Table of Contents
Gas Laws and Kinetic Theory of Gases
1. Introduction
Gases are made up of particles that move continuously and randomly. Their physical behaviour depends on pressure, volume, temperature and the amount of gas present.
Gas laws describe the mathematical relationships among these properties. The kinetic theory of gases explains these relationships by considering the motion and collisions of gas particles.
These concepts are important in physics, chemistry, meteorology, engines, refrigeration, industrial processes and atmospheric science.
2. Basic Properties of Gases
A gas has several distinctive properties:
- It has no fixed shape.
- It has no fixed volume.
- It expands to fill its container.
- It is generally compressible.
- It exerts pressure on the walls of its container.
- Its particles move continuously and randomly.
- Its density is generally much lower than that of solids and liquids under ordinary conditions.
The pressure of a gas results from collisions of its particles with the walls of its container.
3. Gas Variables
The behaviour of a gas is commonly described by four variables.
3.1 Pressure (P)
Pressure is the force exerted per unit area.
[
P=\frac{F}{A}
]
Its SI unit is the pascal (Pa).
[
1\ \text{Pa}=1\ \text{N m}^{-2}
]
3.2 Volume (V)
Volume is the space occupied by a gas.
Its SI unit is the cubic metre (m³), although litres are also commonly used.
[
1\ \text{m}^3=1000\ \text{L}
]
3.3 Temperature (T)
Gas-law calculations use absolute temperature measured in kelvin.
[
T(\text{K})=T(^\circ\text{C})+273.15
]
3.4 Amount of Gas (n)
The amount of gas is measured in moles.
One mole contains approximately:
[
N_A=6.02214076\times10^{23}
]
elementary entities. This number is called the Avogadro constant.
4. Boyle’s Law
Boyle’s law states:
For a fixed amount of gas at constant temperature, pressure is inversely proportional to volume.
[
P\propto\frac{1}{V}
]
Therefore:
[
PV=\text{constant}
]
For two equilibrium states:
[
P_1V_1=P_2V_2
]
4.1 Explanation
When a gas is compressed at constant temperature:
- Its volume decreases.
- Its particles have less space in which to move.
- Collisions with the container walls become more frequent.
- Pressure increases.
When volume increases, pressure decreases under the same temperature and amount-of-gas conditions.
4.2 Applications
- Syringes.
- Bicycle pumps.
- Breathing mechanics.
- Compression systems.
- Gas storage equipment.
5. Charles’s Law
Charles’s law states:
For a fixed amount of gas at constant pressure, volume is directly proportional to its absolute temperature.
[
V\propto T
]
Therefore:
[
\frac{V}{T}=\text{constant}
]
For two states:
[
\frac{V_1}{T_1}=\frac{V_2}{T_2}
]
Temperature must be expressed in kelvin.
5.1 Explanation
When a gas is heated at constant pressure:
- Its particles gain average kinetic energy.
- They move more energetically.
- The gas expands.
- Its volume increases while pressure remains constant.
5.2 Applications
- Hot-air balloons.
- Expansion of gases in flexible containers.
- Temperature-related changes in gas volume.
- Scientific gas experiments.
6. Pressure Law or Gay-Lussac’s Law
The pressure law states:
For a fixed amount of gas at constant volume, pressure is directly proportional to its absolute temperature.
[
P\propto T
]
Therefore:
[
\frac{P}{T}=\text{constant}
]
For two states:
[
\frac{P_1}{T_1}=\frac{P_2}{T_2}
]
6.1 Explanation
When a gas in a sealed rigid container is heated:
- The particles gain average kinetic energy.
- Their collisions with the container walls become more forceful on average.
- Pressure increases.
- Volume remains constant.
6.2 Applications
- Pressure changes in sealed gas containers.
- Industrial gas systems.
- Gas-filled bulbs and rigid vessels.
- Temperature-related pressure monitoring.
7. Avogadro’s Law
Avogadro’s law states:
At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles of gas.
[
V\propto n
]
Therefore:
[
\frac{V}{n}=\text{constant}
]
Equal volumes of different ideal gases at the same temperature and pressure contain equal numbers of molecules.
This principle is known as Avogadro’s hypothesis.
8. Combined Gas Law
The combined gas law combines Boyle’s law, Charles’s law and the pressure law for a fixed amount of gas.
[
\frac{PV}{T}=\text{constant}
]
For two equilibrium states:
[
\frac{P_1V_1}{T_1}=
\frac{P_2V_2}{T_2}
]
This equation is useful when pressure, volume and temperature all change.
9. Ideal Gas Equation
The ideal gas equation combines the relationships among pressure, volume, temperature and the amount of gas.
[
PV=nRT
]
Where:
- (P) = absolute pressure
- (V) = volume
- (n) = amount of gas in moles
- (R) = universal gas constant
- (T) = absolute temperature
The universal gas constant is:
[
R=8.314462618\ \text{J mol}^{-1}\text{K}^{-1}
]
The equation is also written as:
[
PV=Nk_BT
]
Where:
- (N) = number of molecules
- (k_B) = Boltzmann constant
9.1 Characteristics of an Ideal Gas
An ideal gas is a theoretical model in which:
- Gas particles are treated as point-like particles with negligible volume.
- Particles move continuously and randomly.
- Collisions are perfectly elastic.
- No intermolecular forces act between particles except during collisions.
- The particles obey Newtonian mechanics in the classical model.
Real gases approximate ideal-gas behaviour most closely at relatively low pressures and high temperatures, away from conditions where condensation becomes important.
10. Dalton’s Law of Partial Pressures
Dalton’s law states that the total pressure of a mixture of non-reacting gases is equal to the sum of the partial pressures of its component gases.
[
P_{\text{total}}=P_1+P_2+P_3+\cdots
]
The partial pressure of a gas is the pressure it would exert if it alone occupied the entire volume at the same temperature.
10.1 Applications
- Atmospheric science.
- Gas mixtures in industrial processes.
- Respiratory physiology.
- Scuba diving.
- Chemical laboratories.
11. Kinetic Theory of Gases
The kinetic theory explains the macroscopic properties of gases in terms of the motion and collisions of their microscopic particles.
It connects pressure and temperature to molecular motion.
11.1 Main Assumptions
The classical kinetic theory of an ideal gas is based on the following assumptions:
- A gas contains a very large number of tiny particles.
- The particles are in continuous random motion.
- The individual particle volume is negligible compared with the total gas volume.
- Particles exert negligible intermolecular forces except during collisions.
- Collisions between particles and with container walls are perfectly elastic.
- The duration of a collision is negligible compared with the time between collisions.
- Particles obey classical mechanics.
- The motion is random, with no preferred direction in equilibrium.
12. Molecular Explanation of Gas Pressure
Gas particles continuously collide with the walls of their container.
During each collision, the momentum of a particle changes. This produces a force on the wall.
The combined effect of many collisions produces the pressure exerted by the gas.
For an ideal gas:
[
P=\frac{1}{3}\rho c_{\text{rms}}^2
]
Where:
- (P) = gas pressure
- (\rho) = gas density
- (c_{\text{rms}}) = root-mean-square molecular speed
This equation demonstrates that gas pressure is related to the density and average squared speed of its particles.
13. Molecular Kinetic Energy and Temperature
The average translational kinetic energy of an ideal gas molecule is:
[
\overline{E_k}=\frac{3}{2}k_BT
]
This equation establishes a direct relationship between the average translational kinetic energy of gas molecules and absolute temperature.
As absolute temperature increases, the average translational kinetic energy increases.
For one mole of an ideal gas:
[
\overline{E}_{k,\text{mole}}=\frac{3}{2}RT
]
This expression refers to the total translational kinetic energy per mole for a classical ideal gas.
14. Root-Mean-Square Speed
The root-mean-square speed is a statistical measure of molecular speed.
It is calculated by taking the square root of the average of the squared molecular speeds.
For an ideal gas:
[
c_{\text{rms}}=\sqrt{\frac{3k_BT}{m}}
]
Where:
- (k_B) = Boltzmann constant
- (T) = absolute temperature
- (m) = mass of one molecule
In terms of molar mass:
[
c_{\text{rms}}=\sqrt{\frac{3RT}{M}}
]
Where (M) is the molar mass in kg mol(^{-1}).
The equation shows that:
- Molecular speed increases with temperature.
- Molecular speed decreases as molecular mass increases.
15. Degrees of Freedom
Degrees of freedom describe the independent ways in which a molecule can store energy.
Depending on molecular structure and temperature, energy may be associated with:
- Translational motion.
- Rotational motion.
- Vibrational motion.
For a simple monatomic ideal gas in the classical regime, there are three translational degrees of freedom.
For a monatomic ideal gas:
[
U=\frac{3}{2}nRT
]
Where (U) is the internal energy.
For an ideal gas with (f) active quadratic degrees of freedom:
[
U=\frac{f}{2}nRT
]
This general expression applies under the classical equipartition approximation and when the active degrees of freedom can be treated as constant.
16. Real Gases and Deviations from Ideal Behaviour
Real gases do not always behave ideally because:
- Gas molecules have finite volumes.
- Intermolecular attractive and repulsive forces exist.
- Molecular interactions become important at high pressures.
- Deviations increase near condensation and critical conditions.
16.1 Van der Waals Equation
The Van der Waals equation provides a more realistic description of certain real gases.
[
\left(P+\frac{an^2}{V^2}\right)(V-nb)=nRT
]
Where:
- (a) accounts approximately for intermolecular attractions.
- (b) accounts approximately for the finite volume of molecules.
- (n) = number of moles.
- (V) = volume.
- (P) = pressure.
- (T) = absolute temperature.
The Van der Waals equation is an improvement over the ideal gas model, although it is not perfectly accurate for all gases and conditions.
17. Applications of Gas Laws
17.1 Breathing
Changes in chest cavity volume produce pressure changes that help move air into and out of the lungs.
17.2 Hot-Air Balloons
Heating the air inside a balloon lowers its density relative to the surrounding cooler air, producing buoyancy.
17.3 Refrigeration
Gas laws help explain the pressure, temperature and volume changes involved in refrigeration cycles.
17.4 Automobile Tyres
Tyre pressure generally increases as the air inside becomes hotter, provided the tyre volume changes relatively little.
17.5 Scuba Diving
Pressure increases with depth in water, affecting the volume and behaviour of gases in diving equipment and the human body.
17.6 Industrial Gas Storage
Engineers use gas laws to calculate pressures, volumes and safe operating conditions in storage systems.
18. Worked Examples
Example 1: Boyle’s Law
A gas occupies 4 litres at a pressure of 2 atm. Calculate its volume when the pressure increases to 4 atm at constant temperature.
Using:
[
P_1V_1=P_2V_2
]
[
2\times4=4\times V_2
]
[
V_2=\frac{8}{4}
]
[
\boxed{V_2=2\ \text{L}}
]
Example 2: Charles’s Law
A gas occupies 2 litres at 300 K. Calculate its volume at 450 K if pressure remains constant.
Using:
[
\frac{V_1}{T_1}=\frac{V_2}{T_2}
]
[
V_2=\frac{2\times450}{300}
]
[
\boxed{V_2=3\ \text{L}}
]
Example 3: Pressure Law
A gas has a pressure of 100 kPa at 300 K. Calculate its pressure at 450 K if volume remains constant.
Using:
[
\frac{P_1}{T_1}=\frac{P_2}{T_2}
]
[
P_2=\frac{100\times450}{300}
]
[
\boxed{P_2=150\ \text{kPa}}
]
Example 4: Ideal Gas Equation
Calculate the volume occupied by 1 mole of an ideal gas at 300 K and 100,000 Pa.
Using:
[
PV=nRT
]
[
V=\frac{nRT}{P}
]
[
V=\frac{1\times8.314\times300}{100000}
]
[
V=0.024942\ \text{m}^3
]
[
\boxed{V\approx24.94\ \text{L}}
]
Example 5: Root-Mean-Square Speed of Nitrogen
Calculate the approximate root-mean-square speed of nitrogen molecules at 300 K.
Take the molar mass of nitrogen gas as:
[
M=0.028\ \text{kg mol}^{-1}
]
Using:
[
c_{\text{rms}}=\sqrt{\frac{3RT}{M}}
]
[
c_{\text{rms}}=
\sqrt{\frac{3\times8.314\times300}{0.028}}
]
[
\boxed{c_{\text{rms}}\approx517\ \text{m s}^{-1}}
]
19. Important Examination Distinctions
| Law or concept | Main relationship | Constant condition |
|---|---|---|
| Boyle’s law | (P\propto1/V) | Temperature |
| Charles’s law | (V\propto T) | Pressure |
| Pressure law | (P\propto T) | Volume |
| Avogadro’s law | (V\propto n) | Temperature and pressure |
| Combined gas law | (PV/T=\text{constant}) | Fixed amount of gas |
| Ideal gas equation | (PV=nRT) | General ideal-gas relationship |
| Dalton’s law | (P_{\text{total}}=\sum P_i) | Gas mixture |
36 Key Takeaways for Exams
- Gases have no fixed shape or volume.
- Gas pressure is produced by collisions of particles with container walls.
- Pressure is force per unit area.
- The SI unit of pressure is the pascal.
- Gas-law calculations require absolute temperature in kelvin.
- Boyle’s law applies to a fixed amount of gas at constant temperature.
- Boyle’s law states that pressure is inversely proportional to volume.
- Boyle’s law is expressed as (P_1V_1=P_2V_2).
- Charles’s law applies at constant pressure.
- Charles’s law states that gas volume is directly proportional to absolute temperature.
- Charles’s law is expressed as (V_1/T_1=V_2/T_2).
- The pressure law applies at constant volume.
- The pressure law states that gas pressure is directly proportional to absolute temperature.
- The pressure law is expressed as (P_1/T_1=P_2/T_2).
- Avogadro’s law relates gas volume to the number of moles.
- Equal volumes of ideal gases at the same temperature and pressure contain equal numbers of molecules.
- The combined gas law is (P_1V_1/T_1=P_2V_2/T_2).
- The ideal gas equation is (PV=nRT).
- The universal gas constant is approximately 8.314 J mol(^{-1}) K(^{-1}).
- The Boltzmann constant connects molecular energy with temperature.
- One mole contains approximately (6.022\times10^{23}) elementary entities.
- Dalton’s law states that total pressure equals the sum of partial pressures.
- The kinetic theory explains gas behaviour through molecular motion and collisions.
- Ideal-gas particles are assumed to have negligible individual volume.
- Collisions in the ideal-gas model are perfectly elastic.
- The ideal-gas model assumes negligible intermolecular forces except during collisions.
- Gas pressure is related to molecular density and mean squared speed.
- The pressure equation from kinetic theory is (P=\frac{1}{3}\rho c_{\text{rms}}^2).
- Average translational kinetic energy per molecule is (\frac{3}{2}k_BT).
- Root-mean-square molecular speed increases with absolute temperature.
- Root-mean-square molecular speed decreases as molecular mass increases.
- Degrees of freedom describe independent ways of storing energy.
- The internal energy of a monatomic ideal gas is (U=\frac{3}{2}nRT).
- Real gases deviate from ideal behaviour, especially at high pressure and low temperature.
- The Van der Waals equation accounts approximately for molecular volume and intermolecular attraction.
- Gas laws are applied in breathing, hot-air balloons, refrigeration, tyre pressure and industrial gas storage.