MPSC 2023 Prelims Q61: Which of the following mathematical relationships should be

Q: What is the answer to MPSC 2023 Prelims Q61?

(3). 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.

MPSC 2023 Prelims Paper-I · Question 61 of 100

Maharashtra Gazetted Civil Services Combined Preliminary Examination 2023, held 4 June 2023. Transcribed from the commission's own booklet (संच A); answer as per the official MPSC final answer key.

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

(1)pH = log₁₀(1.0 × 10⁻¹)
(2)pH = log₁₀(1.0 × 10¹)
(3)pH = – log₁₀(1.0 × 10⁻¹)
(4)pH = – log₁₀(0.10 × 10¹)

Answer

(3)

Why

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 others are wrong

(1)pH = log₁₀(1.0 × 10⁻¹) — 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.
(2)pH = log₁₀(1.0 × 10¹) — 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.
(4)pH = – log₁₀(0.10 × 10¹) — 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.

MPSC's science block asks chemistry in short computational items of exactly this kind, where the arithmetic is trivial and the whole difficulty lies in handling a definition precisely. This question is unusual and instructive in that it asks for the expression rather than for the number, which removes the option of guessing from a remembered result and forces the candidate to read four printed expressions and judge each against the definition. That format punishes a very common study habit — memorising a formula as a shape rather than as a statement, so that the minus sign in front of the logarithm is remembered as decoration and dropped under pressure. It also rewards fluency in scientific notation, which appears throughout the science section in questions on concentration, on the gas constant and on orders of magnitude, and which candidates from a non-science background most often get wrong in the direction of the exponent. The habit to build is to evaluate every option rather than to stop at the first that looks familiar: here one of the wrong expressions produces the correct numerical value of 1 through two compensating errors, so a candidate who checks only the output and not the structure walks straight into it. Note finally that this paper prints these option expressions in identical Latin form in both the English and the Marathi columns, since mathematical notation is not transliterated, so nothing is being lost in the language column a candidate is not reading.

Key facts

pH is defined as pH = – log₁₀[H⁺], the negative of the base-ten logarithm of the hydrogen ion concentration in moles per litre; the negative sign is part of the definition and converts the negative logarithms of small concentrations into positive scale values.
HCl is a monoprotic strong acid and is taken to be completely dissociated in dilute aqueous solution, so a 0.10 M solution of HCl has a hydrogen ion concentration of 0.10 mole per litre and a pH of 1.
Writing 0.10 in scientific notation gives 1.0 × 10⁻¹; a number smaller than one always carries a negative exponent, because the decimal point must be moved to the right to leave a coefficient between one and ten.
At twenty-five degrees Celsius the ionic product of water Kw is 1.0 × 10⁻¹⁴, pure water has equal H⁺ and OH⁻ concentrations of 1.0 × 10⁻⁷ mole per litre, and hence a neutral pH of 7; the companion relation pH + pOH = 14 holds at that temperature.
Each whole unit on the pH scale corresponds to a tenfold change in hydrogen ion concentration, so a difference of two pH units is a hundredfold difference in acidity.

The item marks an EXPRESSION, so two things must be right at once: the sign and the exponent

pH = – log₁₀[H⁺]→Sørensen's definition. The minus sign is not decoration — ordinary solutions have small [H⁺], their logarithms are negative, and the minus turns them into the positive numbers the scale actually uses

HCl is a monoprotic STRONG acid→taken as completely dissociated in dilute solution, one H⁺ per molecule, so a 0.10 M solution gives [H⁺] = 0.10 mol per litre; water's own ionisation is negligible beside it

0.10 written in scientific notation = 1.0 × 10⁻¹→any number below one carries a NEGATIVE exponent, because the decimal point must move right to leave a coefficient between one and ten

(3) pH = – log₁₀(1.0 × 10⁻¹)→the only expression carrying BOTH the minus sign of the definition and the negative exponent of a sub-unit concentration; it evaluates to – (– 1) = 1

Each whole pH unit is a tenfold change in [H⁺]. At 25 °C, Kw = 1.0 × 10⁻¹⁴, pure water has [H⁺] = [OH⁻] = 1.0 × 10⁻⁷ and pH 7, and pH + pOH = 14.

Study next

The ionic product of water, the relation pH + pOH = 14, and why the neutral point of seven is specific to twenty-five degrees CelsiusCalculating the pH of strong bases from hydroxide concentration through pOH, and the corresponding pOH of strong acidsWeak acids and bases, the dissociation constants Ka and Kb, degree of dissociation and Ostwald's dilution lawBuffer solutions, their composition from a weak acid and its salt, and the Henderson–Hasselbalch equationAcid–base titration, the choice of indicator against the pH at the equivalence point, and the shape of the titration curve

Common traps

Dropping the minus sign from the definition of pH, which turns the pH of every ordinary acidic solution into a negative number and is the single commonest error in this calculation
Selecting an expression because it evaluates to the number you expected, when the question asks which relationship applies and a wrong expression can reach the right value through compensating errors
Moving the decimal point the wrong way in scientific notation, so that a concentration smaller than one acquires a positive exponent
Applying the direct strong-acid calculation to a weak acid, whose hydrogen ion concentration is far below its stated molarity and must be obtained from its dissociation constant
Forgetting that the acid may be polyprotic, so that the hydrogen ion concentration is not simply equal to the molarity of the acid

Acid–base numericals recur in MPSC's science section in a small number of predictable shapes: compute the pH from the molarity of a strong acid, compute it from a strong base by way of pOH, identify which of four solutions is the most acidic, or state how many times more acidic one pH is than another. The Commission favours round concentrations such as 0.1 M, 0.01 M and 0.001 M precisely because the logarithm then comes out whole and the item can be answered in seconds by a candidate who holds the definition and cannot be answered at all by one who does not. The variant used here, which offers four expressions instead of four numbers, is the harder form and appears when the Commission wants to test the definition itself rather than the arithmetic. A candidate who can also write the companion relations — pOH, pH + pOH = 14, and Kw — covers the whole family from a single page of notes.

Related PYQs

No directly related past PYQ was found.

Practice

practice — not a real PYQ
What is the pH of a 0.001 M aqueous solution of a monoprotic strong acid at 25 °C ?
- (a)1
- (b)3
- (c)11
- (d)– 3
Answer(b) 3 — a strong acid is taken to be completely dissociated, so the hydrogen ion concentration equals the molarity, 0.001 M or 1.0 × 10⁻³ M. Applying pH = – log₁₀[H⁺] gives – log₁₀(10⁻³) = – (– 3) = 3. The value 11 would be the pOH of this solution, obtained from pH + pOH = 14, and – 3 is what results from forgetting the negative sign in the definition.
practice — not a real PYQ
A solution of pH 4 is how many times more acidic than a solution of pH 6 ?
- (a)2 times
- (b)20 times
- (c)100 times
- (d)1000 times
Answer(c) 100 times — the pH scale is logarithmic, so each whole unit represents a tenfold change in hydrogen ion concentration. A pH of 4 corresponds to [H⁺] = 10⁻⁴ M and a pH of 6 to [H⁺] = 10⁻⁶ M, a ratio of 10², or one hundred. The answer of two times comes from subtracting the pH values as though the scale were linear, which is the error this question is built to expose.