MPSC 2021 Prelims Q66: 32 g of methane (molar mass 16 g/mol) is mixed with 192 g

MPSC 2021 Prelims Paper-I · Question 66 of 100

State Services Preliminary Examination 2021, held 23 January 2022. Transcribed from the commission's own booklet (संच A); answer as per the official MPSC final answer key.

32 g of methane (molar mass 16 g/mol) is mixed with 192 g of oxygen (molar mass 32 g/mol). Presuming that these gases donot react with each other, what is the mole fraction of methane ?

(1)0.25
(2)6
(3)0.167
(4)4

Answer

(1)

Why

Correct — option (1), 0.25. The calculation has three steps and no shortcuts are needed. First convert each mass to a number of moles by dividing by the molar mass the question itself supplies. Methane: 32 grams divided by 16 grams per mole gives 2 moles. Oxygen: 192 grams divided by 32 grams per mole gives 6 moles. Second, add them to get the total amount of gas present, 2 plus 6, which is 8 moles. Third, apply the definition of mole fraction — the number of moles of the component of interest divided by the total number of moles of all components present. For methane that is 2 divided by 8, which is 0.25. As a check, the mole fraction of oxygen is 6 divided by 8, or 0.75, and the two add to exactly 1, as the mole fractions of all the components of any mixture must. Three features of the answer are worth noticing because they let a candidate reject wrong options without redoing the arithmetic. A mole fraction is a ratio of one amount to a larger amount, so it can never be greater than 1 and can never be negative; any option that is a whole number bigger than one is disqualified on sight. A mole fraction has no unit, because moles have been divided by moles, which is exactly why it is preferred to concentration when a quantity has to be independent of temperature and volume. And the ratio is by amount of substance, not by mass: 192 grams of oxygen weighs six times as much as 32 grams of methane, but it contains only three times as many molecules, because an oxygen molecule is twice as heavy as a methane molecule. Confusing mass with amount is the single most common error in this calculation and it accounts for one of the offered figures directly. The stem's condition that the two gases do not react — printed as 'donot' in the English column — is not decoration. Methane and oxygen react readily on ignition, one molecule of methane consuming two of oxygen to give carbon dioxide and water, and if the reaction were allowed to proceed the amounts of each gas present would change and the question would have no single answer. By stipulating that they are merely mixed, the examiner freezes the composition at the values calculated above.

Why the others are wrong

(2)6 — Six is the number of moles of oxygen in the mixture, obtained by dividing 192 grams by the molar mass of 32 grams per mole. It is a real quantity, and a necessary step on the way to the answer, but it is an amount and not a fraction, and it is the amount of the wrong component. It can be rejected without any arithmetic at all: a mole fraction is a part divided by the whole, so it always lies between 0 and 1, and no mixture can have a mole fraction of six. Options that offer an intermediate result of the calculation as though it were the final answer are a standard device in numerical questions, and the guard against them is to finish by checking that the number has the right form — a mole fraction between zero and one and no unit attached.
(3)0.167 — This figure comes from dividing the two masses directly — 32 grams of methane by 192 grams of oxygen, which is one sixth, or 0.167 — and so from skipping the conversion to moles altogether. That is the error the question is built to catch. Equal masses of two gases do not contain equal numbers of molecules unless the molar masses happen to be equal, and here they are not: the molar mass of oxygen, 32, is twice that of methane, 16, so a given mass of oxygen contains only half as many molecules as the same mass of methane. Working in moles gives 2 and 6, a ratio of one to three; working in grams gives 32 and 192, a ratio of one to six. Notice too that 0.167 is not even the right form of answer if the masses had been the right quantities to use, since a mole fraction is taken over the total and not over the other component alone.
(4)4 — Four is the answer turned upside down: it is the total amount of gas, 8 moles, divided by the amount of methane, 2 moles, instead of the amount of methane divided by the total. The mistake is one of definition rather than of arithmetic, and it is worth fixing the definition firmly for that reason — the mole fraction of a component is the moles of that component over the moles of everything present, so the component's own quantity is always on top and the total always below. Note how the option set has been assembled: 0.25 and 4 are reciprocals of each other, and so are 0.167 and 6, so each of the two ways of misreading the ratio has been printed alongside its inverted form. A candidate who has the definition right and remembers that the value must lie between zero and one is left with only one candidate figure.

Concept

The mole is the chemist's unit of amount of substance: one mole of any substance contains the Avogadro number of particles, about 6.022 times ten to the twenty-third, and has a mass in grams equal to its relative molecular or atomic mass. So one mole of methane weighs 16 grams and one mole of oxygen molecules weighs 32 grams, and converting between mass and amount is always a matter of dividing the mass by the molar mass. Composition of a mixture can then be expressed in several ways, and the examinable point is that they are not interchangeable. Mole fraction is the moles of one component divided by the total moles of all components; it is dimensionless, lies between 0 and 1, and the mole fractions of all components of a mixture sum to 1. Mass fraction, or percentage by mass, uses masses in the same way and gives a different number whenever the molar masses differ. Molarity is moles of solute per litre of solution and depends on temperature because volume does; molality is moles of solute per kilogram of solvent and does not. Mole fraction earns its place in two standard laws. Dalton's law of partial pressures states that in a mixture of gases the partial pressure of a component equals its mole fraction multiplied by the total pressure, so in the mixture of this question methane would exert one quarter of the total pressure and oxygen three quarters. Raoult's law states that the vapour pressure of a component of an ideal solution is its mole fraction multiplied by the vapour pressure of the pure component, and the colligative properties — lowering of vapour pressure, elevation of boiling point, depression of freezing point and osmotic pressure — all follow from it.

MPSC's science section carries a small number of genuinely numerical questions, and mole-concept arithmetic is the commonest of them because it needs no formula sheet: the data required are printed in the stem, as the two molar masses are here, and the work is two divisions and an addition. What the Commission is really testing is whether a candidate knows the definitions well enough to avoid the standard traps rather than whether he can divide. The paper offers, alongside the correct value, an intermediate result, a figure obtained by using masses where amounts were required, and the answer inverted — which is a fair map of how such calculations actually go wrong under time pressure. The habit worth building is to spend the last two seconds on the form of the answer rather than on the arithmetic: a mole fraction between 0 and 1 with no unit, a molarity in moles per litre, a percentage between 0 and 100. That check alone disposes of two of the four options here. The second habit is to read the conditions in the stem, since the stipulation that the gases do not react is what makes a single answer possible at all.

Key facts

One mole of a substance contains the Avogadro number of particles, about 6.022 times ten to the twenty-third, and has a mass in grams numerically equal to its relative molecular mass; the amount in moles is the mass divided by the molar mass.
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 mixture; it is dimensionless, always lies between 0 and 1, and the mole fractions of all components add up to 1.
In this mixture there are 2 moles of methane (32 grams over 16 grams per mole) and 6 moles of oxygen (192 grams over 32 grams per mole), so the total is 8 moles, the mole fraction of methane is 0.25 and that of oxygen is 0.75.
Equal masses of two gases contain equal numbers of molecules only if their molar masses are equal, so a composition worked out from masses is not the same as one worked out from moles.
By Dalton's law of partial pressures, the partial pressure of a gas in a mixture is its mole fraction times the total pressure, which is one of the two standard uses of mole fraction; the other is Raoult's law for the vapour pressure of a solution.

Three steps, no shortcuts, and a last two seconds spent on the FORM of the answer rather than on the arithmetic

n(CH4) = 32 g ÷ 16 g/mol = 2 mol→Convert each mass to an amount first, using the molar mass the stem itself supplies. Nothing in this question can be done in grams, and the conversion is the only place real chemistry enters it

n(O2) = 192 g ÷ 32 g/mol = 6 mol→Here is the whole point of the question in one sentence: 192 grams of oxygen weighs six times as much as 32 grams of methane but contains only three times as many molecules, because an oxygen molecule is twice as heavy as a methane molecule. Equal masses of two gases hold equal numbers of molecules only when their molar masses are equal

x(CH4) = 2 ÷ (2 + 6) = 0.25→The definition, applied exactly as it stands: the amount of the component of interest divided by the total amount of all components present — the component's own quantity always on top, the total always below. That is option (1)

Check: x(O2) = 6 ÷ 8 = 0.75, and 0.25 + 0.75 = 1→The mole fractions of all the components of any mixture add to exactly 1, so the check costs one division and confirms both figures at once

Form check: a mole fraction is dimensionless and lies between 0 and 1→This disqualifies two of the printed figures without any arithmetic at all. Six is the amount of oxygen in moles — a real quantity, a necessary step on the way, but an amount rather than a fraction and the wrong component besides. Four is the ratio inverted, the total of 8 moles divided by the 2 moles of methane. Notice how the set is assembled: 0.25 and 4 are reciprocals of each other, and so are 0.167 and 6, so each way of misreading the ratio has been printed beside its own inverted form

The remaining figure, 0.167, is what comes of dividing the two masses directly — 32 grams by 192 grams, which is one sixth — so it skips the conversion to moles altogether and takes the fraction over the other component instead of over the total, two errors in one number. The stem's condition that the gases do not react, printed as 'donot' in the English column, is not decoration either: methane and oxygen react readily on ignition, one molecule of methane consuming two of oxygen, and if the reaction were allowed to proceed the amounts present would change and the question would have no single answer. Mole fraction earns its place in two standard laws — Dalton's law, by which the partial pressure of a component is its mole fraction times the total pressure, so methane here would exert one quarter of the pressure and oxygen three quarters, and Raoult's law, from which the colligative properties follow.

Study next

The mole concept — Avogadro's number, molar mass, molar volume of a gas at standard conditions, and conversions between mass, moles and number of particlesWays of expressing concentration — mole fraction, mass percentage, molarity, molality and parts per million — and which of them depend on temperatureDalton's law of partial pressures and the other gas laws, including the ideal gas equationRaoult's law and the colligative properties of dilute solutionsStoichiometry of chemical reactions — balancing equations, limiting reagent problems and percentage yield

Common traps

Using masses where amounts in moles are required, which gives the wrong ratio whenever the two molar masses differ
Dividing by the amount of the other component instead of by the total amount of the mixture, so that the fraction is taken over the wrong denominator
Offering an intermediate step, such as the number of moles of one gas, as the final answer to a question that asked for a fraction
Forgetting that a mole fraction is dimensionless and must lie between 0 and 1, a check that disqualifies several wrong options without any calculation

Numerical chemistry in MPSC papers is concentrated on the mole concept and on concentration terms, and the questions are designed to be solvable in under a minute from data given in the stem. Expect to be asked for the number of moles or of molecules in a stated mass, for the mole fraction or the molarity of a mixture, for the volume a gas occupies at standard conditions, or for the mass of a product in a simple stoichiometric reaction. The Commission usually supplies the molar masses, as it does here, so nothing has to be recalled except the definitions themselves; the difficulty is placed entirely in whether the candidate distinguishes mass from amount and knows what each concentration term divides by. It is worth practising until the conversion from grams to moles is automatic, and worth learning the check that the components of a mixture have mole fractions summing to one, since it verifies an answer in a single line.

Related PYQs

No directly related past PYQ was found.

Practice

practice — not a real PYQ
A mixture contains 4 moles of nitrogen and 1 mole of hydrogen. What is the mole fraction of hydrogen in the mixture ?
- (a)0.25
- (b)0.20
- (c)0.80
- (d)4.00
Answer(b) 0.20 — the total amount of gas is 4 plus 1, that is 5 moles, and the mole fraction of hydrogen is its own amount divided by that total, 1 over 5, which is 0.20. The mole fraction of nitrogen is 4 over 5, or 0.80, and the two add to 1 as they must. The figure 0.25 is what results from dividing hydrogen by nitrogen instead of by the total, and 4.00 is not a possible mole fraction at all, since the value must lie between 0 and 1.
practice — not a real PYQ
A vessel contains a mixture of gases in which the mole fraction of carbon dioxide is 0.20, and the total pressure of the mixture is 5 atmospheres. By Dalton's law, the partial pressure of carbon dioxide is
- (a)0.04 atmosphere
- (b)1 atmosphere
- (c)4 atmospheres
- (d)25 atmospheres
Answer(b) 1 atmosphere — Dalton's law of partial pressures states that the partial pressure exerted by a component of a gaseous mixture equals its mole fraction multiplied by the total pressure of the mixture, so here 0.20 multiplied by 5 atmospheres gives 1 atmosphere. The figure 0.04 comes from dividing the mole fraction by the pressure instead of multiplying, and 25 from dividing the pressure by the mole fraction; 4 atmospheres is the partial pressure of everything else in the vessel taken together, since the partial pressures must add up to the total.