Chemistry Physical How to Solve: Gaseous State (Ideal Gas Law, Dalton’s Law, Kinetic Theory, Real Gas – Van der Waals)

How to Solve: Gaseous State (Ideal Gas Law, Dalton’s Law, Kinetic Theory, Real Gas – Van der Waals)

For IIT JEE (Main + Advanced)

Introduction

"Mastering the Gaseous State unlocks 8–10 marks in JEE Main and 12–15 marks in JEE Advanced—enough to push you from a 90th to a 99th percentile rank. Whether it’s calculating the pressure of a gas mixture in a rocket engine or predicting real gas behavior in high-pressure pipelines, these concepts are everywhere. Let’s break them down so you never lose a mark."

WHAT YOU NEED TO KNOW FIRST

1. Basic Gas Properties – Pressure (P), Volume (V), Temperature (T), and moles (n).
2. Boyle’s, Charles’s, and Avogadro’s Laws – The foundation of the Ideal Gas Law.
3. Mole Concept & Stoichiometry – Converting mass to moles and vice versa.

(If you’re shaky on these, pause and review them first.)

KEY TERMS & FORMULAS

1. Ideal Gas Law

Formula: PV = nRT - P = Pressure (atm, Pa, bar) → MEMORISE: 1 atm = 1.013 × 10⁵ Pa - V = Volume (L, m³) → MEMORISE: 1 m³ = 1000 L - n = Moles of gas (mol) - R = Universal gas constant → MEMORISE: R = 0.0821 L·atm·mol⁻¹·K⁻¹ (for atm) OR 8.314 J·mol⁻¹·K⁻¹ (for SI units) - T = Temperature (K) → MEMORISE: K = °C + 273

When to use: When any three of P, V, n, T are given, and you need the fourth.

2. Dalton’s Law of Partial Pressures

Formula: P_total = P₁ + P₂ + P₃ + … - P_total = Total pressure of gas mixture - P₁, P₂, P₃ = Partial pressures of individual gases

Partial Pressure Formula: P_i = χ_i × P_total - χ_i = Mole fraction of gas i = (moles of gas i) / (total moles)

When to use: When gases are mixed and you need total or partial pressure.

3. Kinetic Theory of Gases

Key Equations:
1. Root Mean Square Speed (u_rms): u_rms = √(3RT/M) - M = Molar mass (kg/mol) → Convert g/mol to kg/mol (divide by 1000!) - R = 8.314 J·mol⁻¹·K⁻¹ (SI units)

1. Average Kinetic Energy (KE_avg): KE_avg = (3/2)k_B T
2. k_B = Boltzmann constant = 1.38 × 10⁻²³ J/K (given on exam sheet)

When to use: - u_rms → When asked for speed of gas molecules. - KE_avg → When asked for energy per molecule.

4. Real Gases – Van der Waals Equation

Formula: [P + a(n/V)²] [V – nb] = nRT - a = Attraction correction (given in question) - b = Volume correction (given in question) - n/V = Molar concentration (mol/L)

When to use: When gas is not ideal (high pressure, low temperature).

STEP-BY-STEP METHOD

Step 1: Identify the Given & Required

• Write down all given values (P, V, n, T, etc.).
• Circle what you need to find.

Step 2: Choose the Right Formula

• Ideal Gas? → PV = nRT
• Gas Mixture? → Dalton’s Law
• Molecular Speed? → u_rms = √(3RT/M)
• Real Gas? → Van der Waals

Step 3: Convert Units (CRUCIAL!)

• Pressure: atm → Pa (× 1.013 × 10⁵) or bar → Pa (× 10⁵)
• Volume: L → m³ (÷ 1000) or cm³ → m³ (÷ 10⁶)
• Temperature: °C → K (+ 273)
• Mass: g → kg (÷ 1000) for u_rms

Step 4: Plug & Solve

• Substitute values into the formula.
• Check units match (e.g., R = 0.0821 L·atm·mol⁻¹·K⁻¹ if P is in atm).

Step 5: Verify Answer

• Does the answer make sense? (e.g., pressure can’t be negative)
• Cross-check with another formula if possible.

WORKED EXAMPLES

Example 1 – Basic (Ideal Gas Law)

Question: A 2 L container holds 0.5 mol of O₂ at 27°C. What is the pressure inside?

Step 1: Given & Required - V = 2 L - n = 0.5 mol - T = 27°C = 300 K - Find P

Step 2: Formula PV = nRT

Step 3: Unit Check - V in L, n in mol, T in K → Use R = 0.0821 L·atm·mol⁻¹·K⁻¹

Step 4: Plug & Solve P × 2 = 0.5 × 0.0821 × 300 P = (0.5 × 0.0821 × 300) / 2 P = 6.1575 atm

Step 5: Verify - Pressure is positive → makes sense.

What we did and why: - Used PV = nRT because it’s a single ideal gas. - Converted °C to K because gas laws only work in Kelvin. - Chose R = 0.0821 because pressure was needed in atm.

Example 2 – Medium (Dalton’s Law + Mole Fraction)

Question: A mixture contains 2 g H₂ and 8 g O₂ in a 10 L container at 300 K. Find the partial pressure of H₂.

Step 1: Given & Required - Mass H₂ = 2 g, Mass O₂ = 8 g - V = 10 L, T = 300 K - Find P_H₂

Step 2: Find Moles - n_H₂ = 2 / 2 = 1 mol - n_O₂ = 8 / 32 = 0.25 mol - Total moles = 1 + 0.25 = 1.25 mol

Step 3: Find Mole Fraction (χ_H₂) χ_H₂ = n_H₂ / n_total = 1 / 1.25 = 0.8

Step 4: Find Total Pressure (PV = nRT) P_total × 10 = 1.25 × 0.0821 × 300 P_total = (1.25 × 0.0821 × 300) / 10 = 3.07875 atm

Step 5: Find Partial Pressure (P_H₂ = χ_H₂ × P_total) P_H₂ = 0.8 × 3.07875 = 2.463 atm

What we did and why: - First found moles because Dalton’s Law needs mole fractions. - Calculated total pressure using Ideal Gas Law. - Multiplied mole fraction by total pressure to get partial pressure.

Example 3 – Exam-Style (Real Gas – Van der Waals)

Question: 1 mol of CO₂ is in a 0.5 L container at 300 K. Given a = 3.59 L²·atm·mol⁻², b = 0.0427 L·mol⁻¹, find the pressure. Compare with ideal gas pressure.

Step 1: Given & Required - n = 1 mol, V = 0.5 L, T = 300 K - a = 3.59, b = 0.0427 - Find P (real gas) and P (ideal gas)

Step 2: Van der Waals Equation [P + a(n/V)²] [V – nb] = nRT

Step 3: Plug Values [P + 3.59(1/0.5)²] [0.5 – 1×0.0427] = 1 × 0.0821 × 300 [P + 3.59 × 4] [0.5 – 0.0427] = 24.63 [P + 14.36] [0.4573] = 24.63

Step 4: Solve for P P + 14.36 = 24.63 / 0.4573 P + 14.36 = 53.87 P = 53.87 – 14.36 = 39.51 atm

Step 5: Compare with Ideal Gas PV = nRT → P = nRT / V = (1 × 0.0821 × 300) / 0.5 = 49.26 atm

What we did and why: - Used Van der Waals because CO₂ is a real gas (high pressure). - Compared with Ideal Gas Law to show deviation. - Real gas pressure is lower due to intermolecular attractions (a term).

COMMON MISTAKES

EXAM TRAPS

1-MINUTE RECAP (Night Before Exam)

"Listen up—this is all you need to remember for Gaseous State in JEE:

1. Ideal Gas Law (PV = nRT) – Use when gas is ideal (low P, high T). Convert °C to K!
2. Dalton’s Law (P_total = P₁ + P₂) – For gas mixtures. Partial pressure = mole fraction × total pressure.
3. Kinetic Theory (u_rms = √(3RT/M)) – Molar mass must be in kg/mol!
4. Van der Waals ([P + a(n/V)²][V – nb] = nRT) – For real gases (high P, low T). a = attraction, b = volume correction.
5. Unit conversions are everything – atm → Pa, L → m³, g → kg. Double-check before solving!

If you see a gas mixture, think Dalton. If pressure is high, think Van der Waals. If speed is asked, think u_rms. Now go crush that exam!