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The kinetic-molecular model of gases describes the gaseous state as one in which gas particles are spaced out relative to one another and are moving around with rapid, random motion.

Calculations based on this model reveal a number of interesting facts:

• The average distance between particles is about 10 times the particle diameter – for oxygen, this distance is 3.54 nm.
• There is a wide range of particle speeds within this rapid random motion, and it is possible to calculate an average value, which depends on the temperature of the gas and the mass of the gas particle. For example, at standard temperature (0°C) and pressure (101.3 kPa), referred to as STP, the average speed of an oxygen molecule is 425 m/s and the average speed of a hydrogen molecule is 1845 m/s. These are staggeringly high speeds!
• The mean free path is the distance a given gas particle moves before colliding either with the walls of the container or with another particle. For oxygen at STP, the mean free path is 63 nm. This is an exceptionally small distance.
• The number of collisions occurring per second for a given molecule can be calculated by dividing the average speed by the mean free path. For an oxygen molecule at STP, this is close to 7 billion collisions per second!

## Relationship between gas pressure and gas volume

While studying the compressibility of gases, Robert Boyle (1627–1691) discovered that, for a fixed amount of gas at a given temperature, if the pressure acting on it is doubled, the volume of the gas is halved.

This inverse relationship between the pressure acting on a gas and its volume is known as Boyle’s Law, which takes the mathematical form P1V1=P2V2. In each case, the product of the pressure and the volume is a constant.

## Relationship between gas volume and gas temperature

In 1787, French scientist Jacques Charles discovered that the volume of a fixed amount of gas held at constant pressure decreased with decreasing temperature. If the temperature is measured in kelvin rather than °C, the relationship is a directly proportional one. Charles’s Law takes the mathematical form V1/T1 = V2/T2.

## Combining Boyle’s Law and Charles’s Law

For a fixed amount of gas, these three relationships can be combined to give the form P1V1/T1 = P2V2/T2. This equation is known as the general gas law.

Here is an example of how the general gas law is applied. Weather balloons are filled with helium (or less expensive hydrogen) gas. When released, they move up through the troposphere, and the attached instruments send back information about temperature, pressure, wind speed and humidity.

Suppose a 5000 L balloon is launched when the temperature is 17°C and the pressure 1000 hPa. If it ascends to a height of 35 km where the temperature is -33°C and the pressure 150 hPa, what will the volume of the balloon now be?

Now P1V1/T1 = P2V2/T2 and P1= 1000 hPa; V1 = 5000 L; T1 = 17 + 273 = 290 K

P2= 150 hPa; V2 = ? L; T2 = -33 + 273 = 240 K

Rearranging for V2 gives:

V2 = P1V1T2/P2T1

= (1000 x 5000 x 240)/150 x 290

= 27586.2 L

The balloon has expanded to 5.5 times its original volume. It is most likely that the balloon would have burst before reaching this altitude.

## Nature of science

A scientific law is a statement of fact meant to explain, in simple terms, an action or set of actions. It is generally accepted to be true and universal and can sometimes be expressed in terms of a single mathematical equation. For example, the changing volume of a gas with changing pressure embodied in Boyle’s Law can be expressed as P1V1=P2V2.

Published 29 April 2014 Referencing Hub articles