Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
Chapter Overview
States of Matter is the fifth chapter of CBSE Class 11 Chemistry. It explains why matter exists in three states — solid, liquid, and gas — and how the balance between intermolecular forces and thermal energy determines which state a substance adopts. The chapter's bulk content covers the gaseous state in detail: gas laws, the ideal gas equation, Dalton's law, Graham's law, kinetic molecular theory, and the behaviour of real gases.
This chapter is heavy on numericals and formulae. Every gas law and every speed formula must be memorized along with the conditions under which they apply. The liquid state section covers vapour pressure, surface tension, and viscosity — topics that also appear in board exams.
1. Intermolecular Forces vs Thermal Energy
The state of matter depends on the competition between two opposing factors:
| Factor | Effect | Favours |
|---|---|---|
| Intermolecular forces | Pulls molecules together | Solid/liquid state |
| Thermal energy | Pushes molecules apart | Gaseous state |
| State | Intermolecular Forces | Thermal Energy | Result |
|---|---|---|---|
| Solid | Very strong | Low | Fixed shape and volume |
| Liquid | Moderate | Moderate | Fixed volume, no fixed shape |
| Gas | Very weak | High | No fixed shape or volume |
Types of Intermolecular Forces (in order of increasing strength):
| Force | Description | Example | Relative Strength |
|---|---|---|---|
| London dispersion forces | Temporary dipoles induced in neighbouring molecules due to electron cloud fluctuation | All molecules (even noble gases) | Weakest |
| Dipole-dipole forces | Attraction between permanent dipoles of polar molecules | HCl, CO, SO₂ | Moderate |
| Dipole-induced dipole forces | Polar molecule induces a dipole in a non-polar molecule | HCl dissolved in CCl₄ | Weak-moderate |
| Hydrogen bonding | Strong dipole-dipole interaction where H is bonded to F, O, or N | H₂O, NH₃, HF | Strongest (among intermolecular) |
| Ion-dipole forces | Attraction between an ion and a polar molecule | Na⁺ in water | Very strong |
London dispersion forces increase with:
- Increasing molecular size (more electrons = more polarizable)
- Increasing surface area of contact
- Shape: linear molecules have stronger dispersion forces than branched ones
2. The Gaseous State
Gases are characterized by:
- High compressibility — large spaces between molecules
- Low density — molecules are far apart
- No fixed shape or volume — gases fill the container
- High diffusibility — molecules mix rapidly
- Exert pressure — molecules collide with container walls
Measurable properties of gases:
| Property | Symbol | Unit |
|---|---|---|
| Pressure | P | atm, Pa, mm Hg, torr |
| Volume | V | L, mL, m³ |
| Temperature | T | Kelvin (always use K) |
| Amount | n | moles |
Unit conversions:
- 1 atm = 760 mm Hg = 760 torr = 101.325 kPa = 1.01325 × 10⁵ Pa
- 0°C = 273.15 K
- 1 L = 1 dm³ = 10⁻³ m³
3. Gas Laws
3.1 Boyle's Law (1662)
At constant temperature, the volume of a fixed amount of gas is inversely proportional to its pressure.
Graphical representation:
- P vs V: hyperbola
- PV vs P: horizontal line (constant)
- 1/V vs P: straight line through origin
- P vs 1/V: straight line through origin
3.2 Charles's Law (1787)
At constant pressure, the volume of a fixed amount of gas is directly proportional to its absolute temperature.
Absolute zero: The theoretical temperature at which volume becomes zero: -273.15°C = 0 K. At absolute zero, molecular motion ceases.
3.3 Gay Lussac's Law (Pressure-Temperature Law)
At constant volume, the pressure of a fixed amount of gas is directly proportional to its absolute temperature.
3.4 Avogadro's Law
Equal volumes of all gases at the same temperature and pressure contain the same number of molecules.
- 1 mole of any gas at STP (0°C, 1 atm) occupies 22.4 L and contains 6.022 × 10²³ molecules.
Summary of Gas Laws
| Law | Relation | Constant | Equation |
|---|---|---|---|
| Boyle's | P ∝ 1/V | T, n | P₁V₁ = P₂V₂ |
| Charles's | V ∝ T | P, n | V₁/T₁ = V₂/T₂ |
| Gay Lussac's | P ∝ T | V, n | P₁/T₁ = P₂/T₂ |
| Avogadro's | V ∝ n | T, P | V₁/n₁ = V₂/n₂ |
4. Ideal Gas Equation
Combining all four gas laws:
Where:
- P = pressure (atm)
- V = volume (L)
- n = number of moles
- R = universal gas constant = 0.0821 L atm K⁻¹ mol⁻¹ = 8.314 J K⁻¹ mol⁻¹
- T = temperature (K)
Other forms:
Combined gas law (when n changes):
5. Dalton's Law of Partial Pressures
The total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of each gas.
Application — Pressure of a gas collected over water:
When a gas is collected over water, it mixes with water vapour. The total pressure equals the gas pressure plus the aqueous tension (vapour pressure of water at that temperature).
6. Graham's Law of Diffusion
The rate of diffusion of a gas is inversely proportional to the square root of its molar mass (at constant temperature and pressure).
Also written as:
Applications:
- Separation of isotopes (e.g., ²³⁵U and ²³⁸U as UF₆)
- Comparing rates of effusion through a small hole
- Explaining why lighter gases diffuse faster (H₂ diffuses 4 times faster than O₂)
7. Kinetic Molecular Theory of Gases
Postulates:
1. Gases consist of large numbers of tiny particles (molecules) that are in constant, random, straight-line motion. 2. The volume of gas molecules is negligible compared to the total volume of the container. 3. There are no attractive or repulsive forces between gas molecules (except during collisions). 4. Collisions between molecules and between molecules and the container walls are perfectly elastic (no loss of kinetic energy). 5. The average kinetic energy of gas molecules is directly proportional to the absolute temperature:
6. Pressure exerted by a gas is due to collisions of molecules with the walls of the container.
Derivation of kinetic gas equation:
8. Molecular Speeds
Three important speeds characterize the Maxwell-Boltzmann distribution:
| Speed | Formula | Description |
|---|---|---|
| Most probable speed (ump) | √(2RT/M) | Speed at the peak of distribution |
| Average speed (uavg) | √(8RT/πM) | Mean of all molecular speeds |
| RMS speed (urms) | √(3RT/M) | Square root of mean of squares of speeds |
Order of speeds: urms > uavg > ump
Ratio: urms : uavg : ump = √3 : √(8/π) : √2 = 1.732 : 1.596 : 1.414
Key points:
- All three speeds are proportional to √(T/M).
- Increasing temperature shifts the distribution to higher speeds and broadens it.
- Increasing molecular mass shifts the distribution to lower speeds and narrows it.
Kinetic energy per mole:
Kinetic energy depends only on temperature, not on the nature or mass of the gas.
9. Real Gases and van der Waals Equation
9.1 Deviations from Ideal Behaviour
Real gases deviate from ideal behaviour because: 1. Finite molecular volume: Gas molecules occupy space; the free volume available for movement is less than V. 2. Intermolecular attractions: Molecules attract each other; the pressure exerted on the walls is less than expected.
When do real gases behave ideally?
- At low pressure (molecules far apart, volume and attractions negligible)
- At high temperature (thermal energy overcomes intermolecular forces)
When do real gases deviate most?
- At high pressure and low temperature
9.2 Compressibility Factor (Z)
For real gases:
- At low pressure: Z < 1 (attractions dominate)
- At very high pressure: Z > 1 (volume effect dominates)
- At intermediate pressure: Z passes through a minimum (Boyle temperature)
9.3 van der Waals Equation
Corrects the ideal gas equation for molecular volume and intermolecular forces:
van der Waals constants:
| Constant | Meaning | Unit |
|---|---|---|
| a | Correction for intermolecular attraction | atm L² mol⁻² |
| b | Correction for finite molecular volume (excluded volume) | L mol⁻¹ |
Effect of a: Higher 'a' means stronger intermolecular forces, making the gas easier to liquefy. Effect of b: Higher 'b' means larger molecular size, taking up more space.
van der Waals constants for common gases:
| Gas | a (atm L² mol⁻²) | b (L mol⁻¹) | Ease of Liquefaction |
|---|---|---|---|
| He | 0.034 | 0.0237 | Very difficult |
| H₂ | 0.244 | 0.0267 | Difficult |
| N₂ | 1.39 | 0.0391 | Moderate |
| CO₂ | 3.59 | 0.0427 | Easy |
| NH₃ | 4.17 | 0.0371 | Easy |
| H₂O | 5.46 | 0.0305 | Very easy |
Gases with larger 'a' values are more easily liquefied because they have stronger intermolecular forces.
9.4 Boyle Temperature
The temperature at which a real gas obeys the ideal gas law over a range of pressures.
Above Boyle temperature, Z > 1 at all pressures (repulsive forces dominate). Below Boyle temperature, Z < 1 at low pressures (attractive forces dominate).
10. Liquefaction of Gases
Gases can be liquefied by: 1. Increasing pressure — reduces intermolecular distance. 2. Decreasing temperature — reduces kinetic energy so intermolecular forces can hold molecules together.
Critical temperature (Tc): The highest temperature at which a gas can be liquefied by pressure alone.
Critical pressure (Pc): The minimum pressure required to liquefy a gas at Tc.
Critical volume (Vc): The volume at Tc and Pc.
Gases with high Tc (strong intermolecular forces, large 'a') are easily liquefied. Gases with low Tc (like He, H₂) require very low temperatures.
11. Liquid State
11.1 Vapour Pressure
Vapour pressure is the pressure exerted by the vapour of a liquid in equilibrium with the liquid at a given temperature.
Key points:
- Vapour pressure increases with temperature (more molecules have sufficient energy to escape).
- Boiling occurs when vapour pressure equals atmospheric pressure.
- Liquids with high vapour pressure at room temperature are volatile (e.g., ether, acetone).
- Liquids with low vapour pressure at room temperature are non-volatile (e.g., mercury, glycerol).
11.2 Surface Tension
Surface tension is the force acting along the surface of a liquid perpendicular to any line of unit length on the surface.
Causes: Molecules at the surface experience an unbalanced inward pull (no molecules above), creating a surface film.
Effects of surface tension:
- Spherical shape of liquid drops (spheres minimize surface area for a given volume)
- Capillary rise (liquid rises in narrow tubes)
- Meniscus formation (concave for wetting liquids, convex for non-wetting)
Temperature effect: Surface tension decreases with increasing temperature and becomes zero at the critical temperature.
11.3 Viscosity
Viscosity is the measure of the resistance of a liquid to flow. It is the internal friction between layers of a liquid moving past each other.
Units: Pa s (SI) or poise (CGS); 1 Pa s = 10 poise
Factors affecting viscosity:
- Temperature: Viscosity decreases with increasing temperature (thermal energy overcomes intermolecular forces).
- Intermolecular forces: Stronger forces → higher viscosity (e.g., glycerol has high viscosity due to H-bonding).
- Molecular size: Larger molecules → higher viscosity (e.g., oil is more viscous than water).
12. Solved Numericals
Numerical 1: Boyle's Law
Q: A gas occupies 5.0 L at 2.0 atm pressure. What volume will it occupy at 10.0 atm pressure at the same temperature?
Numerical 2: Ideal Gas Equation
Q: Calculate the pressure exerted by 2.0 moles of an ideal gas at 27°C in a 10 L container.
Numerical 3: Dalton's Law
Q: A mixture of 2.0 moles of N₂ and 3.0 moles of O₂ exerts a total pressure of 5.0 atm. Find the partial pressure of each gas.
Numerical 4: Graham's Law
Q: Calculate the ratio of rates of diffusion of hydrogen and oxygen gases.
Numerical 5: RMS Speed
Q: Calculate the RMS speed of nitrogen molecules at 300 K. (M = 28 × 10⁻³ kg mol⁻¹, R = 8.314 J K⁻¹ mol⁻¹)
Numerical 6: Density from Ideal Gas Equation
Q: Calculate the density of methane (CH₄) at 1 atm and 273 K. (M = 16 g mol⁻¹)
13. Important Definitions
| Term | Definition |
|---|---|
| Intermolecular forces | Forces of attraction and repulsion between molecules |
| London dispersion forces | Temporary dipole-induced dipole forces present in all molecules |
| Dipole-dipole forces | Attraction between permanent dipoles of polar molecules |
| Hydrogen bonding | Strong dipole-dipole attraction where H is bonded to F, O, or N |
| Boyle's law | PV = constant at constant T and n |
| Charles's law | V/T = constant at constant P and n |
| Avogadro's law | Equal volumes of gases at same T and P contain equal molecules |
| Ideal gas | Gas that obeys PV = nRT at all T and P |
| Compressibility factor | Z = PV/nRT; Z=1 for ideal gas |
| van der Waals equation | (P + an²/V²)(V - nb) = nRT |
| Most probable speed | Speed at which maximum number of molecules are moving |
| RMS speed | Square root of mean of squares of speeds |
| Vapour pressure | Pressure exerted by vapour in equilibrium with liquid |
| Surface tension | Force per unit length acting on liquid surface |
| Viscosity | Resistance of a liquid to flow |
| Critical temperature | Highest temperature at which a gas can be liquefied by pressure |
14. Common Mistakes to Avoid
1. Forgetting to convert temperature to Kelvin — All gas law calculations require T in Kelvin. 0°C = 273.15 K, 25°C = 298.15 K. 2. Mixing up R values — Use R = 0.0821 L atm K⁻¹ mol⁻¹ when P is in atm and V is in L. Use R = 8.314 J K⁻¹ mol⁻¹ for energy calculations. Do not mix them. 3. Confusing most probable, average, and RMS speeds — Remember the order: urms > uavg > ump, and their formulas differ by the numerical factor inside the square root. 4. Applying ideal gas equation to real gases without correction — At high pressure and low temperature, use the van der Waals equation instead. 5. Forgetting to subtract aqueous tension — When a gas is collected over water, P_dry gas = P_total - aqueous tension. 6. Confusing 'a' and 'b' in van der Waals equation — 'a' corrects for intermolecular attraction (appears in pressure term); 'b' corrects for molecular volume (appears in volume term). 7. Not using consistent units in Graham's law — Molar mass must be in the same units for both gases (usually g mol⁻¹). The ratio eliminates units, so this works out, but double-check. 8. Confusing surface tension and viscosity — Surface tension is about the surface film (force/length); viscosity is about internal resistance to flow (force × time/area).
15. Exam Tips
- For gas law numericals: Always write the formula first, substitute values with units, and state the answer with correct units and significant figures.
- For ideal gas problems: Use PV = nRT. Identify what is given (3 of P, V, n, T) and solve for the fourth. Convert all units before substituting.
- For Dalton's law: Calculate mole fractions first, then multiply by total pressure.
- For Graham's law: Write the formula r₁/r₂ = √(M₂/M₁). Make sure the heavier gas is in the denominator for the slower rate.
- For molecular speed questions: Remember urms = √(3RT/M), uavg = √(8RT/πM), ump = √(2RT/M). Use R = 8.314 J K⁻¹ mol⁻¹ and M in kg mol⁻¹.
- For van der Waals questions: Identify which correction matters — 'a' for attraction (pressure term), 'b' for volume (volume term).
- Common 1-mark questions: State Boyle's law, define vapour pressure, name the types of intermolecular forces, give the unit of viscosity.
- Common 3-mark questions: Derive the ideal gas equation, explain deviations from ideal behaviour, compare molecular speeds.
- Common 5-mark questions: State and prove Graham's law, explain kinetic theory postulates, derive van der Waals equation, discuss liquefaction of gases.
16. Quick Revision Notes
- Intermolecular forces (increasing strength): London dispersion < dipole-dipole < hydrogen bonding < ion-dipole
- London forces increase with molecular size and surface area
- Boyle's law: P₁V₁ = P₂V₂ (constant T, n)
- Charles's law: V₁/T₁ = V₂/T₂ (constant P, n)
- Gay Lussac's: P₁/T₁ = P₂/T₂ (constant V, n)
- Avogadro's law: V/n = constant (constant T, P)
- Ideal gas: PV = nRT; R = 0.0821 L atm K⁻¹ mol⁻¹
- Dalton's law: P_total = P₁ + P₂ + ... ; Pᵢ = xᵢ × P_total
- Gas collected over water: P_dry = P_total - aqueous tension
- Graham's law: r₁/r₂ = √(M₂/M₁)
- RMS speed = √(3RT/M); Average speed = √(8RT/πM); Most probable = √(2RT/M)
- Order: urms > uavg > ump
- KE per mole = (3/2)RT (depends only on T, not on gas)
- Van der Waals: (P + an²/V²)(V - nb) = nRT
- a = intermolecular attraction correction; b = excluded volume correction
- Z = PV/nRT; Z=1 ideal, Z<1 attractive forces dominate, Z>1 repulsive/volume effect
- Critical temperature: Tc = 8a/(27Rb)
- Vapour pressure increases with temperature
- Surface tension decreases with temperature; zero at critical temperature
- Viscosity decreases with temperature; stronger intermolecular forces = higher viscosity
- 1 atm = 760 mm Hg = 760 torr = 101.325 kPa
- STP: 0°C (273.15 K), 1 atm; 1 mol gas at STP = 22.4 L
17. Practice Questions
1. A gas occupies 200 mL at 27°C and 1 atm. What will be its volume at 327°C and 1 atm? 2. Calculate the density of CO₂ at 1 atm and 0°C. (M = 44 g mol⁻¹) 3. 3.0 g of H₂ and 32.0 g of O₂ are placed in a 10 L container at 27°C. Calculate the total pressure and the partial pressure of each gas. 4. The rate of diffusion of an unknown gas is 0.5 times that of O₂. What is the molar mass of the unknown gas? 5. Calculate the RMS speed of oxygen molecules at 27°C. (M = 32 g mol⁻¹) 6. Using the van der Waals equation, explain why real gases deviate from ideal behaviour at high pressure and low temperature. 7. What is the compressibility factor? What does Z < 1 and Z > 1 indicate? 8. A gas is collected over water at 25°C. The total pressure is 750 mm Hg. If the vapour pressure of water at 25°C is 23.8 mm Hg, what is the pressure of the dry gas? 9. Why does water have a higher boiling point than H₂S despite both being in Group 16? 10. Arrange the following in order of increasing intermolecular forces: CO₂, H₂O, HCl, CH₄.
Quick Answers to Practice Questions
1. V₁/T₁ = V₂/T₂; V₂ = 200 × 600/300 = 400 mL (temperature doubles, volume doubles) 2. d = PM/RT = (1 × 44)/(0.0821 × 273.15) = 44/22.41 = 1.963 g L⁻¹ 3. n(H₂) = 3/2 = 1.5 mol; n(O₂) = 32/32 = 1.0 mol; total n = 2.5 mol; P = nRT/V = (2.5 × 0.0821 × 300)/10 = 6.16 atm; P(H₂) = (1.5/2.5) × 6.16 = 3.70 atm; P(O₂) = (1.0/2.5) × 6.16 = 2.46 atm 4. r₁/r₂ = √(M₂/M₁); 0.5 = √(32/M₁); 0.25 = 32/M₁; M₁ = 128 g mol⁻¹ 5. urms = √(3RT/M) = √(3 × 8.314 × 300 / 0.032) = √(7482.6/0.032) = √233831.25 = 483.6 m s⁻¹ 6. At high pressure: molecular volume becomes significant (b correction) and molecules are forced close together where attraction matters (a correction). At low temperature: kinetic energy decreases so intermolecular attractions become significant. Both cause deviations from PV = nRT. 7. Z = PV/nRT. Z < 1: attractive forces dominate, gas is more compressible than ideal. Z > 1: repulsive forces/volume effect dominate, gas is less compressible than ideal. 8. P_dry = P_total - P_water = 750 - 23.8 = 726.2 mm Hg 9. Water has strong intermolecular hydrogen bonding (O-H···O) which requires much more energy to break. H₂S has weaker dipole-dipole interactions because S is less electronegative than O and forms weaker hydrogen bonds. 10. CH₄ < CO₂ < HCl < H₂O (London forces < dipole-dipole < stronger dipole-dipole < hydrogen bonding)
18. Final Revision Checklist
Before the exam, make sure you can:
- List and compare types of intermolecular forces with examples
- State and apply all four gas laws with mathematical forms
- Use PV = nRT to solve numerical problems
- Apply Dalton's law to find partial pressures and correct for aqueous tension
- Apply Graham's law to compare rates of diffusion
- Calculate RMS, average, and most probable speeds
- Explain the five postulates of kinetic molecular theory
- Explain deviations from ideal behaviour using van der Waals equation
- Define and interpret compressibility factor (Z)
- Explain vapour pressure, surface tension, and viscosity with their temperature dependences
- Convert between units of pressure (atm, mm Hg, Pa, kPa) and temperature (°C, K)