If one mole of an ideal gas occupies 22.4 Lit at Standard Temperature and Pressure, then the value of the universal gas constant R is

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Correct — option (1). The stem hands over every quantity needed to derive the constant rather than to recall it. The ideal gas equation is PV = nRT, so R = PV ÷ nT, and the conditions described are one mole of gas occupying 22.4 litres at standard temperature and pressure — that is, n = 1 mol, V = 22.4 Lit, P = 1 atmosphere and T = 273 K, the standard temperature being the melting point of ice expressed on the Kelvin scale. Substituting gives R = (1 × 22.4) ÷ (1 × 273) = 0.0821, and the units follow mechanically from the same substitution: pressure in atmospheres multiplied by volume in litres, divided by moles and by kelvin, gives atm Lit mol⁻¹ K⁻¹. That is exactly what option (1) prints, both in figure and in unit, and it is the answer. The point the item is really testing is that R is not a single number but a single physical quantity that takes different numerical values in different systems of units, so a value and its unit must always travel together. In SI units, with pressure in pascals and volume in cubic metres, the same constant is 8.314 J K⁻¹ mol⁻¹; in the CGS system, where energy is measured in ergs and one joule is 10⁷ ergs, it becomes 8.314 × 10⁷ erg K⁻¹ mol⁻¹; in thermochemical work it is about 1.987, or roughly 2, calories K⁻¹ mol⁻¹. Each of these is the same constant, and quoting the figure from one system with the unit from another is simply a false statement. That is precisely how the three wrong options here have been constructed: each pairs a genuine numerical value of R with a unit belonging to a different system, so a candidate who has memorised loose digits without their units cannot separate them. R itself is the universal gas constant, universal because its value does not depend on the identity of the gas, and it is related to the Boltzmann constant by R = N_A × k_B, the gas constant being the Boltzmann constant scaled up from a single molecule to a mole. Option (1) is the answer.

Why the other options are wrong — (b) The figure and the unit here belong to different systems and contradict each other. In joules the universal gas constant is 8.314 J K⁻¹ mol⁻¹, with no power of ten attached; the multiplier of 10⁷ belongs to the CGS expression of the same constant in ergs, since one joule equals 10⁷ ergs. As printed, this option overstates R in SI units by a factor of ten million. It is a well-made distractor because both of its pieces are things a candidate has genuinely seen — the digits 8.314 and the factor 10⁷ both appear in correct statements of R — and the error lies only in having combined them. It is also the mirror image of option (4), which carries the joule figure with the erg unit, and noticing that the two cannot both be right is the fastest way to see that the pair is testing unit systems rather than memory of digits.  (c) The unit is right for the conditions given but the number is a thousand times too large. Dividing 22.4 litre-atmospheres by 273 kelvin gives 0.0821 and not 82.1, and the arithmetic can be checked in a moment: 82.1 multiplied by 273 would imply a molar volume of over twenty-two thousand litres at standard conditions, which is absurd for a gas measured in litres. The digits themselves are not invented — 82.1 is the correct value of R when the volume is expressed in millilitres or cubic centimetres rather than litres, that is 82.1 mL atm K⁻¹ mol⁻¹ — so this option is the same trick as options (2) and (4) applied to the volume unit instead of to the energy unit. Whenever a printed value of R looks familiar in its digits, the safe move is to substitute it back into PV = nRT with the stated conditions and see whether the molar volume comes out at 22.4 litres.  (d) This is the complement of option (2): it takes the SI figure of 8.314 and attaches the CGS unit, the erg. In ergs the constant is 8.314 × 10⁷, because an erg is a very small unit of energy — one ten-millionth of a joule — so the same physical quantity requires a correspondingly larger number when expressed in it. As printed, this option understates R by a factor of ten million. The general rule the pair illustrates is worth stating explicitly, since it applies well beyond this constant: when a quantity is re-expressed in a smaller unit its numerical value grows, and when it is re-expressed in a larger unit the value shrinks. A candidate who applies that check to the digits 8.314 sitting beside 'erg' will see at once that a number this small cannot be correct in so small a unit.

Concept: The ideal gas equation PV = nRT draws together the three simple gas laws — Boyle's law relating pressure and volume at constant temperature, Charles's law relating volume and temperature at constant pressure, and Avogadro's law relating volume to the number of moles — into one relation, and R is the constant of proportionality that makes them consistent. It is called the universal gas constant because its value is the same for every gas that behaves ideally, unlike the specific gas constant, which is R divided by the molar mass and therefore differs from gas to gas. R has the dimensions of energy per mole per kelvin, which is why its value changes with the units chosen for energy: 8.314 joules, or 8.314 × 10⁷ ergs, or about 1.987 calories, all per kelvin per mole, and 0.0821 litre-atmospheres per kelvin per mole when pressure and volume are measured in atmospheres and litres rather than converted to energy units. The constant also connects the macroscopic and molecular descriptions of a gas: R = N_A × k_B, so the gas constant is simply the Boltzmann constant, which governs the energy of a single molecule, multiplied by Avogadro's number to scale it to a mole. The molar volume of 22.4 litres used in this question follows from the equation itself rather than being an independent fact, being the volume one mole occupies at 273 K and one atmosphere. It is worth knowing that the definition of standard conditions has changed: the older convention of 0 °C and one atmosphere, which gives 22.4 litres, is the one Indian syllabi and this question use, while IUPAC's current recommendation sets standard pressure at one bar, under which the molar volume is about 22.7 litres. The ideal gas equation holds strictly only for a hypothetical gas of point particles with no intermolecular forces, and real gases deviate from it at high pressure and low temperature, which is what the van der Waals equation corrects for.

State PCS 2023 Q66 — Science & Technology PYQ with Answer

State PCS 2023 Q66 — Science & Technology PYQ with Answer