. The everyday examples are worth keeping: a hot air balloon rises because warm air expands and becomes less dense, as Charles's law requires; a tyre's pressure increases on a long drive, as the pressure law requires; a pressure cooker raises the boiling point by raising the pressure; and the smell of a leaking cooking-gas cylinder reaches you quickly because diffusion is fast for light molecules, as Graham's law requires.