States of Matter and Gaseous State — JEE Main & Advanced Notes

Use ideal gas law, partial pressures, kinetic theory and real-gas corrections to solve fast numerical problems.

Priority: Scoring. Unit: Physical Chemistry. Level: Foundation.

How the uploaded material was used: Mapped from gas laws, kinetic theory and real-gas numerical worksheets. The final student-facing notes and questions are original, rewritten and copyright-safe.

These are the ideas that decide most correct answers in States of Matter and Gaseous State.

• The ideal gas model and its assumptions: An ideal gas is one where (1) molecules have negligible volume compared to the container, and (2) there are no intermolecular attractive or repulsive forces — only perfectly elastic collisions. Under these assumptions, PV = nRT holds exactly. Real gases approach ideal behaviour at low pressure (molecules far apart, attractions negligible) and high temperature (high kinetic energy overcomes attractions). The ideal gas equation can also be written as PV/nT = R = constant.
• Dalton's law and partial pressures: In a mixture of non-reacting gases, each component exerts its own partial pressure as if it were alone in the container. The partial pressure Pi = xi × P_total, where xi = ni/n_total is the mole fraction. Total pressure = sum of all partial pressures. Applications: collecting gases over water (subtract vapour pressure of water), problems involving gas composition, respiratory physiology. The mole fraction xi is the key conversion between partial pressure and total pressure.
• Kinetic molecular theory — microscopic explanation of gas laws: KMT postulates: (1) gas molecules are point masses in continuous random motion; (2) collisions are perfectly elastic (no energy loss); (3) there are no intermolecular forces between molecules; (4) the average kinetic energy = (3/2)kT (proportional to absolute temperature only). From these postulates, all gas laws follow: Boyle's law (P ∝ 1/V at constant T), Charles's law (V ∝ T at constant P), Avogadro's law (V ∝ n at constant P, T).
• Three types of molecular speeds: The Maxwell-Boltzmann distribution gives three characteristic speeds. Most probable speed (u_mp): the speed possessed by the maximum number of molecules = sqrt(2RT/M). Average speed (u_avg): arithmetic mean of all speeds = sqrt(8RT/piM). Root-mean-square speed (u_rms): square root of mean of squared speeds = sqrt(3RT/M). Their ratio is always u_rms > u_avg > u_mp ≈ 1.22 : 1.13 : 1. All three are proportional to sqrt(T/M): faster at higher T, faster for lighter molecules. Heating shifts the entire distribution to the right and flattens it.
• Real gases and deviations from ideality — the van der Waals equation: Real molecules have finite size (excluded volume correction b, units L/mol) and experience intermolecular attractions (pressure correction a/V², units atm·L²/mol²). The van der Waals equation (P + an²/V²)(V - nb) = nRT corrects both defects. At high pressure (small V), the volume correction b dominates and Z > 1 (molecules repel). At moderate pressure and low temperature, the attraction correction dominates and Z < 1. At the Boyle temperature T_B = a/Rb, the two corrections cancel over a wide pressure range and the gas behaves approximately ideally.
• Compressibility factor Z — reading deviations: Z = PV/nRT. For an ideal gas, Z = 1 always. Real gas behaviour: Z < 1 means the gas is more compressible than ideal (attractive forces dominate — gas liquefies more easily). Z > 1 means the gas is less compressible than ideal (repulsive forces dominate — at very high pressure). At very low pressure, all gases approach Z = 1 (nearly ideal). The Z vs P curve for each gas has a characteristic shape that depends on a and b.
• Graham's law of effusion and diffusion: The rate of effusion (gas escaping through a tiny hole into a vacuum) is inversely proportional to sqrt(M): r1/r2 = sqrt(M2/M1). The lighter gas effuses faster. Diffusion (mixing of gases) follows the same relation. Applications: isotope separation (e.g., ²³⁵UF₆ vs ²³⁸UF₆), identification of unknown gases, leak detection. Note: diffusion rate also depends on concentration gradient (Fick's law) but in JEE problems Graham's law is almost always the relevant relation.
• Liquefaction and critical constants: A gas can be liquefied only below its critical temperature Tc. Above Tc, no amount of pressure can liquefy the gas. The critical constants for a van der Waals gas are: Tc = 8a/(27Rb), Pc = a/(27b²), Vc = 3b. These relationships link observable critical behaviour to the molecular parameters a and b. For permanent gases (H₂, He, N₂), Tc is very low; they must be cooled first before pressure can liquefy them. For easily liquefiable gases (CO₂, NH₃), Tc is near or above room temperature.

• Ideal gas law: PV = nRT (R = 8.314 J/mol·K = 0.0821 L·atm/mol·K — units must match P and V)
• Dalton law of partial pressures: P_total = P1 + P2 + ... = sum of all partial pressures
• Partial pressure: Pi = xi × P_total where xi = mole fraction of component i = ni/n_total
• Kinetic molecular theory speeds: u_rms = sqrt(3RT/M); u_avg = sqrt(8RT/piM); u_mp = sqrt(2RT/M)
• Speed ratio: u_rms > u_avg > u_mp in ratio sqrt(3) : sqrt(8/pi) : sqrt(2) = 1.732 : 1.596 : 1.414
• Average kinetic energy of ideal gas: KE = (3/2)nRT per mole = (3/2)kT per molecule (depends only on T)
• Compressibility factor: Z = PV/nRT (Z=1 ideal; Z>1 repulsive dominates; Z<1 attractive dominates)
• van der Waals equation: (P + an²/V²)(V - nb) = nRT (a corrects for attractions; b corrects for finite volume)
• Boyle temperature: T_B = a/Rb (at T_B, gas behaves ideally over a wide pressure range)
• Graham law of effusion: rate of effusion proportional to 1/sqrt(M); r1/r2 = sqrt(M2/M1)
• Critical constants: Vc = 3b, Pc = a/(27b²), Tc = 8a/(27Rb)

Derivation / logic hint: Do not plug values blindly. Start from conservation of mass/charge, equilibrium definition, energy balance, electron movement, structure-property relation, or stability of the product/intermediate.

Most negative marks in this chapter come from condition errors, not lack of memory.

• Using Celsius instead of Kelvin: all gas law equations (PV=nRT, speed formulas, etc.) require the absolute temperature in Kelvin. T(K) = T(°C) + 273.15. Using 25°C instead of 298 K is a guaranteed wrong answer.
• Mixing units of R with mismatched P and V units: R = 8.314 J/mol·K is used when P is in Pa and V is in m³. R = 0.0821 L·atm/mol·K is used when P is in atm and V in litres. Mixing (e.g., P in atm but R in J/mol·K) gives a nonsensical numerical answer.
• Ignoring mole fraction when computing partial pressures: Pi = xi × P_total, not ni × P_total. Always compute the mole fraction first. If given only volumes (at same T and P), volume fraction = mole fraction.
• Assuming all gases follow Z=1 regardless of conditions: real gases deviate significantly from ideal behaviour at high pressure or low temperature. The ideal gas approximation is valid only at low P and high T. At high P, Z > 1 (real volume > ideal volume); at moderate P and low T, Z < 1.
• Confusing the three speed types (u_rms, u_avg, u_mp): u_rms is NOT the most common speed — u_mp is. u_rms is always the largest of the three. For a JEE problem that asks which speed a gas molecule is most likely to have, answer u_mp.

For JEE Main, prioritise direct formula use, NCERT-aligned facts, named-reaction recognition, trend comparison and quick elimination. Target 60–90 seconds per question.

• Ideal gas equation and gas laws
• Dalton law of partial pressures
• Kinetic molecular theory
• Maxwell-Boltzmann speed distribution

For JEE Advanced, combine ideas. Expect assertion-reason, integer, multiple-correct, paragraph-style and hidden-condition problems. Before finalising, ask which assumption the question is testing.

• Kinetic molecular theory
• Maxwell-Boltzmann speed distribution
• van der Waals equation
• Compressibility factor Z
• Graham law of effusion
• Critical constants and liquefaction

Use this block in the final 24–48 hours before a mock.

• Always convert temperature to Kelvin before any gas law calculation. Match R units to P and V units (0.0821 for L·atm; 8.314 for J).
• Partial pressure: Pi = xi × P_total where xi = ni/n_total. Collect conditions precisely.
• Speed order: u_rms > u_avg > u_mp. All proportional to sqrt(T/M). Higher T or lower M → faster speeds.
• Z < 1: attractive forces dominate (gas easier to compress than ideal). Z > 1: repulsive forces dominate (high pressure). Z = 1: ideal gas.
• Van der Waals: a corrects for attraction (reduces pressure), b corrects for volume (reduces available volume). Boyle temperature = a/Rb.