Which of the following mathematical relationships should be used to calculate the pH of a 0.10M aqueous HCl solution ?

Question

UnlockIAS · Accepted Answer

Correct — option (3). Read what the question actually asks for: it does not ask for the value of the pH, it asks which mathematical relationship should be used to calculate it, so the answer is an expression and it is marked right or wrong on two things — the sign in front of the logarithm and the way the concentration has been written in scientific notation. Both have to be right together. The definition of pH, introduced by Sørensen, is pH = – log₁₀[H⁺]: the negative of the base-ten logarithm of the hydrogen ion concentration in moles per litre. The minus sign is part of the definition and not an optional decoration. It is there because the hydrogen ion concentrations of ordinary solutions are small numbers written as negative powers of ten, so their logarithms are negative, and taking the negative of the logarithm turns them into the convenient small positive numbers that the pH scale actually uses. Now the chemistry of this particular solution. HCl is a strong acid, which means that in dilute aqueous solution it is taken to be completely dissociated into H⁺ and Cl⁻, and each molecule releases exactly one hydrogen ion because the acid is monoprotic. A 0.10 M solution of HCl therefore supplies a hydrogen ion concentration of 0.10 mol per litre, and the small quantity of H⁺ contributed by the self-ionisation of water is negligible beside it. The last step is purely notational: 0.10 written in scientific notation is 1.0 × 10⁻¹, because the decimal point has to be moved one place to the right to give a number between one and ten, and moving it right makes the exponent negative. Substituting that into the definition gives pH = – log₁₀(1.0 × 10⁻¹), which is exactly what option (3) prints, and evaluating it gives – (– 1) = 1. So the keyed expression carries both the negative sign the definition demands and the correct scientific-notation form of a concentration smaller than one, and option (3) is the answer.

Why the other options are wrong — (a) The concentration is written correctly here — 1.0 × 10⁻¹ is indeed 0.10 — but the minus sign that the definition of pH requires has been dropped, and that single omission is fatal. Evaluated as printed, this expression gives log₁₀(1.0 × 10⁻¹) = – 1, so it claims the pH of the solution is minus one. A candidate should be able to reject that on sight without any arithmetic at all: a decimolar solution of a common strong acid is an everyday laboratory reagent, not something at the extreme edge of the scale, and negative pH values arise only in very concentrated strong acid where the ordinary treatment breaks down anyway. This option exists precisely to catch the candidate who remembers that pH involves a logarithm of the hydrogen ion concentration but has stored the formula without its sign.  (b) This is the most dangerous option on the page, because it evaluates to the right number by way of two errors that cancel. It drops the minus sign, as option (1) does, and it also writes the concentration as 1.0 × 10¹, which is ten moles per litre — a hundred times the concentration the question actually gives. Since log₁₀(1.0 × 10¹) = 1, the expression happens to produce the value 1, which is the true pH of 0.10 M HCl. A candidate who quietly computes each option and stops at the first one yielding the expected answer will select this and lose the mark, because the question asks which relationship should be used, and a relationship that reaches the right number through a wrong sign and a wrong concentration is not the relationship for this solution. The defence is to check the expression against the definition rather than against the answer you expect.  (d) The negative sign is present, so this option gets the definition of pH right, but the concentration inside the logarithm is wrong. The expression 0.10 × 10¹ evaluates to 1.0, which is one mole per litre, ten times the concentration stated in the question, and it is also not a legitimate scientific-notation form since the leading number in that notation must lie between one and ten. Worked through, this option gives pH = – log₁₀(1.0) = 0, the pH of a molar solution of a strong acid, not of a decimolar one. The error being tested is the direction in which the exponent moves: converting a number smaller than one into scientific notation requires shifting the decimal point to the right and therefore a negative exponent, and a candidate who shifts it the wrong way turns 0.10 into 10 instead of into 1.0 × 10⁻¹.

Concept: pH is a compressed way of reporting hydrogen ion concentration, defined as pH = – log₁₀[H⁺] with the concentration expressed in moles per litre. Because acidic and basic solutions span concentrations from roughly one mole per litre down to 10⁻¹⁴ mole per litre, reporting them directly would require fourteen orders of magnitude; the logarithm collapses that range onto a scale of about zero to fourteen, and the negative sign makes the resulting numbers positive for ordinary solutions. The scale is anchored on water itself. Water ionises very slightly into H⁺ and OH⁻, and at twenty-five degrees Celsius the product of the two concentrations, the ionic product Kw, is 1.0 × 10⁻¹⁴. In pure water the two concentrations are equal at 1.0 × 10⁻⁷ mole per litre, so pure water has a pH of seven and that is what neutrality means. Taking negative logarithms of the whole Kw expression gives the companion relation pH + pOH = 14 at that temperature, which is the quickest route from a known concentration of a strong base to a pH. Each unit of pH is a factor of ten in hydrogen ion concentration, so a solution at pH 3 is a hundred times more acidic than one at pH 5, not slightly more. The calculation is direct only for strong acids and strong bases, which can be assumed fully dissociated so that the acid concentration is the ion concentration; for a weak acid only a fraction dissociates and the hydrogen ion concentration must be obtained from the dissociation constant before the logarithm can be taken. Two boundary conditions are worth holding: the neutral pH of seven belongs to twenty-five degrees Celsius alone, since Kw rises with temperature, and in very concentrated or very dilute solutions the simple treatment fails because activity departs from concentration in the first case and the water's own ions cease to be negligible in the second.

State PCS 2023 Q61 — Science & Technology PYQ with Answer

State PCS 2023 Q61 — Science & Technology PYQ with Answer