NDA (II) 2018 General Ability Test · Question 63 of 146

Correct — D, 16·5 u. This calls for a weighted average, not a plain one. In a 3 : 1 proportion, three atoms out of every four have mass 16 u and one out of every four has mass 18 u, so the fractional abundances are 3/4 and 1/4. The average atomic mass is therefore (3 × 16 + 1 × 18) ÷ 4 = (48 + 18) ÷ 4 = 66 ÷ 4 = 16·5 u. A quick sanity check confirms it without any arithmetic: the answer has to lie between 16 and 18, and because the lighter isotope is three times as common it has to lie much nearer 16 than 18 — which leaves 16·5 u as the only possibility among the four options.

• (a)17·5 u — This is the weighted mean you get if the ratio is read backwards, with the 18 u isotope taken as the abundant one — (1 × 16 + 3 × 18) ÷ 4 = 17·5 u. Read which isotope the 3 belongs to before multiplying.
• (b)17 u — The plain arithmetic mean of 16 and 18, which throws away the 3 : 1 proportion altogether. It would be the answer only if the two isotopes were equally abundant.
• (c)16 u — The mass of the commoner isotope on its own. The presence of any atoms of the heavier isotope must pull the average above 16, so this cannot be right whatever the proportion is.

Isotopes are atoms of the same element — the same number of protons, and so the same chemistry — that differ in the number of neutrons and therefore in mass. An element found in nature is usually a fixed mixture of its isotopes, and the atomic mass printed in the periodic table is the mean of the isotopic masses weighted by how abundant each one is. That is why so many atomic masses are not whole numbers even though individual isotopic masses are close to whole numbers.

Set the calculation out as fractions of the total rather than as a ratio, because that is where marks are lost. A 3 : 1 ratio means four parts in all, so the multipliers are 3/4 and 1/4, and the divisor is 4 rather than 2. The same item recurs with chlorine, where isotopes of 35 u and 37 u in a 3 : 1 ratio give the familiar 35·5 u, and with boron, where the answer runs the other way and you are given the average and asked for the abundances. Learn the one formula and all three versions follow from it.

• Isotopes have the same atomic number but different mass numbers, because they differ in neutron count.
• Average atomic mass = (mass₁ × fraction₁) + (mass₂ × fraction₂), with the fractions adding to 1.
• Chlorine's atomic mass of 35·5 u comes from Cl-35 and Cl-37 in roughly a 3 : 1 ratio, by exactly this calculation.
• The weighted mean always lies between the extreme isotopic masses and closer to the more abundant one.
• Isotopes of an element are chemically alike because chemistry depends on electrons, which the neutron count does not change.

• Dividing by 2 instead of by the total number of parts in the ratio.
• Assigning the ratio to the wrong isotope and getting 17·5 u.
• Taking the simple average of the two masses and ignoring the abundances.

NDA sets this every couple of years — two isotopic masses and a ratio, asking for the average; sometimes reversed, giving the average and asking for the abundances.

• practice — not a real PYQ

  An element has two isotopes of masses 10 u and 12 u present in the ratio 1 : 3. Its average atomic mass is

  • (a)11·0 u
  • (b)11·5 u
  • (c)10·5 u
  • (d)12·0 u

  Answer(b) 11·5 u — (1 × 10 + 3 × 12) ÷ 4 = 46 ÷ 4 = 11·5 u, the mean pulled towards the more abundant heavier isotope.

• practice — not a real PYQ

  The atomic mass of chlorine is given as 35·5 u rather than a whole number because

  • (a)a chlorine atom contains half a neutron
  • (b)it is the weighted average of two isotopes of masses 35 u and 37 u
  • (c)chlorine is a diatomic gas
  • (d)the mass of the electron has been included

  Answer(b) it is the weighted average of two isotopes of masses 35 u and 37 u — present in about a 3 : 1 ratio, which averages to 35·5 u.