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APEX CLASS ESTD 2018

IITJEE CHEMISTRY: States of Matter, Compressibility (Z) & Van der Waals Gases

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Ashish Kumar Bahekar

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APEX CLASS • IITJEE Physical Chemistry Lab

States of Matter, Compressibility (Z) & Van der Waals Gases

Master Ideal Gas Graphs, Graham’s Law, Z vs P Deviation Curves, and Critical Constants (Tc, Pc)!

🔬 PV = nRT. Select an Ideal Gas Concept:
P V → T₁ T₂ T₃ Isotherms: T₃ > T₂ > T₁ V T → P₃ P₂ P₁ Isobars: P₃ > P₂ > P₁ Ideal Gas Laws Boyle’s (T Const): P₁V₁ = P₂V₂ Charles’s (P Const): V₁ / T₁ = V₂ / T₂ Gay-Lussac (V Const): P₁ / T₁ = P₂ / T₂ 1. Graham’s Law of Effusion Pinhole Start Lighter Gas (He) Heavier (O₂) r₁ / r₂ = √(M₂ / M₁) = √(d₂ / d₁) Rate ∝ 1 / √(Molar Mass) 2. Dalton’s Law of Partial Pressures For non-reacting gases at constant V and T: Ptotal = P₁ + P₂ + P₃ … • Partial Pressure Formula: Pi = Xi × Ptotal (Xi = Mole Fraction = ni / ntotal) • Trap: Pdry gas = Ptotal – Aqueous Tension! The Kinetic Gas Equation & Translational Energy PV = ⅓ m N c² • m = Mass of one molecule • N = Total number of molecules • c = Root Mean Square (RMS) Speed • Total Translational Kinetic Energy: Ek = 3/2 n R T = 3/2 P V Energy depends ONLY on absolute Temp (T)! ⚡ Average KE per molecule = 3/2 kB T (where kB = R / NA is the Boltzmann constant). Density & Molar Mass of Ideal Gases (d = PM / RT) Derived from PV = nRT ⇒ PV = (w/M) RT ⇒ P = (w/V) (RT/M) d = P M / R T At Constant T and P: Density (d) ∝ Molar Mass (M) Vapor Density (V.D.): Molar Mass = 2 × V.D. 🎯 Payloader Balloon Problem: Payload = (Mass of displaced air) – (Mass of balloon + Mass of gas inside)!
Root Mean Square (c)
√(3RT / M)
Highest Speed Value (1.224)
Average Speed
√(8RT / πM)
Middle Speed Value (1.128)
Most Probable Speed
√(2RT / M)
Lowest Speed Value (1.000)
Avogadro’s Law
V ∝ n (Const T, P)
Equal volumes contain equal molecules
🛑 Z = PV / nRT. Select a Real Gas Concept:
Z Pressure (P) → Z=1 Ideal Gas H₂, He Repulsion dominates N₂ CO₂, CH₄ Attraction dominates Compressibility Factor Z Z = (P V) / (n R T) = Vreal / Videal • If Z = 1: Gas behaves Ideally. • If Z < 1 (Negative Deviation): Easily liquefiable. Attractive forces win. • If Z > 1 (Positive Deviation): Repulsive forces win. [ P + (a n² / V²) ] × ( V – n b ) = n R T Constant ‘a’ (Pressure Correction) Attraction • Magnitude of Intermolecular Forces. • Higher ‘a’ = Easier to Liquefy! • Units: atm L² mol⁻² Constant ‘b’ (Volume Correction) Excluded Vol • Excluded Volume / Co-volume. • b = 4 × NA × (4/3 π r³) • Units: L mol⁻¹ 🏆 Gas Liquefaction Order: NH₃ > CO₂ > O₂ > N₂ > H₂ > He (Directly proportional to ‘a’)! Mathematical Variations of Z for 1 Mole of Gas 1. At Very Low Pressure (Ideal Behavior): Both corrections ‘a’ and ‘b’ are negligible. PV = RT ⇒ Z = 1 2. At Moderate Pressure (Attraction Dominates): Volume is large enough to ignore ‘b’, but ‘a’ matters. (P + a/V²)V = RT ⇒ PV + a/V = RT ⇒ Z = 1 – a/(RTV) (Negative Deviation, Z < 1) 3. At High Pressure (Size Dominates): Pressure is too high, ‘a/V²’ becomes negligible. P(V – b) = RT ⇒ PV – Pb = RT Z = 1 + Pb/(RT) (Positive Deviation, Z > 1) ⚡ Note: For H₂ and He, intermolecular attraction ‘a’ is incredibly small, so they ALWAYS follow Z = 1 + Pb/RT (Z > 1) at all pressures! P V → Liquid + Vapor T > Tc T = Tc Pc Vc Critical Constants Formulas 1. Critical Temp (Tc) = 8a / 27Rb Gas cannot be liquefied above Tc no matter the P! 2. Critical Pressure (Pc) = a / 27b² 3. Critical Volume (Vc) = 3b Universal Zc = PcVc / RTc = 3/8 = 0.375
Boyle Temperature (Tb)
Tb = a / (Rb)
Gas behaves ideally over large P range
Inversion Temperature (Ti)
Ti = 2a / (Rb) = 2 Tb
Joule-Thomson effect flips sign here
Unit of constant ‘a’
atm L² mol⁻²
Measures attractive magnitude
Unit of constant ‘b’
L mol⁻¹
Effective volume of molecules
🔥 High-Yield JEE Real Gas Trick:
The relationship between the Critical Temperature and Boyle Temperature is exactly Tc = (8/27) Tb. Furthermore, because H₂ and He have extraordinarily low ‘a’ values, their Boyle and Inversion temperatures are extremely low—meaning they actually heat up instead of cooling down upon sudden expansion at room temperature (Joule-Thomson heating)!
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