Real Gases and van der Waals Equation

• Identify why real gases deviate from ideal behaviour at high pressure and low temperature
• Apply the van der Waals equation with constants a and b
• Interpret the compressibility factor Z and its deviation from 1
• Understand Boyle temperature and its physical significance
• Derive and use critical constants T_c, P_c, and V_c in terms of a and b
• Compare ideal vs real pressure in numerical problems

Level 1 — Foundations

Why ideal gas fails: The ideal gas model assumes:

1. Zero molecular volume (negligible size)
2. No intermolecular forces

Both assumptions fail at:

• High pressure: molecules are forced close together → their finite volume matters; repulsions are strong
• Low temperature: molecules move slowly → intermolecular attractions become significant relative to KE

Real behaviour: at high P, gases occupy less volume than ideal (attractions) or more (repulsions dominate at very high P).

Level 2 — JEE depth

van der Waals Equation (for n moles):

(P + an²/V²)(V − nb) = nRT

• a (L² atm mol⁻²): corrects for intermolecular attractions; higher a = stronger attractions (e.g., CO₂ > N₂ > He)
• b (L mol⁻¹): corrects for finite molecular volume (excluded volume); b ≈ 4 × actual volume per mole of molecules

For 1 mole: (P + a/V²)(V − b) = RT

The term an²/V² is the internal pressure — molecules near the wall are pulled back by interior molecules, reducing the effective pressure.

Compressibility Factor Z: Z = PV/nRT (= PV_m/RT for 1 mole)

At low P: Z → 1 for all gases (both corrections negligible) At intermediate P: Z dips below 1 (attractions win) At very high P: Z > 1 (volume correction dominates)

Boyle Temperature (T_B): The temperature at which a real gas behaves most like an ideal gas over a wide range of pressures (Z ≈ 1 for large P range):

T_B = a/(bR)

For N₂: a = 1.39 L² atm/mol², b = 0.0391 L/mol → T_B = 1.39/(0.0391 × 0.0821) ≈ 433 K

Critical Constants: At the critical point, the gas-liquid distinction disappears (single phase).

T_c = 8a / (27Rb) P_c = a / (27b²) V_c = 3nb (for n moles; V_c = 3b for 1 mole)

At critical point: Z_c = P_cV_c/(RT_c) = 3/8 = 0.375 (universal for van der Waals gas)

From a and b you can calculate all three critical constants, and vice versa: a = 3P_cV_c² (with V_c per mole) = 27R²T_c²/(64P_c) b = V_c/3 = RT_c/(8P_c)

JEE trap: The van der Waals equation is cubic in V — for a given P and T it may have up to three real roots. Below T_c the three roots correspond to vapour volume, unstable region, and liquid volume.

JEE trap: When comparing P_ideal and P_real: P_real = RT/(V−b) − a/V² and P_ideal = RT/V. Since the a/V² term decreases P, and the (V−b) term increases P, which effect wins depends on conditions.

CO₂ (a = 3.59 L² atm mol⁻², b = 0.0427 L/mol): 1 mol, V = 2 L, T = 300 K. Compare P from ideal vs vdW.

Ideal gas:
P_ideal = nRT/V = (1 × 0.0821 × 300) / 2 = 24.63 / 2 = 12.315 atm

van der Waals:
P_vdW = [RT/(V − b)] − a/V²
      = [0.0821 × 300 / (2 − 0.0427)] − 3.59/2²
      = [24.63 / 1.9573] − 3.59/4
      = 12.583 − 0.8975
      = 11.686 atm

P_vdW < P_ideal because at these conditions (moderate V),
intermolecular attractions reduce pressure compared to ideal.

Difference: 12.315 − 11.686 = 0.629 atm (about 5% error from ideal)

Find Boyle temperature for N₂ (a = 1.39 L² atm mol⁻², b = 0.0391 L/mol)

T_B = a / (bR)
    = 1.39 / (0.0391 × 0.0821)
    = 1.39 / 0.003210
    = 433 K ≈ 160°C

At 433 K, N₂ behaves most like an ideal gas over a wide pressure range.
Note: This is well above room temperature, which is why N₂ is
approximately ideal under ordinary lab conditions.

• Q1: For a gas with a = 4.0 and b = 0.05, calculate the Boyle temperature.
• Q2: At very high pressure, is Z for a real gas greater than, less than, or equal to 1? Explain.
• Q3: Which gas has larger 'a': N₂ or NH₃? Why?
• Q4: Calculate T_c for CO₂ using a = 3.59 L² atm/mol², b = 0.0427 L/mol, R = 0.0821 L atm/mol·K.
• Stretch: Q5: Show that for a van der Waals gas, Z_c = P_cV_c/(RT_c) = 3/8, using the critical constant expressions.

NCERT Chapter 5 link: Chapter 5 (Class 11) — Section 5.8 covers deviation from ideal behaviour, compressibility factor plots (Z vs P for different gases), van der Waals equation, and critical constants. The Z vs P plots are directly tested in JEE as interpretation questions.

Exam connections: JEE Mains frequently tests: (a) whether Z > 1 or < 1 at given conditions, (b) comparing P_ideal and P_vdW, (c) which van der Waals constant is larger for a given pair of gases, (d) calculating one critical constant given a and b. JEE Advanced may require deriving critical constants from the vdW equation.

Study strategy: Sketch the Z vs P graph for N₂, CO₂, and H₂ at room temperature — this visual locks in which effect (attraction vs repulsion) dominates at each pressure range. Then do 5 numerical problems each for P_vdW vs P_ideal comparisons and critical constant calculations.

• Use the platform-native live simulation or PhET-style tool for this topic.
• Mirror / body / home activity: observe that a sealed syringe is harder to compress as you push it — this is the excluded-volume effect. Describe what you feel and relate it to the b constant.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.