1. Introduction

Matter exists in three primary states: Solid, Liquid, and Gas. The state depends on the balance between:

Intermolecular Forces: Tend to keep molecules together (Cohesive forces).
Thermal Energy: Tends to keep molecules apart (Random motion).

Gas: Weakest forces, Highest thermal energy.
Liquid: Intermediate forces.
Solid: Strongest forces, Lowest thermal energy.

2. The Gaseous State

2.1 Gas Laws

1. Boyle's Law (P-V Relation):

At constant T, Pressure is inversely proportional to Volume.

P ∝ 1/V or P₁V₁ = P₂V₂

2. Charles' Law (V-T Relation):

At constant P, Volume is directly proportional to Absolute Temperature.

V ∝ T or V₁/T₁ = V₂/T₂

3. Gay-Lussac's Law (P-T Relation):

At constant V, Pressure is directly proportional to Absolute Temperature.

P ∝ T or P₁/T₁ = P₂/T₂

4. Avogadro's Law (V-n Relation):

Equal volumes of all gases at same T and P contain equal number of moles.

V ∝ n

2.2 Ideal Gas Equation

PV = nRT

R (Universal Gas Constant):
• 8.314 J K^-1 mol^-1 (SI Units)
• 0.0821 L atm K^-1 mol^-1 (Common for Pressure/Volume)
• 2 cal K^-1 mol^-1 (CGS)

3. Kinetic Theory of Gases (KTG)

Key Assumptions

Gases consist of large number of identical particles (atoms/molecules).
Volume of molecules is negligible compared to container volume.
No force of attraction or repulsion between molecules.
Collisions are perfectly elastic.

Pressure Equation

P = (1/3) ρ v_rms²

4. Molecular Speeds

Maxwell-Boltzmann distribution defines three types of speeds:

1. Root Mean Square (V_rms):

√(3RT / M)

2. Average Speed (V_avg):

√(8RT / πM)

3. Most Probable Speed (V_mp):

√(2RT / M)

Ratio: V_mp : V_avg : V_rms :: 1 : 1.128 : 1.224

5. Real Gases

Real gases deviate from ideal behavior at High Pressure and Low Temperature.

Compressibility Factor (Z)

Z = PV / nRT

Z = 1 (Ideal Gas)
Z < 1 (Negative Deviation, Attractive forces dominate)
Z > 1 (Positive Deviation, Repulsive forces dominate)

[Image of Compressibility factor Z vs Pressure graph]

Van der Waals Equation

(P + an²/V²) (V - nb) = nRT

'a': Measure of intermolecular attraction (Unit: atm L² mol^-2).
'b': Measure of effective size of molecule (Unit: L mol^-1).

6. Critical Constants

Critical Temp (T_c): 8a / 27Rb (Gas cannot be liquefied above this T).
Critical Pressure (P_c): a / 27b².
Critical Volume (V_c): 3b.

7. Liquid Properties

7.1 Surface Tension (γ)

Force acting per unit length perpendicular to the line drawn on the surface.

γ = Force / Length

Causes drops to be spherical (Minimizing surface area).

Excess Pressure:
Liquid Drop: P_ex = 2γ / r
Soap Bubble: P_ex = 4γ / r

7.2 Viscosity (η)

Internal resistance to flow.

F = η A (dv/dx)

Depends on temperature (Liquids: T↑ η↓; Gases: T↑ η↑).

7.3 Capillarity

Rise or fall of liquid in a narrow tube.

h = (2γ cosθ) / (r ρ g)

(θ = Contact angle)

Numericals & HOTS

Q1. Density Calculation

Calculate the density of CO₂ gas at 27°C and 2 atm pressure. (R = 0.0821 L atm K^-1 mol^-1)

Solution:
Formula: d = PM / RT
Molar Mass (M) of CO₂ = 44 g/mol
T = 27 + 273 = 300 K
P = 2 atm

d = (2 × 44) / (0.0821 × 300)
d = 88 / 24.63
d = 3.57 g/L

Q2. Bubble Expansion (Combined Law)

An air bubble of volume 1.0 mL rises from the bottom of a lake 40 m deep at a temperature of 12°C. To what volume does it grow when it reaches the surface, which is at 35°C? (1 atm = 10 m of water column).

Solution:
At Bottom (State 1):
Depth = 40m. Pressure P₁ = P_atm + 40m water.
Since 10m water = 1 atm, 40m = 4 atm.
P₁ = 1 atm (surface) + 4 atm = 5 atm.
V₁ = 1 mL, T₁ = 12+273 = 285 K.

At Surface (State 2):
P₂ = 1 atm.
T₂ = 35+273 = 308 K.

Formula: P₁V₁/T₁ = P₂V₂/T₂
(5 × 1) / 285 = (1 × V₂) / 308
V₂ = (5 × 308) / 285
V₂ = 5.4 mL

Q3. Diffusion Ratio

Find the ratio of the rate of diffusion of Hydrogen (H₂) to that of Oxygen (O₂).

Solution:
Molar Mass H₂ = 2 g/mol
Molar Mass O₂ = 32 g/mol

Formula: r_H2 / r_O2 = √(M_O2 / M_H2)
Ratio = √(32 / 2)
Ratio = √16 = 4
Answer: 4:1

Q4. RMS Speed (Unit Trap)

Calculate the root mean square speed (V_rms) of Oxygen molecules at 27°C. (R = 8.314 J K^-1 mol^-1)

Solution:
Trap: Molar mass must be in kg/mol for SI units.
M(O₂) = 32 g/mol = 0.032 kg/mol.
T = 300 K.

V_rms = √(3RT / M)
V_rms = √(3 × 8.314 × 300 / 0.032)
V_rms = √(7482.6 / 0.032)
V_rms = √(233831)
V_rms = 483.5 m/s

Q5. Kinetic Energy Concept

Calculate the ratio of average Kinetic Energy of 1 mole of H₂ and 1 mole of CH₄ at 27°C.

Solution:
Average K.E. per mole = (3/2) RT.
It depends ONLY on Temperature, not on Molar Mass or nature of gas.
Since T is same (300 K) for both:
Ratio = 1:1

Q6. Partial Pressure

A mixture contains 16g of O₂ and 20g of Ne. Total pressure is 10 atm. Calculate the partial pressure of O₂. (Molar mass Ne = 20)

Solution:
Moles O₂ = 16/32 = 0.5 mol.
Moles Ne = 20/20 = 1.0 mol.
Total Moles = 1.5 mol.

Mole Fraction of O₂ (X_O2) = 0.5 / 1.5 = 1/3.

P_O2 = X_O2 × P_total
P_O2 = (1/3) × 10
P_O2 = 3.33 atm

Q7. Van der Waals (HOTS)

Calculate the pressure exerted by 1 mole of methane (CH₄) in a 0.25 L container at 300K using Van der Waals equation. Given: a = 2.25 L² atm mol^-2, b = 0.04 L mol^-1.

Solution:
Formula: (P + an²/V²)(V - nb) = nRT

Step 1: RT = 0.0821 × 300 = 24.63
Step 2: (V - nb) = 0.25 - 0.04 = 0.21 L
Step 3: an²/V² = 2.25(1)² / (0.25)² = 2.25 / 0.0625 = 36 atm

(P + 36)(0.21) = 24.63
P + 36 = 24.63 / 0.21 = 117.28
P = 117.28 - 36
P = 81.28 atm

Q8. Nature of Gas (Z)

At 500 atm and 300 K, the density of a gas (M=28) is 0.7 g/mL. What is the value of Compressibility Factor (Z)?

Solution:
Ideal density d_ideal = PM / RT
d_ideal = (500 × 28) / (0.0821 × 300)
d_ideal = 14000 / 24.63 = 568.4 g/L = 0.568 g/mL.

Real density d_real = 0.7 g/mL.
Z = d_ideal / d_real (since V_real < V_ideal)
Wait, simpler way: Z = PV / nRT = P(M/d) / RT
Z = P M / (d RT)
Z = (500 × 28) / (700 g/L × 0.0821 × 300)
Z = 14000 / 17241
Z = 0.81 (Attractive forces dominate)

Q9. Soap Bubble Pressure

Calculate the excess pressure inside a soap bubble of radius 2 mm if surface tension of soap solution is 0.03 N/m.

Solution:
For soap bubble (2 surfaces), P_ex = 4γ / r.
γ = 0.03 N/m.
r = 2 mm = 2 × 10^-3 m.

P_ex = (4 × 0.03) / (2 × 10^-3)
P_ex = 0.12 / 0.002
P_ex = 60 Pa

Q10. Capillary Rise Ratio

Two capillaries of radii r₁ and r₂ are dipped in the same liquid. If r₁ = 2r₂, calculate the ratio of heights h₁ : h₂.

Solution:
Capillary rise formula: h = 2γcosθ / (rρg)
Thus, h ∝ 1/r (assuming same liquid).

h₁ / h₂ = r₂ / r₁
Given r₁ = 2r₂
h₁ / h₂ = r₂ / (2r₂) = 1/2
Ratio = 1:2

Important Formulae

1. Ideal Gas Laws

Ideal Gas Equation:

PV = nRT = (w/M)RT

Density Form:

d = PM / RT

Combined Gas Law:

P₁V₁ / T₁ = P₂V₂ / T₂

2. Mixtures & Diffusion

Dalton's Law of Partial Pressure:

P_total = P_A + P_B + ...
P_A = x_A × P_total

Graham's Law of Diffusion:

r ∝ 1 / √M or r ∝ 1 / √d

r₁/r₂ = √(M₂/M₁)

3. Molecular Speeds (KTG)

V_rms: √(3RT/M) = √(3PV/M)
V_avg: √(8RT/πM)
V_mp: √(2RT/M)

V_mp < V_avg < V_rms

4. Real Gases (Van der Waals)

(P + an²/V²) (V - nb) = nRT

Critical Temp (T_c)	Critical Pressure (P_c)	Critical Vol (V_c)
8a / 27Rb	a / 27b²	3b

Boyle Temp (T_B): a / Rb

5. Surface Tension & Viscosity

Capillary Rise:

h = (2γ cosθ) / (r ρ g)

Viscous Force:

F = η A (dv/dx)

20 Golden Facts (NEET)

1. Kinetic Energy: The average kinetic energy of gas molecules depends ONLY on absolute temperature. K.E. ∝ T. It is independent of the nature of the gas.
2. Unit of 'a': The Van der Waals constant 'a' represents attractive forces. Unit: atm L² mol^-2. Higher 'a' means easier liquefaction (e.g., NH₃ > N₂).
3. Unit of 'b': The constant 'b' represents excluded volume (size). Unit: L mol^-1. Value is 4 times the actual volume of molecules ($b = 4 \times V_m$).
4. Z at Low Pressure: For real gases at low pressure, attractive forces dominate. Z < 1. Equation becomes: Z = 1 - a/(VRT).
5. Z at High Pressure: Repulsive forces dominate. Z > 1. Equation becomes: Z = 1 + Pb/RT.
6. H₂ and He Exception: Hydrogen and Helium show Z > 1 at almost all conditions because their intermolecular attraction ('a') is negligible.
7. Dalton's Condition: Dalton's Law of Partial Pressure is applicable ONLY for non-reacting gases. (e.g., N₂ + O₂ is valid, but NH₃ + HCl is invalid as they form solid NH₄Cl).
8. Wet vs Dry Gas: P_{dry gas} = P_total - Aqueous Tension (Vapor pressure of water). Always subtract aq. tension in numericals.
9. Boiling Point vs Pressure: Boiling occurs when Vapor Pressure = External Pressure. On mountains (low P), B.P. decreases (cooking takes longer). In pressure cookers (high P), B.P. increases.
10. Mean Free Path (λ): Distance traveled between two collisions. $\lambda \propto T/P$. Decreases with high pressure.
11. Surface Tension vs Temp: Surface tension decreases as temperature increases (Hot soup tastes better because it spreads easily on the tongue).
12. Viscosity vs Temp: Viscosity of liquids decreases with Temp, but viscosity of GASES increases with Temp (due to increased collision randomness).
13. Absolute Zero: At -273.15°C (0 K), the volume of an ideal gas theoretically becomes zero, and molecular motion ceases.
14. Diffusion Rate: Lighter gases diffuse faster. Rate of diffusion of H₂ is 4 times that of O₂ (since Mass is 1/16th).
15. Falling Drops: Raindrops are spherical due to Surface Tension (minimizing surface area/energy).
16. Compressibility Factor Z: At the Critical Point, Z = 3/8 (approx 0.375) for a Van der Waals gas.
17. Laminar Flow: In streamline flow of liquids, the velocity is maximum at the center of the tube and minimum (zero) at the walls.
18. Units of R: In energy calculations (K.E., work), use R = 8.314 J. In Equation of State (PV=nRT), use R = 0.0821 L atm.
19. Avogadro's Hypothesis: The density of a gas is directly proportional to its molar mass at constant T and P ($d \propto M$).
20. Capillary Action: If cohesive forces (liquid-liquid) > adhesive forces (liquid-solid), the liquid will fall (convex meniscus, e.g., Mercury). If adhesive > cohesive, it rises (concave meniscus, e.g., Water).

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