Isotopes and Relative Atomic Mass – Structure of an Atom and Mass Spectrum (Senior 4 Chemistry) | Ganzaa

Structure of an Atom and Mass Spectrum

Lessons

Senior 4 · Chemistry

Structure of an Atom and Mass Spectrum

4 lessons

Lesson 2 of 4

Isotopes and Relative Atomic Mass

25 min

Why is chlorine 35.5?

Look up chlorine in the periodic table and you find a relative atomic mass of 35.5. Yet a chlorine atom contains whole numbers of protons and neutrons, each of relative mass very close to 1. No single chlorine atom has a mass of 35.5. The number is an average: natural chlorine is a mixture of two kinds of atom, one with mass 35 and one with mass 37, in a ratio of about three to one. This lesson explains how such atoms differ, and how to calculate the average that the periodic table prints.

Definition

Atomic number, Z: the number of protons in the nucleus of an atom. It identifies the element: every atom with 17 protons is chlorine.

Mass number, A: the total number of protons and neutrons (together called nucleons) in the nucleus. The number of neutrons is $N = A - Z$ .

Nuclide notation: ${}^{A}_{Z}\mathrm{X}$ , for example ${}^{35}_{17}\mathrm{Cl}$ . An ion's charge is written at the top right, as in ${}^{37}_{17}\mathrm{Cl^{-}}$ .

Definition

Isotopes: atoms of the same element, so with the same number of protons, that have different numbers of neutrons and therefore different mass numbers. Example: ${}^{35}_{17}\mathrm{Cl}$ (18 neutrons) and ${}^{37}_{17}\mathrm{Cl}$ (20 neutrons). An isotope is named by its mass number: chlorine-35, carbon-14.

Same chemistry, different mass

Chemical reactions are made and broken by electrons. Isotopes of one element have the same number of protons, so their neutral atoms have the same number of electrons arranged in the same way. They therefore have the same chemical properties: chlorine-35 and chlorine-37 both form NaCl, HCl and AgCl in exactly the same reactions.

Their physical properties that depend on mass differ slightly: density, rate of diffusion, melting and boiling points. The difference is largest for hydrogen, where deuterium doubles the mass of the atom. Water made from deuterium, $\mathrm{D_2O}$ or "heavy water", has a density of $1.107\,\mathrm{g\,cm^{-3}}$ at 25 °C compared with $0.997\,\mathrm{g\,cm^{-3}}$ for ordinary water; it melts at 3.8 °C and boils at 101.4 °C. An ice cube of heavy water sinks in ordinary water.

Table 2.1: Some natural isotopes (abundances rounded)

Isotope	Protons	Neutrons	Natural abundance / %	Note
${}^{1}_{1}\mathrm{H}$	1	0	99.99	protium
${}^{2}_{1}\mathrm{H}$	1	1	0.01	deuterium, D
${}^{12}_{6}\mathrm{C}$	6	6	98.9	the standard for atomic masses
${}^{13}_{6}\mathrm{C}$	6	7	1.1	stable
${}^{14}_{6}\mathrm{C}$	6	8	trace	radioactive; used in dating
${}^{35}_{17}\mathrm{Cl}$	17	18	75.8	stable
${}^{37}_{17}\mathrm{Cl}$	17	20	24.2	stable
${}^{235}_{92}\mathrm{U}$	92	143	0.72	nuclear fuel
${}^{238}_{92}\mathrm{U}$	92	146	99.27	radioactive

Worked example 1: counting particles

Give the numbers of protons, neutrons and electrons in (a) ${}^{37}_{17}\mathrm{Cl^{-}}$ , (b) ${}^{56}_{26}\mathrm{Fe^{3+}}$ , (c) ${}^{2}_{1}\mathrm{H^{+}}$ .

Solution. Protons = Z. Neutrons = A − Z. Electrons = Z − charge (a negative charge adds electrons).

(a) 17 protons; $37 - 17 = 20$ neutrons; $17 + 1 = 18$ electrons.
(b) 26 protons; $56 - 26 = 30$ neutrons; $26 - 3 = 23$ electrons.
(c) 1 proton; $2 - 1 = 1$ neutron; $1 - 1 = 0$ electrons. This is a bare deuterium nucleus.

Relative atomic mass

Single atoms are far too light to weigh in kilograms conveniently: a carbon-12 atom has a mass of $1.993 \times 10^{-26}\,\mathrm{kg}$ . Chemists therefore compare masses with a standard. Since 1961 the standard has been the carbon-12 atom, given a mass of exactly 12. One twelfth of its mass is the unified atomic mass unit, $1\,\mathrm{u} = 1.6605 \times 10^{-27}\,\mathrm{kg}$ .

Definition

Relative isotopic mass: the mass of an atom of one isotope compared with one twelfth of the mass of a carbon-12 atom. Example: chlorine-37 has a relative isotopic mass of 36.966.

Relative atomic mass, $A_{r}$ : the weighted mean mass of the atoms of an element, in its natural mixture of isotopes, compared with one twelfth of the mass of an atom of carbon-12. It has no units.

A_r = \frac{\sum \left(\text{relative isotopic mass} \times \text{percentage abundance}\right)}{100}

Worked example 2: the relative atomic mass of chlorine

Natural chlorine is 75.76% chlorine-35 (relative isotopic mass 34.969) and 24.24% chlorine-37 (36.966). Calculate $A_r(\mathrm{Cl})$ to 4 significant figures.

Solution.

A_r = \frac{(34.969 \times 75.76) + (36.966 \times 24.24)}{100} = \frac{2649.25 + 896.06}{100} = 35.45

Answer: 35.45, closer to 35 than to 37 because three quarters of the atoms are chlorine-35. Using mass numbers 35 and 37 instead gives 35.48, which rounds to the familiar 35.5.

Worked example 3: finding an abundance from $A_{r}$

Copper has two isotopes, copper-63 (relative isotopic mass 62.93) and copper-65 (64.93). $A_r(\mathrm{Cu}) = 63.55$ . Find the percentage of copper-63.

Solution. Let the percentage of copper-63 be $x$ ; then copper-65 is $100 - x$ .

\frac{62.93x + 64.93(100 - x)}{100} = 63.55

62.93x + 6493 - 64.93x = 6355 \quad\Rightarrow\quad 2.00x = 138 \quad\Rightarrow\quad x = 69.0

Answer: 69.0% copper-63 and 31.0% copper-65. (With mass numbers 63 and 65 you would get 72.5%: approximate masses give a poor abundance when the two isotopes differ by only 2.)

Warning

Mass number is not relative atomic mass. A mass number counts particles in one nucleus and is always a whole number. A relative atomic mass is an average over many atoms and is usually not a whole number. Also, the percentages of all the isotopes must add up to 100; check this before you divide.

Tip

Before you calculate, predict. $A_{r}$ must lie between the lightest and the heaviest isotope, and nearer to the most abundant one. If your answer for chlorine is 36.5, you have swapped the abundances.

FuseSchool: How To Calculate Relative Atomic Mass (4 min). Watch how each isotope's mass is multiplied by its abundance before the products are added, and how the answer is checked against the most abundant isotope.

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Practice

Define isotopes and give one pair of isotopes not named in this lesson's examples.
Complete for each species the numbers of protons, neutrons and electrons: ${}^{40}_{20}\mathrm{Ca^{2+}}$ , ${}^{32}_{16}\mathrm{S^{2-}}$ , ${}^{14}_{6}\mathrm{C}$ .
Explain why ${}^{12}\mathrm{C}$ and ${}^{14}\mathrm{C}$ burn in oxygen to give the same product, but ${}^{14}\mathrm{CO_2}$ diffuses more slowly than ${}^{12}\mathrm{CO_2}$ .
Gallium is 60.1% gallium-69 (68.93) and 39.9% gallium-71 (70.92). Calculate $A_r(\mathrm{Ga})$ to 4 significant figures.
Silicon is 92.23% silicon-28 (27.98), 4.68% silicon-29 (28.98) and the rest silicon-30 (29.97). Calculate $A_r(\mathrm{Si})$ .
Bromine has two isotopes, bromine-79 and bromine-81, and $A_{r} = 79.9$ . Using mass numbers, find the percentage of each.
An ion X²⁺ has 24 electrons and 32 neutrons. Identify X and write its nuclide notation.

Show answersHide answers

Atoms of the same element (same number of protons) with different numbers of neutrons. Example: uranium-235 and uranium-238.
$\mathrm{Ca^{2+}}$ : 20 p, 20 n, 18 e. $\mathrm{S^{2-}}$ : 16 p, 16 n, 18 e. ${}^{14}\mathrm{C}$ : 6 p, 8 n, 6 e.
Both have 6 electrons arranged in the same way, so the chemistry is the same; ${}^{14}\mathrm{CO_2}$ molecules are heavier (relative mass 46 against 44), so at the same temperature they move more slowly and diffuse more slowly.
$\frac{(68.93 \times 60.1) + (70.92 \times 39.9)}{100} = \frac{4142.7 + 2829.7}{100} = 69.72$ .
Silicon-30 is $100 - 92.23 - 4.68 = 3.09\%$ . $A_r = \frac{(27.98 \times 92.23) + (28.98 \times 4.68) + (29.97 \times 3.09)}{100} = 28.09$ .
$79x + 81(100 - x) = 7990 \Rightarrow 2x = 110 \Rightarrow x = 55$ : 55% bromine-79 and 45% bromine-81. (The true values are close to 51% and 49%; the error comes from using mass numbers.)
Z = 24 + 2 = 26, so X is iron; A = 26 + 32 = 58: ${}^{58}_{26}\mathrm{Fe^{2+}}$ .

Quick check

Key points

Z = protons; A = protons + neutrons; neutrons = A − Z; for an ion, electrons = Z − charge.
Isotopes have the same Z but different A. They have the same chemical properties but slightly different mass-dependent physical properties.
$A_{r}$ is the weighted mean mass of an element's atoms relative to one twelfth of the mass of a carbon-12 atom: $A_r = \frac{\sum(\text{mass} \times \%)}{100}$ .
With two isotopes and a known $A_{r}$ , let one abundance be $x$ and the other $100 - x$ , then solve.

Sources and credits

Written by Ganzaa. Isotopic masses and abundances are rounded from the tables of the IUPAC Commission on Isotopic Abundances and Atomic Weights; the properties of heavy water are standard reference values, as listed in the CRC Handbook of Chemistry and Physics.
Video: How To Calculate Relative Atomic Mass, FuseSchool - Global Education, embedded from YouTube.
The diagrams, table, worked examples and Practice items were written and drawn by Ganzaa.

1Why is chlorine 35.5?

2Same chemistry, different mass

3Relative atomic mass

Why is chlorine 35.5?

Same chemistry, different mass

Relative atomic mass