Isotopes and Atomic Masses

1.6 Isotopes and Atomic Masses

Learning Objective

To know the meaning of isotopes and atomic masses.

Rutherford’s nuclear model of the atom helped explain why atoms of different elements exhibit different chemical behavior. The identity of an element is defined by its atomic number (Z)The number of protons in the nucleus of an atom of an element., the number of protons in the nucleus of an atom of the element. The atomic number is therefore different for each element. The known elements are arranged in order of increasing Z in the periodic tableA chart of the chemical elements arranged in rows of increasing atomic number so that the elements in each column (group) have similar chemical properties. (Figure 1.24 "The Periodic Table Showing the Elements in Order of Increasing "; also see Chapter 32 "Appendix H: Periodic Table of Elements"),We will explain the rationale for the peculiar format of the periodic table in Chapter 7 "The Periodic Table and Periodic Trends". in which each element is assigned a unique one-, two-, or three-letter symbol. The names of the elements are listed in the periodic table, along with their symbols, atomic numbers, and atomic masses. The chemistry of each element is determined by its number of protons and electrons. In a neutral atom, the number of electrons equals the number of protons.

Figure 1.24 The Periodic Table Showing the Elements in Order of Increasing Z

As described in Section 1.7 "Introduction to the Periodic Table", the metals are on the bottom left in the periodic table, and the nonmetals are at the top right. The semimetals lie along a diagonal line separating the metals and nonmetals.

In most cases, the symbols for the elements are derived directly from each element’s name, such as C for carbon, U for uranium, Ca for calcium, and Po for polonium. Elements have also been named for their properties [such as radium (Ra) for its radioactivity], for the native country of the scientist(s) who discovered them [polonium (Po) for Poland], for eminent scientists [curium (Cm) for the Curies], for gods and goddesses [selenium (Se) for the Greek goddess of the moon, Selene], and for other poetic or historical reasons. Some of the symbols used for elements that have been known since antiquity are derived from historical names that are no longer in use; only the symbols remain to remind us of their origin. Examples are Fe for iron, from the Latin ferrum; Na for sodium, from the Latin natrium; and W for tungsten, from the German wolfram. Examples are in Table 1.4 "Element Symbols Based on Names No Longer in Use". As you work through this text, you will encounter the names and symbols of the elements repeatedly, and much as you become familiar with characters in a play or a film, their names and symbols will become familiar.

Table 1.4 Element Symbols Based on Names No Longer in Use

Element	Symbol	Derivation	Meaning
antimony	Sb	stibium	Latin for “mark”
copper	Cu	cuprum	from Cyprium, Latin name for the island of Cyprus, the major source of copper ore in the Roman Empire
gold	Au	aurum	Latin for “gold”
iron	Fe	ferrum	Latin for “iron”
lead	Pb	plumbum	Latin for “heavy”
mercury	Hg	hydrargyrum	Latin for “liquid silver”
potassium	K	kalium	from the Arabic al-qili, “alkali”
silver	Ag	argentum	Latin for “silver”
sodium	Na	natrium	Latin for “sodium”
tin	Sn	stannum	Latin for “tin”
tungsten	W	wolfram	German for “wolf stone” because it interfered with the smelting of tin and was thought to devour the tin

Recall from Section 1.5 "The Atom" that the nuclei of most atoms contain neutrons as well as protons. Unlike protons, the number of neutrons is not absolutely fixed for most elements. Atoms that have the same number of protons, and hence the same atomic number, but different numbers of neutrons are called isotopesAtoms that have the same numbers of protons but different numbers of neutrons.. All isotopes of an element have the same number of protons and electrons, which means they exhibit the same chemistry. The isotopes of an element differ only in their atomic mass, which is given by the mass number (A)The number of protons and neutrons in the nucleus of an atom of an element., the sum of the numbers of protons and neutrons.

The element carbon (C) has an atomic number of 6, which means that all neutral carbon atoms contain 6 protons and 6 electrons. In a typical sample of carbon-containing material, 98.89% of the carbon atoms also contain 6 neutrons, so each has a mass number of 12. An isotope of any element can be uniquely represented as ${}_{Z}^{A}X,$ where X is the atomic symbol of the element. The isotope of carbon that has 6 neutrons is therefore ${}_{6}^{12}C .$ The subscript indicating the atomic number is actually redundant because the atomic symbol already uniquely specifies Z. Consequently, ${}_{6}^{12}C$ is more often written as ¹²C, which is read as “carbon-12.” Nevertheless, the value of Z is commonly included in the notation for nuclear reactions because these reactions involve changes in Z, as described in Chapter 20 "Nuclear Chemistry".

In addition to ¹²C, a typical sample of carbon contains 1.11% ${}_{6}^{13}C$ (¹³C), with 7 neutrons and 6 protons, and a trace of ${}_{6}^{14}C$ (¹⁴C), with 8 neutrons and 6 protons. The nucleus of ¹⁴C is not stable, however, but undergoes a slow radioactive decay that is the basis of the carbon-14 dating technique used in archaeology (see Chapter 14 "Chemical Kinetics"). Many elements other than carbon have more than one stable isotope; tin, for example, has 10 isotopes. The properties of some common isotopes are in Table 1.5 "Properties of Selected Isotopes".

Table 1.5 Properties of Selected Isotopes

Element	Symbol	Atomic Mass (amu)	Isotope Mass Number	Isotope Masses (amu)	Percent Abundances (%)
hydrogen	H	1.0079	1	1.007825	99.9855
hydrogen	H	1.0079	2	2.014102	0.0115
boron	B	10.81	10	10.012937	19.91
boron	B	10.81	11	11.009305	80.09
carbon	C	12.011	12	12 (defined)	99.89
carbon	C	12.011	13	13.003355	1.11
oxygen	O	15.9994	16	15.994915	99.757
			17	16.999132	0.0378
			18	17.999161	0.205
iron	Fe	55.845	54	53.939611	5.82
			56	55.934938	91.66
			57	56.935394	2.19
			58	57.933276	0.33
uranium	U	238.03	234	234.040952	0.0054
			235	235.043930	0.7204
			238	238.050788	99.274

Sources of isotope data: G. Audi et al., Nuclear Physics A 729 (2003): 337–676; J. C. Kotz and K. F. Purcell, Chemistry and Chemical Reactivity, 2nd ed., 1991.

Example 5

An element with three stable isotopes has 82 protons. The separate isotopes contain 124, 125, and 126 neutrons. Identify the element and write symbols for the isotopes.

Given: number of protons and neutrons

Asked for: element and atomic symbol

Strategy:

A Refer to the periodic table (see Chapter 32 "Appendix H: Periodic Table of Elements") and use the number of protons to identify the element.

B Calculate the mass number of each isotope by adding together the numbers of protons and neutrons.

C Give the symbol of each isotope with the mass number as the superscript and the number of protons as the subscript, both written to the left of the symbol of the element.

Solution:

A The element with 82 protons (atomic number of 82) is lead: Pb.

B For the first isotope, A = 82 protons + 124 neutrons = 206. Similarly, A = 82 + 125 = 207 and A = 82 + 126 = 208 for the second and third isotopes, respectively. The symbols for these isotopes are ${}_{82}^{206}P b,$ ${}_{82}^{207}P b,$ and ${}_{82}^{208}P b,$ which are usually abbreviated as ²⁰⁶Pb, ²⁰⁷Pb, and ²⁰⁸Pb.

Exercise

Identify the element with 35 protons and write the symbols for its isotopes with 44 and 46 neutrons.

Answer: ${}_{35}^{79}B r$ and ${}_{35}^{81}B r$ or, more commonly, ⁷⁹Br and ⁸¹Br.

Although the masses of the electron, the proton, and the neutron are known to a high degree of precision (Table 1.3 "Properties of Subatomic Particles*"), the mass of any given atom is not simply the sum of the masses of its electrons, protons, and neutrons. For example, the ratio of the masses of ¹H (hydrogen) and ²H (deuterium) is actually 0.500384, rather than 0.49979 as predicted from the numbers of neutrons and protons present. Although the difference in mass is small, it is extremely important because it is the source of the huge amounts of energy released in nuclear reactions (Chapter 20 "Nuclear Chemistry").

Because atoms are much too small to measure individually and do not have a charge, there is no convenient way to accurately measure absolute atomic masses. Scientists can measure relative atomic masses very accurately, however, using an instrument called a mass spectrometer. The technique is conceptually similar to the one Thomson used to determine the mass-to-charge ratio of the electron. First, electrons are removed from or added to atoms or molecules, thus producing charged particles called ionsA charged particle produced when one or more electrons is removed from or added to an atom or molecule.. When an electric field is applied, the ions are accelerated into a separate chamber where they are deflected from their initial trajectory by a magnetic field, like the electrons in Thomson’s experiment. The extent of the deflection depends on the mass-to-charge ratio of the ion. By measuring the relative deflection of ions that have the same charge, scientists can determine their relative masses (Figure 1.25 "Determining Relative Atomic Masses Using a Mass Spectrometer"). Thus it is not possible to calculate absolute atomic masses accurately by simply adding together the masses of the electrons, the protons, and the neutrons, and absolute atomic masses cannot be measured, but relative masses can be measured very accurately. It is actually rather common in chemistry to encounter a quantity whose magnitude can be measured only relative to some other quantity, rather than absolutely. We will encounter many other examples later in this text. In such cases, chemists usually define a standard by arbitrarily assigning a numerical value to one of the quantities, which allows them to calculate numerical values for the rest.

Figure 1.25 Determining Relative Atomic Masses Using a Mass Spectrometer

Chlorine consists of two isotopes, ³⁵Cl and ³⁷Cl, in approximately a 3:1 ratio. (a) When a sample of elemental chlorine is injected into the mass spectrometer, electrical energy is used to dissociate the Cl₂ molecules into chlorine atoms and convert the chlorine atoms to Cl⁺ ions. The ions are then accelerated into a magnetic field. The extent to which the ions are deflected by the magnetic field depends on their relative mass-to-charge ratios. Note that the lighter ³⁵Cl⁺ ions are deflected more than the heavier ³⁷Cl⁺ ions. By measuring the relative deflections of the ions, chemists can determine their mass-to-charge ratios and thus their masses. (b) Each peak in the mass spectrum corresponds to an ion with a particular mass-to-charge ratio. The abundance of the two isotopes can be determined from the heights of the peaks.

The arbitrary standard that has been established for describing atomic mass is the atomic mass unit (amu)One-twelfth of the mass of one atom of ${}^{12}C$ ; $1 amu = 1.66 \times 1 0^{- 24} g$ ., defined as one-twelfth of the mass of one atom of ¹²C. Because the masses of all other atoms are calculated relative to the ¹²C standard, ¹²C is the only atom listed in Table 1.5 "Properties of Selected Isotopes" whose exact atomic mass is equal to the mass number. Experiments have shown that 1 amu = 1.66 × 10⁻²⁴ g.

Mass spectrometric experiments give a value of 0.167842 for the ratio of the mass of ²H to the mass of ¹²C, so the absolute mass of ²H is

\frac{mass of {}^{2}H}{mass of {}^{12}C} \times mass of {}^{12}C = 0 .167842 \times 12 amu = 2 .104104 amu

The masses of the other elements are determined in a similar way.

The periodic table (see Chapter 32 "Appendix H: Periodic Table of Elements") lists the atomic masses of all the elements. If you compare these values with those given for some of the isotopes in Table 1.5 "Properties of Selected Isotopes", you can see that the atomic masses given in the periodic table never correspond exactly to those of any of the isotopes. Because most elements exist as mixtures of several stable isotopes, the atomic mass of an element is defined as the weighted average of the masses of the isotopes. For example, naturally occurring carbon is largely a mixture of two isotopes: 98.89% ¹²C (mass = 12 amu by definition) and 1.11% ¹³C (mass = 13.003355 amu). The percent abundance of ¹⁴C is so low that it can be ignored in this calculation. The average atomic mass of carbon is then calculated as

(0.9889 × 12 amu) + (0.0111 × 13.003355 amu) = 12.01 amu

Carbon is predominantly ¹²C, so its average atomic mass should be close to 12 amu, which is in agreement with our calculation.

The value of 12.01 is shown under the symbol for C in the periodic table (see Chapter 32 "Appendix H: Periodic Table of Elements"), although without the abbreviation amu, which is customarily omitted. Thus the tabulated atomic mass of carbon or any other element is the weighted average of the masses of the naturally occurring isotopes.

Example 6

Naturally occurring bromine consists of the two isotopes listed in the following table:

Isotope	Exact Mass (amu)	Percent Abundance (%)
⁷⁹Br	78.9183	50.69
⁸¹Br	80.9163	49.31

Calculate the atomic mass of bromine.

Given: exact mass and percent abundance

Asked for: atomic mass

Strategy:

A Convert the percent abundances to decimal form to obtain the mass fraction of each isotope.

B Multiply the exact mass of each isotope by its corresponding mass fraction (percent abundance ÷ 100) to obtain its weighted mass.

C Add together the weighted masses to obtain the atomic mass of the element.

D Check to make sure that your answer makes sense.

Solution:

A The atomic mass is the weighted average of the masses of the isotopes. In general, we can write

atomic mass of element = [(mass of isotope 1 in amu) (mass fraction of isotope 1)] + [(mass of isotope 2) (mass fraction of isotope 2)] + …

Bromine has only two isotopes. Converting the percent abundances to mass fractions gives

\begin{array}{l} {}^{79}B r: \frac{50.69}{100} = 0.5069 \\ {}^{81}B r: \frac{49.31}{100} = 0.4931 \end{array}

B Multiplying the exact mass of each isotope by the corresponding mass fraction gives the isotope’s weighted mass:

⁷⁹Br: 79.9183 amu × 0.5069 = 40.00 amu ⁸¹Br: 80.9163 amu × 0.4931 = 39.90 amu

C The sum of the weighted masses is the atomic mass of bromine is

40.00 amu + 39.90 amu = 79.90 amu

D This value is about halfway between the masses of the two isotopes, which is expected because the percent abundance of each is approximately 50%.

Exercise

Magnesium has the three isotopes listed in the following table:

Isotope	Exact Mass (amu)	Percent Abundance (%)
²⁴Mg	23.98504	78.70
²⁵Mg	24.98584	10.13
²⁶Mg	25.98259	11.17

Use these data to calculate the atomic mass of magnesium.

Answer: 24.31 amu

Summary

Each atom of an element contains the same number of protons, which is the atomic number (Z). Neutral atoms have the same number of electrons and protons. Atoms of an element that contain different numbers of neutrons are called isotopes. Each isotope of a given element has the same atomic number but a different mass number (A), which is the sum of the numbers of protons and neutrons. The relative masses of atoms are reported using the atomic mass unit (amu), which is defined as one-twelfth of the mass of one atom of carbon-12, with 6 protons, 6 neutrons, and 6 electrons. The atomic mass of an element is the weighted average of the masses of the naturally occurring isotopes. When one or more electrons are added to or removed from an atom or molecule, a charged particle called an ion is produced, whose charge is indicated by a superscript after the symbol.

Key Takeaway

The mass of an atom is a weighted average that is largely determined by the number of its protons and neutrons, whereas the number of protons and electrons determines its charge.

Conceptual Problems

Complete the following table for the missing elements, symbols, and numbers of electrons.

Element	Symbol	Number of Electrons
molybdenum
		19
titanium
	B
		53
	Sm
helium
		14

Complete the following table for the missing elements, symbols, and numbers of electrons.

Element	Symbol	Number of Electrons
lanthanum
	Ir
aluminum
		80
sodium
	Si
		9
	Be

Is the mass of an ion the same as the mass of its parent atom? Explain your answer.
What isotopic standard is used for determining the mass of an atom?
Give the symbol ${}_{Z}^{A}X$ for these elements, all of which exist as a single isotope.
1. beryllium
2. ruthenium
3. phosphorus
4. aluminum
5. cesium
6. praseodymium
7. cobalt
8. yttrium
9. arsenic
Give the symbol ${}_{Z}^{A}X$ for these elements, all of which exist as a single isotope.
1. fluorine
2. helium
3. terbium
4. iodine
5. gold
6. scandium
7. sodium
8. niobium
9. manganese
Identify each element, represented by X, that have the given symbols.
1. ${}_{26}^{55}X$
2. ${}_{33}^{74}X$
3. ${}_{12}^{24}X$
4. ${}_{53}^{127}X$
5. ${}_{18}^{40}X$
6. ${}_{63}^{152}X$

Numerical Problems

Please be sure you are familiar with the topics discussed in Essential Skills 1 (Section 1.9 "Essential Skills 1") before proceeding to the Numerical Problems.

The isotopes ¹³¹I and ⁶⁰Co are commonly used in medicine. Determine the number of neutrons, protons, and electrons in a neutral atom of each.
Determine the number of protons, neutrons, and electrons in a neutral atom of each isotope:
1. ⁹⁷Tc
2. ¹¹³In
3. ⁶³Ni
4. ⁵⁵Fe
Both technetium-97 and americium-240 are produced in nuclear reactors. Determine the number of protons, neutrons, and electrons in the neutral atoms of each.
The following isotopes are important in archaeological research. How many protons, neutrons, and electrons does a neutral atom of each contain?
1. ²⁰⁷Pb
2. ¹⁶O
3. ⁴⁰K
4. ¹³⁷Cs
5. ⁴⁰Ar

Copper, an excellent conductor of heat, has two isotopes: ⁶³Cu and ⁶⁵Cu. Use the following information to calculate the average atomic mass of copper:

Isotope	Percent Abundance (%)	Atomic Mass (amu)
⁶³Cu	69.09	62.9298
⁶⁵Cu	30.92	64.9278

Silicon consists of three isotopes with the following percent abundances:

Isotope	Percent Abundance (%)	Atomic Mass (amu)
²⁸Si	92.18	27.976926
²⁹Si	4.71	28.976495
³⁰Si	3.12	29.973770

Calculate the average atomic mass of silicon.

Complete the following table for neon. The average atomic mass of neon is 20.1797 amu.

Isotope	Percent Abundance (%)	Atomic Mass (amu)
²⁰Ne	90.92	19.99244
²¹Ne	0.257	20.99395
²²Ne

Are ${}_{28}^{63}X$ and ${}_{29}^{62}X$ isotopes of the same element? Explain your answer.

Complete the following table:

Isotope	Number of Protons	Number of Neutrons	Number of Electrons
²³⁸X			95
²³⁸U
	75	112

Complete the following table:

Isotope	Number of Protons	Number of Neutrons	Number of Electrons
⁵⁷Fe
⁴⁰X		20
³⁶S

Using a mass spectrometer, a scientist determined the percent abundances of the isotopes of sulfur to be 95.27% for ³²S, 0.51% for ³³S, and 4.22% for ³⁴S. Use the atomic mass of sulfur from the periodic table (see Chapter 32 "Appendix H: Periodic Table of Elements") and the following atomic masses to determine whether these data are accurate, assuming that these are the only isotopes of sulfur: 31.972071 amu for ³²S, 32.971459 amu for ³³S, and 33.967867 amu for ³⁴S.
The percent abundances of two of the three isotopes of oxygen are 99.76% for ¹⁶O, and 0.204% for ¹⁸O. Use the atomic mass of oxygen given in the periodic table (see Chapter 32 "Appendix H: Periodic Table of Elements") and the following data to determine the mass of ¹⁷O: 15.994915 amu for ¹⁶O and 17.999160 amu for ¹⁸O.
Which element has the higher proportion by mass in NaI?
Which element has the higher proportion by mass in KBr?