180 g of acetic acid (CH₃COOH) is mixed with 180 g of water (H₂O). The mole fraction of acetic acid in the resultant solution is
[H = 1, C = 12, O = 16]

Question

UnlockIAS · Accepted Answer

Correct — option (1). Mole fraction counts particles, not grams, so the first and decisive move is to convert both masses into moles; the atomic masses printed in square brackets under the stem are supplied for exactly that purpose. Acetic acid, CH₃COOH, has the molecular formula C₂H₄O₂, so its molar mass is 2 × 12 for carbon plus 4 × 1 for hydrogen plus 2 × 16 for oxygen, that is 24 + 4 + 32 = 60 g per mole. Water, H₂O, has a molar mass of 2 × 1 + 16 = 18 g per mole. Dividing the given masses by these gives 180 ÷ 60 = 3 moles of acetic acid and 180 ÷ 18 = 10 moles of water. Notice what has happened: the two components were supplied in equal masses, yet they are present in very unequal numbers of moles, because a molecule of acetic acid is more than three times as heavy as a molecule of water. That asymmetry is the entire point of the question. The mole fraction of a component is the number of moles of that component divided by the total number of moles of all components in the solution. The total here is 3 + 10 = 13 moles, so the mole fraction of acetic acid is 3 ÷ 13 = 0.2307…, which to two decimal places is 0.23 as printed in option (1). Two checks confirm the result. First, the mole fraction of water must be 10 ÷ 13 = 0.77, and the two mole fractions sum to 1 as they must, since every mole in the solution belongs to one component or the other. Second, the answer must lie below 0.5, because acetic acid is present in the smaller number of moles, and simply noticing that the heavier molecule must be the scarcer one at equal mass eliminates two of the four options before any division is performed. Option (4), 'None of the above', has no work to do here because the computed value matches a printed option exactly. Option (1) is therefore the answer.

Why the other options are wrong — (b) A mole fraction of 1.0 would mean that every mole in the container belonged to acetic acid — that is, that the substance was pure and no water was present at all. Since the question explicitly puts 180 g of water into the mixture, the value is impossible on the face of it, and this option can be struck out without any calculation. The value arises from one specific error: dividing the mass of acetic acid by the mass of water, 180 ÷ 180, and reporting the result as a fraction. That is a ratio of masses of the two components rather than a fraction of the total moles, and it is doubly wrong — wrong in using masses instead of moles, and wrong in dividing by one component instead of by the whole. The definition to hold is that a mole fraction always has the total moles of the solution in its denominator, so it can never exceed one and can equal one only for a pure substance.  (c) This is the trap the question was built around, and it is the answer produced by reading '180 g and 180 g' as 'equal amounts' and concluding that each component must therefore make up half of the mixture. Equal masses are not equal numbers of moles unless the two substances happen to have the same molar mass, and here they are far apart: 60 g per mole for acetic acid against 18 g per mole for water, so the same 180 g yields 3 moles of the one and 10 moles of the other. A mole fraction of 0.5 would require 3 moles of water alongside the 3 moles of acid, which is 54 g of water and not 180 g. The equal masses in the stem are deliberate bait, chosen so that a candidate who skips the conversion to moles arrives at a clean and plausible number rather than at obvious nonsense.  (d) This escape option is offered on exactly three questions in this paper — Q57, Q65 and Q85 — while the companion escape, 'all of the above', is offered on thirteen; and the discipline it demands is to compute the answer first and consult the option list afterwards. Here the calculation gives 3 ÷ 13 = 0.2307…, which rounds to 0.23 and matches option (1) as printed, so there is nothing for 'None of the above' to cover. The one circumstance in which such an option deserves serious thought is when a carefully checked calculation produces a value that no printed option approximates; the mere fact that the computed figure is a recurring decimal rather than a round number is not such a circumstance, since 0.23 is plainly the two-decimal form of it. Selecting this option because the arithmetic felt untidy is a common way of converting a correct calculation into a lost mark and, under this paper's negative marking, into a deduction.

Concept: Mole fraction expresses the composition of a mixture by counting particles rather than by weighing them: the mole fraction of a component is the moles of that component divided by the total moles of all components present. It is a pure number without units, it always lies between zero and one, and the mole fractions of all the components of a mixture necessarily sum to one — a relation that provides a free check on any calculation involving two components, since finding one immediately gives the other. Its two great advantages over molarity are that it is independent of temperature, because no volume enters the definition and volumes expand while masses and mole counts do not, and that it treats solute and solvent symmetrically instead of privileging one of them. Those properties make it the natural measure wherever the physical behaviour of a solution depends on the proportion of molecules present rather than on their mass. Raoult's law states that the partial vapour pressure of a component of an ideal solution equals its mole fraction multiplied by the vapour pressure of the pure component, and the whole treatment of colligative properties — relative lowering of vapour pressure, elevation of boiling point, depression of freezing point and osmotic pressure — follows from it. Dalton's law of partial pressures uses the same measure on the gas side, the partial pressure of a gas in a mixture being its mole fraction multiplied by the total pressure. Molality, the moles of solute per kilogram of solvent, shares the temperature independence of mole fraction and is used for the same reason in colligative-property work, while molarity, defined per litre of solution, is convenient for volumetric analysis but varies with temperature.

State PCS 2023 Q65 — Science & Technology PYQ with Answer

State PCS 2023 Q65 — Science & Technology PYQ with Answer