1. Introduction to Gaseous State
Matter exists mainly in three common physical states: solid, liquid and gas. Among these, gases have the greatest freedom of molecular movement.
In a gas, the molecules are separated by relatively large distances compared with their molecular size. They move continuously and randomly in all directions.
The gaseous state can be described using measurable variables:
- Pressure (P)
- Volume (V)
- Temperature (T)
- Amount of gas (n)
The relationship between these variables is studied through gas laws and the ideal gas equation.
2. Characteristics of Gases
Important properties of gases include:
- Gases have no fixed shape.
- Gases have no fixed volume and occupy the entire available volume of the container.
- Gases are highly compressible.
- Gases have low density compared with solids and liquids.
- Gases exert pressure on the walls of their container.
- Gases mix rapidly with one another.
- Gas molecules are in continuous random motion.
- The attractive forces between gas molecules are relatively weak under ordinary conditions.
Why gases are highly compressible
A large part of the volume of a gas is empty space. When pressure is applied, the molecules can be brought closer together.
3. Pressure of a Gas
Gas molecules continuously collide with the walls of their container. These collisions produce pressure.
Pressure is defined as force acting per unit area.
Common units of pressure
| Unit | Relationship |
|---|---|
| 1 atmosphere | 1 atm = 101325 Pa |
| 1 bar | 1 bar = 100000 Pa |
| 1 atm | 1 atm ≈ 1.01325 bar |
| 1 atm | 760 mm Hg |
| 1 atm | 760 torr |
Temperature conversion
Gas-law calculations generally require absolute temperature.
4. Boyle's Law
Boyle's law describes the relationship between pressure and volume of a fixed amount of gas at constant temperature.
It states that at constant temperature, the volume of a fixed mass of gas is inversely proportional to its pressure.
Therefore:
For two conditions:
Interpretation
If pressure is increased while temperature remains constant, the gas volume decreases.
If pressure is decreased:
Graph
A plot of V against P gives a rectangular hyperbola.
5. Charles' Law
Charles' law describes the relationship between volume and absolute temperature at constant pressure.
It states that the volume of a fixed mass of gas is directly proportional to its absolute temperature at constant pressure.
Therefore:
For two conditions:
Interpretation
This relationship is valid when pressure and amount of gas remain constant.
6. Gay-Lussac's Law
For a fixed amount of gas at constant volume, pressure is directly proportional to absolute temperature.
For two conditions:
Therefore:
A sealed gas container can develop higher pressure when heated because the molecules move faster and collide more frequently and forcefully with the container walls.
7. Avogadro's Law
At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles of gas.
For two conditions:
Thus, increasing the amount of gas increases the volume when temperature and pressure remain constant.
Avogadro's hypothesis
Equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.
8. Combined Gas Law
Boyle's law, Charles' law and Gay-Lussac's law can be combined into one relationship for a fixed amount of gas.
Therefore, for two states:
This equation is useful when pressure, volume and temperature all change simultaneously.
Example of use
Suppose a gas changes from one set of P, V and T conditions to another. Insert all known quantities into:
Then solve for the unknown variable.
9. Ideal Gas Equation
Combining the gas laws gives:
Introducing the proportionality constant R:
This is the ideal gas equation.
Meaning of symbols
| Symbol | Meaning |
|---|---|
| P | Pressure |
| V | Volume |
| n | Number of moles |
| R | Gas constant |
| T | Absolute temperature in Kelvin |
Number of moles
Therefore:
This form is useful for calculations involving mass and molar mass.
10. Gas Constant R
The numerical value of R depends on the units used for pressure and volume.
| Units | Value of R |
|---|---|
| L atm mol⁻¹ K⁻¹ | 0.0821 |
| L bar mol⁻¹ K⁻¹ | 0.08314 |
| J mol⁻¹ K⁻¹ | 8.314 |
| m³ Pa mol⁻¹ K⁻¹ | 8.314 |
For example, if pressure is in atm and volume is in litres, use:
11. Density and Molar Mass of a Gas
Start from the ideal gas equation:
Since:
We get:
Rearranging:
Since density:
Therefore:
Hence:
Relative density and molar mass
Under the same temperature and pressure, the density of a gas is proportional to its molar mass.
Therefore, for gases compared under the same conditions:
When a gas is compared with hydrogen:
Thus:
12. Dalton's Law of Partial Pressures
Dalton's law applies to a mixture of non-reacting gases.
It states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of all the gases.
Partial pressure
Partial pressure is the pressure that a particular gas would exert if it alone occupied the entire volume of the mixture at the same temperature.
Relation with mole fraction
For gas i:
where Xi is the mole fraction of that gas.
Gas collected over water
When a gas is collected over water, the measured pressure contains both the pressure of the gas and water vapour.
Therefore:
13. Kinetic Theory of Gases
The kinetic molecular theory explains the behaviour of gases in terms of molecular motion.
Main assumptions
- A gas consists of a very large number of small particles called molecules.
- The molecules are in continuous random motion.
- The volume occupied by the molecules themselves is negligible compared with the total volume of the gas.
- There are no significant intermolecular attractive or repulsive forces in an ideal gas.
- Gas molecules collide with one another and with the walls of the container.
- Collisions are perfectly elastic.
- The time of collision is very small compared with the time between collisions.
- The average kinetic energy of gas molecules depends on absolute temperature.
Pressure according to kinetic theory
Gas pressure results from the continuous collisions of molecules with the walls of the container.
14. Kinetic Gas Equation
According to kinetic theory, the pressure of an ideal gas can be related to molecular motion by:
Where:
- P = pressure
- V = volume
- N = number of molecules
- m = mass of one molecule
- c² = mean square speed
The equation can also be written in terms of density:
Therefore:
Connection with temperature
From kinetic theory:
Thus, absolute temperature is directly related to the average kinetic energy of gas molecules.
15. Molecular Velocities
Gas molecules do not all move with the same speed. At any instant, some molecules are moving slowly while others are moving much faster.
Three useful velocity terms are:
- Most probable velocity
- Average velocity
- Root mean square velocity
Most probable velocity
Average velocity
Root mean square velocity
Order of velocities
16. Temperature and Molecular Kinetic Energy
The average translational kinetic energy of one molecule of an ideal gas is:
Where k is Boltzmann constant.
For one mole of an ideal gas:
Therefore, increasing temperature increases the average kinetic energy of gas molecules.
Important point
At the same temperature, different ideal gases have the same average translational kinetic energy per molecule, although their molecular speeds may be different because their masses are different.
17. Graham's Law of Diffusion
Graham's law states that the rate of diffusion or effusion of a gas is inversely proportional to the square root of its molar mass.
Therefore, for two gases:
A lighter gas diffuses faster than a heavier gas under the same conditions.
Example
Hydrogen has a much lower molar mass than oxygen, so hydrogen diffuses faster than oxygen.
18. Diffusion and Effusion
Diffusion
Diffusion is the spontaneous mixing of gas molecules due to their random molecular motion.
Example: the smell of a volatile substance spreading through a room.
Effusion
Effusion is the escape of gas molecules through a very small opening into a region of lower pressure.
Difference
| Diffusion | Effusion |
|---|---|
| Mixing of gas molecules | Escape through a tiny opening |
| Occurs during mixing | Occurs through a small hole |
| Example: gases mixing in air | Gas escaping through a pinhole |
Graham's law can be applied to both diffusion and effusion.
19. Real Gases
An ideal gas is a theoretical model. Real gases do not obey the ideal gas equation perfectly under all conditions.
Real-gas behaviour becomes especially important at:
- High pressure
- Low temperature
- Conditions near liquefaction
Why real gases deviate
There are two important assumptions of the ideal gas model that are not completely true for real gases.
- Gas molecules occupy a finite volume.
- Intermolecular attractive forces exist.
At high pressure
Molecules are brought closer together, so the actual volume of molecules becomes significant.
At low temperature
Molecular kinetic energy decreases, so intermolecular attraction becomes more important.
Low temperature → intermolecular attraction matters.
20. Compressibility Factor
The deviation of a real gas from ideal behaviour can be described using the compressibility factor Z.
For an ideal gas:
For real gases, Z may be greater than or less than 1.
When Z < 1
Attractive forces are important. The gas is more compressible than an ideal gas.
When Z > 1
Repulsive effects and finite molecular volume become important. The gas is less compressible than an ideal gas.
| Value of Z | Meaning |
|---|---|
| Z = 1 | Ideal gas behaviour |
| Z < 1 | Attractive forces dominate |
| Z > 1 | Repulsive/volume effects dominate |
21. van der Waals Equation
The ideal gas equation does not account for molecular volume and intermolecular attraction. van der Waals modified the equation to account for both effects.
For one mole of a real gas:
For n moles:
Meaning of a
The constant a represents the correction for intermolecular attractive forces.
Because attractions reduce the observed pressure, a pressure correction is added:
Meaning of b
The constant b represents the excluded volume of gas molecules. The volume actually available for molecular motion is therefore smaller than the container volume.
Physical meaning
| Constant | Correction |
|---|---|
| a | Intermolecular attraction |
| b | Finite molecular volume |
b → bulk/excluded volume.
22. Critical Constants
A gas can be liquefied by suitable combinations of temperature and pressure. However, above a certain temperature, increasing pressure alone cannot liquefy the gas.
Critical temperature
Critical temperature is the highest temperature at which a gas can be liquefied by applying pressure.
Above Tc, a gas cannot be liquefied by pressure alone.
Critical pressure
Critical pressure is the minimum pressure required to liquefy a gas at its critical temperature.
Critical volume
Critical volume is the molar volume of a substance at its critical temperature and critical pressure.
23. Liquefaction of Gases
Liquefaction means conversion of a gas into a liquid.
Two important conditions favour liquefaction:
- Lowering the temperature
- Increasing the pressure
Lowering temperature reduces molecular kinetic energy, while increasing pressure brings molecules closer together.
Effect of critical temperature
A gas must be below its critical temperature before pressure can liquefy it.
Joule-Thomson effect
The Joule-Thomson effect refers to the temperature change observed when a real gas expands from a high-pressure region to a low-pressure region through a porous plug or throttling valve under suitable conditions.
This effect is important in gas-cooling and liquefaction processes.
Applications
- Production of liquid nitrogen
- Production of liquid oxygen
- Industrial gas separation
- Refrigeration systems
- Cryogenic applications
24. Worked Numerical Problems
A gas occupies 4.0 L at a pressure of 2.0 atm. What volume will it occupy at 4.0 atm if temperature remains constant?
Given:
Formula:
Therefore:
The volume decreases to 2.0 L.
A gas has a volume of 2.0 L at 300 K. Calculate its volume at 450 K at constant pressure.
Formula:
Therefore:
Therefore, the volume becomes 3.0 L.
Calculate the volume occupied by 2.0 mol of an ideal gas at 300 K and 1 atm pressure.
Formula:
Therefore:
Substitution:
Therefore, the volume is approximately 49.3 L.
The density of a gas is 1.25 g L⁻¹ at 1 atm and 300 K. Calculate its molar mass.
Formula:
Given:
Substitution:
Therefore, the molar mass is approximately 30.8 g mol⁻¹.
A gas mixture contains three gases with partial pressures of 0.20 atm, 0.35 atm and 0.15 atm. Calculate the total pressure.
Formula:
Therefore, the total pressure is 0.70 atm.
A gas has a mole fraction of 0.25 in a mixture whose total pressure is 4 atm. Calculate its partial pressure.
Formula:
Therefore, the partial pressure is 1.0 atm.
Compare the rates of diffusion of H₂ and O₂. Their molar masses are 2 and 32 g mol⁻¹ respectively.
Formula:
Substitution:
Therefore, hydrogen diffuses 4 times faster than oxygen under the same conditions.
Write the expression for the RMS velocity of an ideal gas.
This shows that:
Therefore, increasing temperature increases molecular speed, while increasing molar mass decreases molecular speed at the same temperature.
25. Common Mistakes
1. Using Celsius instead of Kelvin
Always convert temperature before using gas-law equations.
2. Using the wrong gas constant
Check the pressure and volume units before selecting R.
3. Forgetting that Boyle's law requires constant temperature
P₁V₁ = P₂V₂ applies when the amount of gas and temperature remain constant.
4. Confusing Charles' law and Gay-Lussac's law
5. Forgetting water vapour correction
For gas collected over water:
6. Confusing diffusion and effusion
Diffusion is mixing of gases, while effusion is passage through a very small opening.
7. Thinking real gases behave ideally at every condition
Real-gas deviations become important particularly at high pressure and low temperature.
8. Forgetting the meaning of van der Waals constants
26. TU Examination Preparation
Important 2-Mark Questions
- State Boyle's law.
- State Charles' law.
- State Avogadro's law.
- Define ideal gas.
- Write the ideal gas equation.
- Define partial pressure.
- State Dalton's law of partial pressures.
- State Graham's law.
- Define diffusion and effusion.
- What is compressibility factor?
- Define critical temperature.
- What are van der Waals constants a and b?
Important 5-Mark Questions
- Explain Boyle's law with equation and graph.
- Explain Charles' law and its mathematical expression.
- Derive the ideal gas equation.
- Explain Dalton's law of partial pressures.
- Explain the assumptions of kinetic theory of gases.
- Derive the kinetic gas equation.
- Explain Graham's law of diffusion.
- Explain the deviation of real gases from ideal behaviour.
- Explain the significance of compressibility factor.
Important 10-Mark Questions
- Derive the ideal gas equation from Boyle's, Charles' and Avogadro's laws.
- Discuss the kinetic theory of gases and derive the kinetic gas equation.
- Explain real gases and derive the van der Waals equation.
- Explain critical constants and the liquefaction of gases.
- Explain Dalton's law and its applications in gas mixtures.
- Derive Graham's law and discuss diffusion and effusion.
- Discuss the deviation of real gases from ideal behaviour using compressibility factor and van der Waals equation.
27. Quick Revision
Gas laws
Ideal gas
Gas mixtures
Kinetic theory
Diffusion
Real gases
Boyle → P and V
Charles → V and T
Gay-Lussac → P and T
Avogadro → V and n
Dalton → partial pressures
Graham → diffusion
van der Waals → real gases