The gas laws were developed in the late 1800s when the scientists understood the relationship between the pressure, volume, and temperature for a sample of gas. These relationships would, in turn, be, approximately, valid for all the gases. Nonetheless, all the gases behave similarly. Gases have widely spaced individual particles. The ‘Kinetic Theory of Gases’ derives the ‘Equation of State’ for an ideal gas. Different scientists did numerous experiments and hence, put forth different gas laws which relate to different state variables of a gas. The state variables of the gas are:
- Pressure, P (mmHg, atm, kPa, and Torr)
- Volume, V (L)
- Temperature, T (K)
- Amount of Substance, n
Boyle’s Law: The Pressure-Volume Law
Boyle’s Law states that at a constant temperature, the volume of a given mass of a gas is inversely proportional to the pressure; i.e., at constant temperature V ∝ 1/P or PV= constant.
Proof: From the Kinetic Theory of gases, we know:
where c is the root mean square velocity of the molecules, m1 is the mass of a molecule, V is the volume, and N is the number of molecules. Now, at a constant temperature, c, N, and m1 are constants; hence,
When the pressure increases, the volume of a gas decreases; and vice-versa. The following equation can be derived from the Boyle’s Law:
Example: When compressed air is filled in a tire, the pressure measurements are taken into consideration. As the tire is inflated with more and more air at the same temperature, all the molecules of gas are forced to pack together, reduce their volume, and increase the pressure on the walls of the tire.
Other examples are:
- Spray-paint can
- Syringe
- Soda can
Charles Law: The Temperature-Volume Law
Charles Law or Law of Volume states that at constant pressure, the volume of a given mass of a gas is directly proportional to its absolute temperature; i.e., at constant pressure, V ∝ T or V/T= constant.
Proof: Now as c2 ∝ T, thus at a constant pressure for a given mass of a gas, V ∝ T.
As the temperature of a gas increases, the volume of the gas also increases. Moreover, the initial and final temperature, as well as the volume of a gas, can be easily determined;
Example: Leaving a basketball out during the cold months deflates it. You will notice that when the ball is left under colder conditions, it starts losing the air inside it or its volume starts decreasing. This proves that under constant pressure conditions, if there is a fall in temperature, the volume also decreases.
Gay Lussac’s Law: The Pressure-Temperature Law
Gay Lussac’s Law states that at a constant volume, the pressure of a given mass of a gas is directly proportional to its absolute temperature; i.e., at constant volume, P ∝ T or P/T= constant.
Proof: From the Kinetic Theory of gases, we know:
Now as c^2 ∝ T, thus at a constant volume, P ∝ T for a given mass of gas.
As the temperature increases, the pressure also increases. Under the similar condition, the initial and final pressure and temperature for a given volume of gas can be calculated;
Example: The working of a pressure cooker follows the Gay Lussac’s law. As the temperature increases, the pressure inside the pressure cooker also increases, which makes the food cook faster.
Other examples are:
- Car tires in hot weather
- Aerosol can
Avogadro’s Law: The Volume Amount Law
Avogadro’s law states that for constant temperature, pressure, and volume, all the gases contain an equal number of molecules. 1 mole of any gas at NTP occupies a volume of 22.4L. It is important for determining the relationship between the amount of gas (N) and the volume of the gas (V).
Proof: From the Kinetic Theory of gases, we know;
Now as c^2 ∝ T, thus at a constant V, P, & T, N= constant, for a given mass of a gas.
If the number of molecules of a gas increases, the volume of the gas also increases;
If the temperature and pressure remain constant, the volume-amount fraction will be constant;
Example: As you inhale air, your lungs expand. Similarly, the volume of your lungs decreases as you exhale.
The Combined Gas Law
Now, we can easily combine the Boyle’s law, Charles law, and the Guy Lussac’s law to a ‘Combined Gas Law Equation’ or the ‘General Gas Equation.’ It determines the relationship between the pressure, volume, and temperature for a given quantity of gas.
The given volume of gas is directly proportional to the Kelvin temperature and inversely proportional to the pressure.
The initial and final volume and temperature can also be calculated;
The Ideal Gas Law
The ideal gas law is obtained by the addition of the Avogadro’s law to the combined gas law:
where;
- P= pressure,
- V= volume,
- n= number of moles,
- R= universal gas constant, 8.3144598 (kPa∙L)/(mol∙K), and
- T= temperature (K)
Another formulation of the ideal gas law can be;
where,
- P= pressure,
- V= volume,
- N= number of gas molecules,
- k= Boltzmann constant, 1.381×10−23 J·K−1 in SI units, and
- T= temperature (K)