Ideal Gas Equation Numericals
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Direct answer
PV = nRT ties pressure, volume, amount and temperature into one working equation for every gas numerical: pressure in atmospheres with R = 0.0821 litre atm per kelvin per mole, or pascals with R = 8.314 J per kelvin per mole. One mole of any ideal gas occupies 22.4 litre at 273.15 K and 1 atm — but 22.7 litre at the 1 bar standard NCERT now prefers, a distinction the paper has tested directly. For before-and-after problems the combined law P1V1/T1 = P2V2/T2 avoids computing n altogether, and the density form M = dRT/P extracts molar mass from a single measurement. Real gases hold to this only at high temperature and low pressure; the van der Waals equation adds the correction terms a/V² for attraction and b for molecular volume.
What you must remember
- Three R values, three situations: 0.0821 litre atm K−1 mol−1 (atm problems), 8.314 J K−1 mol−1 (SI, also written Pa m³), and 8.314 × 10−2 litre bar K−1 mol−1 (bar problems).
- Two standard volumes: 22.4 litre per mole at 273.15 K and 1 atm; 22.7 litre per mole at 273.15 K and 1 bar — NCERT's bar-based standard state since the SI revision.
- Molar mass from density: M = dRT/P with d in gram per litre; density is directly proportional to molar mass at fixed T and P.
- Mole counting: n = m/M; molecules = n × 6.022 × 10^23; at STP, moles = volume in litre ÷ 22.4.
- Kelvin discipline: every T must be kelvin (add 273.15, or 273 in exam arithmetic); 27 °C is 300 K, the commonest value set.
- van der Waals equation: (P + an²/V²)(V − nb) = nRT; a carries units litre² atm mol−² (attraction strength), b litre mol−¹ (excluded volume, about four times the molecular volume).
- Compressibility factor: Z = PV/nRT; Z less than 1 means attractive forces dominate (easier liquefaction, as with NH3 and CO2), Z greater than 1 means the gas is harder to compress than ideal at very high pressure.
A full gas numerical, start to finish
Pressure of 8.0 g oxygen in a 5 litre vessel at 27 °C. Convert first: n = 8/32 = 0.25 mol, T = 300 K. Then P = nRT/V = (0.25 × 0.0821 × 300)/5 = 1.23 atm. Notice the sequence — grams to moles, celsius to kelvin, then substitute — because nearly every lost mark here is a skipped conversion, not a hard sum.
Two other costumes of the same equation. Density form: the density of oxygen at 1 atm and 273 K is d = PM/RT = 32/(0.0821 × 273) = 1.43 g per litre. Two-state form: a 2.0 litre balloon at 27 °C and 1 atm is heated to 127 °C at constant pressure; V2 = V1T2/T1 = 2 × 400/300 = 2.67 litre, no n ever needed. When a problem gives a gas collected over water, subtract the aqueous tension from the total pressure before any of this arithmetic begins.
The traps that cost the numerical mark
Unit mismatch destroys more answers than chemistry does. Millilitres fed to R in joules, atmospheres fed to 8.314, or 25 used instead of 298 — each produces a wrong option planted in the list. The 22.4 versus 22.7 litre question is a deliberate NCERT-based trap: read whether the problem says 1 atm or 1 bar before deciding. Conceptual traps cluster around ideality: candidates quote "real gases always deviate" when the deviation's direction matters — at moderate pressure attraction dominates and Z dips below 1, while at very high pressure the finite molecular volume pushes Z above 1. Hydrogen and helium, with negligible attraction, show Z above 1 almost throughout — a favourite assertion-reason pairing.
Frequently asked questions
Which value of R is used when pressure is in atmosphere?
R = 0.0821 litre atm K−1 mol−1; using 8.314 with atmospheres is the single commonest unit error in gas numericals.
Why does NCERT sometimes quote 22.7 litre instead of 22.4 litre?
22.4 litre applies at 1 atm, 22.7 litre at 1 bar — the standard pressure NCERT adopted with the SI convention, and the paper checks which one the question specifies.
How is molar mass found from gas density?
Rearrange the ideal equation to M = dRT/P; measure density at known temperature and pressure, and the molar mass follows directly.
Under what conditions do real gases deviate most from ideality?
At low temperature and high pressure; attraction pulls Z below 1 at moderate pressures, while excluded volume drives Z above 1 at very high pressures.
How is a gas collected over water handled?
Subtract the aqueous tension (vapour pressure of water at the collection temperature) from the total pressure; the difference is the dry gas pressure for further calculation.