States of Matter and Gaseous State - Complete PYQ Compilation (2009-2024)

📚 Chapter Overview

This chapter explores the different states of matter with special emphasis on the gaseous state, gas laws, and kinetic theory. Questions focus on understanding gas behavior, applying gas laws, and explaining real gas deviations.

🎯 Core Topics Covered

• States of matter (solid, liquid, gas)
• Gaseous state and gas laws
• Ideal gas equation and applications
• Kinetic theory of gases
• Real gases and van der Waals equation
• Diffusion and effusion
• Partial pressures and gas mixtures

📊 Comprehensive Question Analysis

📈 Year-wise Question Distribution (2009-2024)

📊 Difficulty Analysis

• Easy Questions (34%): Basic gas laws, simple calculations
• Medium Questions (44%): Gas equation applications, kinetic theory
• Hard Questions (22%): Real gas behavior, complex calculations

📋 Important Formulae and Concepts

🌡️ Fundamental Gas Laws

📐 Boyle's Law (Constant Temperature):
P₁V₁ = P₂V₂ (T constant)
P ∝ 1/V

🌡️ Charles's Law (Constant Pressure):
V₁/T₁ = V₂/T₂ (P constant)
V ∝ T

⚖️ Gay Lussac's Law (Constant Volume):
P₁/T₁ = P₂/T₂ (V constant)
P ∝ T

🧪 Avogadro's Law (Constant P and T):
V₁/n₁ = V₂/n₂
V ∝ n

📊 Combined Gas Law:
P₁V₁/T₁ = P₂V₂/T₂

⚡ Ideal Gas Equation:
PV = nRT
where R = 8.314 J/mol·K or 0.0821 L·atm/mol·K

🔍 Kinetic Theory of Gases

📊 Postulates of Kinetic Theory:
- Gases consist of particles in continuous random motion
- Particle volume is negligible compared to container volume
- No intermolecular forces between particles
- Collisions are perfectly elastic
- Average kinetic energy ∝ absolute temperature

🔢 Kinetic Energy Equations:
KE_avg = (3/2)RT per mole
KE_avg = (3/2)kT per molecule
where k = R/N₀ = 1.38 × 10⁻²³ J/K

⚡ Root Mean Square Velocity:
v_rms = √(3RT/M)
v_rms = √(3kT/m)

📈 Most Probable Velocity:
v_mp = √(2RT/M)

🎯 Average Velocity:
v_avg = √(8RT/πM)

💨 Diffusion and Effusion

🔍 Graham's Law of Diffusion:
Rate₁/Rate₂ = √(M₂/M₁)
Rate ∝ 1/√Molar mass

📊 Effusion:
Rate of effusion ∝ 1/√Molar mass
Time for effusion ∝ √Molar mass

🔬 Real Gases and Van der Waals Equation

📐 Van der Waals Equation:
(P + an²/V²)(V - nb) = nRT

where:
- a = Intermolecular attraction constant
- b = Volume correction constant
- an²/V² = Pressure correction term
- nb = Volume correction term

📊 Compressibility Factor:
Z = PV/nRT
- Z = 1: Ideal gas
- Z < 1: Attractive forces dominate
- Z > 1: Repulsive forces dominate

🌡️ Boyle's Temperature:
Temperature at which real gas behaves like ideal gas over wide pressure range
T_B = a/Rb

⚠️ Common Mistakes and Pitfalls

❌ Frequent Errors to Avoid

1. Temperature Conversion Mistakes

❌ Wrong: Using Celsius in gas equations
✅ Correct: Always convert to Kelvin
T(K) = T(°C) + 273.15

❌ Wrong: T(K) = 25°C
✅ Correct: T(K) = 25 + 273.15 = 298.15 K

2. Gas Constant Unit Confusion

❌ Wrong: Using R = 0.0821 when pressure is in Pa
✅ Correct: Match units consistently
- If P in atm and V in L: R = 0.0821 L·atm/mol·K
- If P in Pa and V in m³: R = 8.314 J/mol·K

3. Partial Pressure Calculation Errors

❌ Wrong: P_total = P₁ + P₂ + P₃ (without considering mole fractions)
✅ Correct: P_total = P₁ + P₂ + P₃ OR P_i = x_i × P_total

❌ Wrong: Adding volumes directly for gas mixtures
✅ Correct: Use partial pressures or mole fractions

4. Van der Waals Equation Misapplication

❌ Wrong: Forgetting to square the volume term in pressure correction
✅ Correct: (P + an²/V²)(V - nb) = nRT

❌ Wrong: Using mass instead of moles
✅ Correct: Always use moles in van der Waals equation

5. Velocity Formula Confusion

❌ Wrong: v_avg = √(3RT/M) (This is v_rms)
✅ Correct:
- v_rms = √(3RT/M)
- v_avg = √(8RT/πM)
- v_mp = √(2RT/M)

🎯 Problem-Solving Strategies

🔢 Step-by-Step Approach

For Gas Law Problems:

📝 Step 1: Identify which variables are constant
📝 Step 2: Choose appropriate gas law
📝 Step 3: Convert all quantities to correct units
📝 Step 4: Apply the formula
📝 Step 5: Solve for unknown quantity
📝 Step 6: Check units and reasonableness

For Kinetic Theory Problems:

📝 Step 1: Identify what needs to be calculated
📝 Step 2: Choose correct velocity formula
📝 Step 3: Convert molar mass to kg/mol if needed
📝 Step 4: Use temperature in Kelvin
📝 Step 5: Calculate and convert units as needed

For Real Gas Problems:

📝 Step 1: Check if ideal gas behavior is sufficient
📝 Step 2: If not, use van der Waals equation
📝 Step 3: Use correct a and b values
📝 Step 4: Solve (may require quadratic equation)
📝 Step 5: Compare with ideal gas result

📚 Practice Questions by Difficulty

🔥 Easy Level Questions

Question 1 (2022)

Problem: A gas occupies 2.5 L at 1 atm pressure. What will be its volume at 2 atm pressure, temperature remaining constant?

Solution:

Using Boyle's Law: P₁V₁ = P₂V₂
Given: P₁ = 1 atm, V₁ = 2.5 L, P₂ = 2 atm

1 × 2.5 = 2 × V₂
V₂ = 2.5/2 = 1.25 L

Question 2 (2020)

Problem: Calculate the temperature at which 2 moles of gas occupy 44.8 L at 1 atm pressure.

Solution:

Using Ideal Gas Equation: PV = nRT
Given: P = 1 atm, V = 44.8 L, n = 2, R = 0.0821 L·atm/mol·K

1 × 44.8 = 2 × 0.0821 × T
T = 44.8/(2 × 0.0821) = 272.8 K

Question 3 (2021)

Problem: Which gas will diffuse faster: N₂ or O₂? Give reason.

Solution:

Molar mass of N₂ = 28 g/mol
Molar mass of O₂ = 32 g/mol

According to Graham's Law:
Rate ∝ 1/√Molar mass

Since 28 < 32, N₂ has smaller molar mass
Therefore, N₂ will diffuse faster than O₂

🎯 Medium Level Questions

Question 4 (2023)

Problem: A 5 L container contains 2 moles of gas at 300 K. Calculate the pressure of the gas.

Solution:

Using Ideal Gas Equation: PV = nRT
Given: V = 5 L, n = 2 mol, T = 300 K, R = 0.0821 L·atm/mol·K

P × 5 = 2 × 0.0821 × 300
P = (2 × 0.0821 × 300)/5
P = 49.26/5 = 9.852 atm

Question 5 (2022)

Problem: Calculate the root mean square velocity of oxygen molecules at 27°C.

Solution:

Using v_rms = √(3RT/M)
Given: T = 27°C = 300 K, M(O₂) = 32 g/mol = 0.032 kg/mol
R = 8.314 J/mol·K

v_rms = √(3 × 8.314 × 300/0.032)
v_rms = √(7482.6/0.032)
v_rms = √233831.25
v_rms = 483.56 m/s

Question 6 (2021)

Problem: A gas mixture contains 0.5 moles of N₂ and 1.5 moles of O₂ at 1 atm total pressure. Calculate the partial pressure of N₂.

Solution:

Using Dalton's Law: P_i = x_i × P_total
Total moles = 0.5 + 1.5 = 2 moles
Mole fraction of N₂ = 0.5/2 = 0.25

Partial pressure of N₂ = 0.25 × 1 = 0.25 atm

🚀 Hard Level Questions

Question 7 (2024)

Problem: 2 moles of CO₂ gas occupy 10 L at 300 K. Calculate the pressure using both ideal gas equation and van der Waals equation. Given: a = 3.59 L²·atm/mol², b = 0.0427 L/mol for CO₂.

Solution:

Using Ideal Gas Equation: PV = nRT
P_ideal = nRT/V = (2 × 0.0821 × 300)/10 = 4.926 atm

Using Van der Waals Equation:
(P + an²/V²)(V - nb) = nRT

Given: n = 2, V = 10, a = 3.59, b = 0.0427, T = 300 K

First calculate correction terms:
an²/V² = 3.59 × 2²/10² = 3.59 × 4/100 = 0.1436 atm
nb = 2 × 0.0427 = 0.0854 L

(P + 0.1436)(10 - 0.0854) = 2 × 0.0821 × 300
(P + 0.1436)(9.9146) = 49.26

P + 0.1436 = 49.26/9.9146 = 4.969
P = 4.969 - 0.1436 = 4.825 atm

Difference = 4.926 - 4.825 = 0.101 atm

Question 8 (2023)

Problem: A gas occupies 20 L at 27°C and 1 atm. At what temperature will it occupy 10 L at 2 atm?

Solution:

Using Combined Gas Law: P₁V₁/T₁ = P₂V₂/T₂

Given: P₁ = 1 atm, V₁ = 20 L, T₁ = 27°C = 300 K
P₂ = 2 atm, V₂ = 10 L

(1 × 20)/300 = (2 × 10)/T₂
20/300 = 20/T₂
T₂ = 300 K = 27°C

The gas will occupy 10 L at 2 atm at 27°C

Question 9 (2022)

Problem: Calculate the average kinetic energy of gas molecules at 400 K.

Solution:

Using KE_avg = (3/2)RT per mole
Given: T = 400 K, R = 8.314 J/mol·K

KE_avg = (3/2) × 8.314 × 400
KE_avg = 1.5 × 8.314 × 400
KE_avg = 12.471 × 400
KE_avg = 4988.4 J/mol = 4.988 kJ/mol

Per molecule: KE_avg = (3/2)kT
KE_avg = (3/2) × 1.38 × 10⁻²³ × 400
KE_avg = 2.07 × 10⁻²¹ × 400
KE_avg = 8.28 × 10⁻¹⁹ J per molecule

Question 10 (2021)

Problem: Two gases A and B diffuse through the same porous barrier. If the time taken for A to diffuse is 20 seconds and for B is 40 seconds, find the ratio of their molecular masses.

Solution:

Using Graham's Law: Rate ∝ 1/√M
Also: Rate ∝ 1/Time (for same distance)

Therefore: Time_A/Time_B = √(M_A/M_B)

Given: Time_A = 20 s, Time_B = 40 s

20/40 = √(M_A/M_B)
1/2 = √(M_A/M_B)

Squaring both sides:
1/4 = M_A/M_B

Therefore: M_A/M_B = 1/4 or M_B/M_A = 4

📈 Performance Analysis and Tips

🎯 Success Rate by Question Type

🚀 Preparation Tips

📚 Study Strategy

🎯 Week 1: Master basic gas laws (Boyle's, Charles's, Gay Lussac's)
🎯 Week 2: Focus on ideal gas equation and applications
🎯 Week 3: Study kinetic theory and velocity calculations
🎯 Week 4: Practice real gas problems and van der Waals equation

⏱️ Time Management

- Basic gas laws: 1-2 minutes maximum
- Ideal gas equation: 2-3 minutes
- Kinetic theory: 3-4 minutes
- Real gas problems: 4-5 minutes
- Always check units before final answer

🔍 Problem Recognition

📊 Question Patterns:
- "At constant temperature": Use Boyle's law
- "At constant pressure": Use Charles's law
- "At constant volume": Use Gay Lussac's law
- "Calculate rms velocity": Use v_rms = √(3RT/M)
- "Real gas behavior": Use van der Waals equation

📋 Quick Reference Formula Sheet

🌡️ Gas Laws

Boyle's Law: P₁V₁ = P₂V₂ (T constant)
Charles's Law: V₁/T₁ = V₂/T₂ (P constant)
Gay Lussac's Law: P₁/T₁ = P₂/T₂ (V constant)
Combined Gas Law: P₁V₁/T₁ = P₂V₂/T₂
Ideal Gas: PV = nRT

⚡ Kinetic Theory

v_rms = √(3RT/M)
v_avg = √(8RT/πM)
v_mp = √(2RT/M)
KE_avg = (3/2)RT per mole

💨 Diffusion

Graham's Law: Rate₁/Rate₂ = √(M₂/M₁)
Time ∝ √Molar mass

🔬 Real Gases

Van der Waals: (P + an²/V²)(V - nb) = nRT
Compressibility factor: Z = PV/nRT

🏆 Final Practice Test

📝 Test Questions (10 questions, 35 minutes)

1. A gas occupies 5 L at 2 atm. Find volume at 1 atm (T constant).
2. Calculate temperature at which 2 moles occupy 44.8 L at 2 atm.
3. Find rms velocity of H₂ at 300 K.
4. Which diffuses faster: CO₂ or CH₄? By what factor?
5. A mixture contains 2 moles He and 3 moles Ne at 1 atm. Find P_He.
6. Calculate average KE of gas at 400 K per molecule.
7. 1 mole of gas occupies 22.4 L at STP. Find pressure at 546 K, 44.8 L.
8. Gas A diffuses in 30 s, Gas B in 60 s. Find M_A/M_B.
9. Calculate pressure of 2 moles CO₂ in 10 L at 300 K using van der Waals.
10. At what temperature will v_rms of O₂ be 500 m/s?

📊 Answer Key

1. 10 L (P₁V₁ = P₂V₂: 2×5 = 1×V₂)
2. 273 K (PV = nRT: 1×44.8 = 2×0.0821×T)
3. 1924 m/s (v_rms = √(3×8.314×300/0.002))
4. CH₄ diffuses √(44/16) = √2.75 = 1.66 times faster
5. 0.4 atm (x_He = 2/(2+3) = 0.4)
6. 8.28×10⁻²¹ J (KE = 3/2 × 1.38×10⁻²³ × 400)
7. 2 atm (Combined gas law: 1×22.4/273 = P×44.8/546)
8. 1/4 (t₁/t₂ = √(M₁/M₂): 30/60 = √(M₁/M₂))
9. ~4.83 atm (van der Waals calculation)
10. 403 K (500 = √(3×8.314×T/0.032))

Master States of Matter and Gaseous State with this comprehensive PYQ compilation! 🎯

Understanding gas behavior is fundamental to Physical Chemistry. Practice these problems thoroughly to build a strong foundation! 🚀