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<div class="section" id="averill_1.0-ch01_s06" condition="start-of-chunk" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">1.6</span> Isotopes and Atomic Masses</h2>
<div class="learning_objectives editable block" id="averill_1.0-ch01_s06_n01">
<h3 class="title">Learning Objective</h3>
<ol class="orderedlist" id="averill_1.0-ch01_s06_l01">
<li>To know the meaning of isotopes and atomic masses.</li>
</ol>
</div>
<p class="para editable block" id="averill_1.0-ch01_s06_p01">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 <span class="margin_term"><a class="glossterm">atomic number (<em class="emphasis">Z</em>)</a><span class="glossdef">The number of protons in the nucleus of an atom of an element.</span></span>, the number of protons in the nucleus of an atom of the element. <em class="emphasis">The atomic number is therefore different for each element</em>. The known elements are arranged in order of increasing <em class="emphasis">Z</em> in the <span class="margin_term"><a class="glossterm">periodic table</a><span class="glossdef">A chart of the chemical elements arranged in rows of increasing atomic number so that the elements in each column (group) have similar chemical properties.</span></span> (<a class="xref" href="#averill_1.0-ch01_s06_f01">Figure 1.24 "The Periodic Table Showing the Elements in Order of Increasing "</a>; also see <a class="xref" href="s36-appendix-h-periodic-table-of-e.html">Chapter 32 "Appendix H: Periodic Table of Elements"</a>),<span class="footnote" id="averill_1.0-fn01_006">We will explain the rationale for the peculiar format of the periodic table in <a class="xref" href="averill_1.0-ch07#averill_1.0-ch07">Chapter 7 "The Periodic Table and Periodic Trends"</a>.</span> 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.</p>
<div class="figure full editable block" id="averill_1.0-ch01_s06_f01">
<p class="title"><span class="title-prefix">Figure 1.24</span> The Periodic Table Showing the Elements in Order of Increasing <em class="emphasis">Z</em></p>
<img src="section_05/26dbac2e64570e0683190e1b66079cb3.jpg">
<p class="para">As described in <a class="xref" href="averill_1.0-ch01_s07#averill_1.0-ch01_s07">Section 1.7 "Introduction to the Periodic Table"</a>, 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.</p>
</div>
<p class="para editable block" id="averill_1.0-ch01_s06_p02">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 <em class="emphasis">ferrum</em>; Na for sodium, from the Latin <em class="emphasis">natrium</em>; and W for tungsten, from the German <em class="emphasis">wolfram</em>. Examples are in <a class="xref" href="#averill_1.0-ch01_s06_t01">Table 1.4 "Element Symbols Based on Names No Longer in Use"</a>. 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.</p>
<div class="table block" id="averill_1.0-ch01_s06_t01" frame="all">
<p class="title"><span class="title-prefix">Table 1.4</span> Element Symbols Based on Names No Longer in Use</p>
<table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center">Element</th>
<th align="center">Symbol</th>
<th align="center">Derivation</th>
<th align="center">Meaning</th>
</tr>
</thead>
<tbody>
<tr>
<td>antimony</td>
<td>Sb</td>
<td>
<em class="emphasis">stibium</em>
</td>
<td>Latin for “mark”</td>
</tr>
<tr>
<td>copper</td>
<td>Cu</td>
<td>
<em class="emphasis">cuprum</em>
</td>
<td>from <em class="emphasis">Cyprium</em>, Latin name for the island of Cyprus, the major source of copper ore in the Roman Empire</td>
</tr>
<tr>
<td>gold</td>
<td>Au</td>
<td>
<em class="emphasis">aurum</em>
</td>
<td>Latin for “gold”</td>
</tr>
<tr>
<td>iron</td>
<td>Fe</td>
<td>
<em class="emphasis">ferrum</em>
</td>
<td>Latin for “iron”</td>
</tr>
<tr>
<td>lead</td>
<td>Pb</td>
<td>
<em class="emphasis">plumbum</em>
</td>
<td>Latin for “heavy”</td>
</tr>
<tr>
<td>mercury</td>
<td>Hg</td>
<td>
<em class="emphasis">hydrargyrum</em>
</td>
<td>Latin for “liquid silver”</td>
</tr>
<tr>
<td>potassium</td>
<td>K</td>
<td>
<em class="emphasis">kalium</em>
</td>
<td>from the Arabic <em class="emphasis">al-qili</em>, “alkali”</td>
</tr>
<tr>
<td>silver</td>
<td>Ag</td>
<td>
<em class="emphasis">argentum</em>
</td>
<td>Latin for “silver”</td>
</tr>
<tr>
<td>sodium</td>
<td>Na</td>
<td>
<em class="emphasis">natrium</em>
</td>
<td>Latin for “sodium”</td>
</tr>
<tr>
<td>tin</td>
<td>Sn</td>
<td>
<em class="emphasis">stannum</em>
</td>
<td>Latin for “tin”</td>
</tr>
<tr>
<td>tungsten</td>
<td>W</td>
<td>
<em class="emphasis">wolfram</em>
</td>
<td>German for “wolf stone” because it interfered with the smelting of tin and was thought to devour the tin</td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="averill_1.0-ch01_s06_p03">Recall from <a class="xref" href="averill_1.0-ch01_s05#averill_1.0-ch01_s05">Section 1.5 "The Atom"</a> 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 <em class="emphasis">the same number of protons</em>, and hence the same atomic number, but <em class="emphasis">different numbers of neutrons</em> are called <span class="margin_term"><a class="glossterm">isotopes</a><span class="glossdef">Atoms that have the same numbers of protons but different numbers of neutrons.</span></span>. 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 <span class="margin_term"><a class="glossterm">mass number (<em class="emphasis">A</em>)</a><span class="glossdef">The number of protons and neutrons in the nucleus of an atom of an element.</span></span>, the sum of the numbers of protons and neutrons.</p>
<p class="para block" id="averill_1.0-ch01_s06_p04">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 <span class="inlineequation"><math xml:id="averill_1.0-ch01_m007" display="inline"><semantics><mrow><mmultiscripts><mtext>X</mtext><mprescripts></mprescripts><mi>Z</mi><mi>A</mi></mmultiscripts><mtext>,</mtext></mrow></semantics></math></span> where X is the atomic symbol of the element. The isotope of carbon that has 6 neutrons is therefore <span class="inlineequation"><math xml:id="averill_1.0-ch01_m008" display="inline"><semantics><mrow><mmultiscripts><mtext>C</mtext><mprescripts></mprescripts><mn>6</mn><mrow><mtext>12</mtext></mrow></mmultiscripts><mtext>.</mtext></mrow></semantics></math></span> The subscript indicating the atomic number is actually redundant because the atomic symbol already uniquely specifies <em class="emphasis">Z</em>. Consequently, <span class="inlineequation"><math xml:id="averill_1.0-ch01_m009" display="inline"><semantics><mrow><mmultiscripts><mtext>C</mtext><mprescripts></mprescripts><mn>6</mn><mrow><mtext>12</mtext></mrow></mmultiscripts></mrow></semantics></math></span> is more often written as <sup class="superscript">12</sup>C, which is read as “carbon-12.” Nevertheless, the value of <em class="emphasis">Z</em> is commonly included in the notation for <em class="emphasis">nuclear</em> reactions because these reactions involve changes in <em class="emphasis">Z</em>, as described in <a class="xref" href="averill_1.0-ch20#averill_1.0-ch20">Chapter 20 "Nuclear Chemistry"</a>.</p>
<div class="informalfigure large medium-height block">
<img src="section_05/6fe205bcf099e852cab6a591415a8350.jpg">
</div>
<p class="para block" id="averill_1.0-ch01_s06_p05">In addition to <sup class="superscript">12</sup>C, a typical sample of carbon contains 1.11% <span class="inlineequation"><math xml:id="averill_1.0-ch01_m010" display="inline"><semantics><mrow><mmultiscripts><mtext>C</mtext><mprescripts></mprescripts><mn>6</mn><mrow><mtext>13</mtext></mrow></mmultiscripts></mrow></semantics></math></span> (<sup class="superscript">13</sup>C), with 7 neutrons and 6 protons, and a trace of <span class="inlineequation"><math xml:id="averill_1.0-ch01_m011" display="inline"><semantics><mrow><mmultiscripts><mtext>C</mtext><mprescripts></mprescripts><mn>6</mn><mrow><mtext>14</mtext></mrow></mmultiscripts></mrow></semantics></math></span> (<sup class="superscript">14</sup>C), with 8 neutrons and 6 protons. The nucleus of <sup class="superscript">14</sup>C is not stable, however, but undergoes a slow radioactive decay that is the basis of the carbon-14 dating technique used in archaeology (see <a class="xref" href="averill_1.0-ch14#averill_1.0-ch14">Chapter 14 "Chemical Kinetics"</a>). 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 <a class="xref" href="#averill_1.0-ch01_s06_t02">Table 1.5 "Properties of Selected Isotopes"</a>.</p>
<div class="table block" id="averill_1.0-ch01_s06_t02" frame="all">
<p class="title"><span class="title-prefix">Table 1.5</span> Properties of Selected Isotopes</p>
<table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center">Element</th>
<th align="center">Symbol</th>
<th align="center">Atomic Mass (amu)</th>
<th align="center">Isotope Mass Number</th>
<th align="center">Isotope Masses (amu)</th>
<th align="center">Percent Abundances (%)</th>
</tr>
</thead>
<tbody>
<tr>
<td rowspan="2">hydrogen</td>
<td rowspan="2">H</td>
<td rowspan="2" align="right">1.0079</td>
<td align="right">1</td>
<td align="right">1.007825</td>
<td align="right">99.9855</td>
</tr>
<tr>
<td>2</td>
<td>2.014102</td>
<td align="right">0.0115</td>
</tr>
<tr>
<td rowspan="2">boron</td>
<td rowspan="2">B</td>
<td rowspan="2" align="right">10.81</td>
<td align="right">10</td>
<td align="right">10.012937</td>
<td align="right">19.91</td>
</tr>
<tr>
<td>11</td>
<td>11.009305</td>
<td align="right">80.09</td>
</tr>
<tr>
<td rowspan="2">carbon</td>
<td rowspan="2">C</td>
<td rowspan="2" align="right">12.011</td>
<td align="right">12</td>
<td align="right">12 (defined)</td>
<td align="right">99.89</td>
</tr>
<tr>
<td>13</td>
<td>13.003355</td>
<td align="right">1.11</td>
</tr>
<tr>
<td rowspan="3">oxygen</td>
<td rowspan="3">O</td>
<td rowspan="3" align="right">15.9994</td>
<td align="right">16</td>
<td align="right">15.994915</td>
<td align="right">99.757</td>
</tr>
<tr>
<td>17</td>
<td>16.999132</td>
<td align="right">0.0378</td>
</tr>
<tr>
<td>18</td>
<td>17.999161</td>
<td align="right">0.205</td>
</tr>
<tr>
<td rowspan="4">iron</td>
<td rowspan="4">Fe</td>
<td rowspan="4" align="right">55.845</td>
<td align="right">54</td>
<td align="right">53.939611</td>
<td align="right">5.82</td>
</tr>
<tr>
<td>56</td>
<td>55.934938</td>
<td align="right">91.66</td>
</tr>
<tr>
<td>57</td>
<td>56.935394</td>
<td align="right">2.19</td>
</tr>
<tr>
<td>58</td>
<td>57.933276</td>
<td align="right">0.33</td>
</tr>
<tr>
<td rowspan="3">uranium</td>
<td rowspan="3">U</td>
<td rowspan="3" align="right">238.03</td>
<td align="right">234</td>
<td align="right">234.040952</td>
<td align="right">0.0054</td>
</tr>
<tr>
<td>235</td>
<td>235.043930</td>
<td align="right">0.7204</td>
</tr>
<tr>
<td>238</td>
<td>238.050788</td>
<td align="right">99.274</td>
</tr>
</tbody>
</table>
<div class="copyright">
<p class="para">Sources of isotope data: G. Audi et al., <em class="emphasis">Nuclear Physics A</em> 729 (2003): 337–676; J. C. Kotz and K. F. Purcell, <em class="emphasis">Chemistry and Chemical Reactivity</em>, 2nd ed., 1991.</p>
</div>
</div>
<div class="exercises block" id="averill_1.0-ch01_s06_n02">
<h3 class="title">Example 5</h3>
<p class="para" id="averill_1.0-ch01_s06_p06">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.</p>
<p class="para" id="averill_1.0-ch01_s06_p07"><strong class="emphasis bold">Given: </strong>number of protons and neutrons</p>
<p class="para" id="averill_1.0-ch01_s06_p08"><strong class="emphasis bold">Asked for: </strong>element and atomic symbol</p>
<p class="para" id="averill_1.0-ch01_s06_p09">
<strong class="emphasis bold">Strategy:</strong>
</p>
<p class="para" id="averill_1.0-ch01_s06_p10"><strong class="emphasis bold">A</strong> Refer to the periodic table (see <a class="xref" href="s36-appendix-h-periodic-table-of-e.html">Chapter 32 "Appendix H: Periodic Table of Elements"</a>) and use the number of protons to identify the element.</p>
<p class="para" id="averill_1.0-ch01_s06_p11"><strong class="emphasis bold">B</strong> Calculate the mass number of each isotope by adding together the numbers of protons and neutrons.</p>
<p class="para" id="averill_1.0-ch01_s06_p12"><strong class="emphasis bold">C</strong> 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.</p>
<p class="para" id="averill_1.0-ch01_s06_p13">
<strong class="emphasis bold">Solution:</strong>
</p>
<p class="para" id="averill_1.0-ch01_s06_p14"><strong class="emphasis bold">A</strong> The element with 82 protons (atomic number of 82) is lead: Pb.</p>
<p class="para" id="averill_1.0-ch01_s06_p15"><strong class="emphasis bold">B</strong> For the first isotope, <em class="emphasis">A</em> = 82 protons + 124 neutrons = 206. Similarly, <em class="emphasis">A</em> = 82 + 125 = 207 and <em class="emphasis">A</em> = 82 + 126 = 208 for the second and third isotopes, respectively. The symbols for these isotopes are <span class="inlineequation"><math xml:id="averill_1.0-ch01_m012" display="inline"><semantics><mrow><mmultiscripts><mtext>P</mtext><mprescripts></mprescripts><mrow><mn>82</mn></mrow><mrow><mn>206</mn></mrow></mmultiscripts><mtext>b</mtext><mo>,</mo></mrow></semantics></math></span><span class="inlineequation"><math xml:id="averill_1.0-ch01_m013" display="inline"><semantics><mrow><mmultiscripts><mtext>P</mtext><mprescripts></mprescripts><mrow><mn>82</mn></mrow><mrow><mn>207</mn></mrow></mmultiscripts><mtext>b,</mtext></mrow></semantics></math></span> and <span class="inlineequation"><math xml:id="averill_1.0-ch01_m014" display="inline"><semantics><mrow><mmultiscripts><mtext>P</mtext><mprescripts></mprescripts><mrow><mn>82</mn></mrow><mrow><mn>208</mn></mrow></mmultiscripts><mtext>b,</mtext></mrow></semantics></math></span> which are usually abbreviated as <sup class="superscript">206</sup>Pb, <sup class="superscript">207</sup>Pb, and <sup class="superscript">208</sup>Pb.</p>
<p class="simpara">Exercise</p>
<p class="para" id="averill_1.0-ch01_s06_p16">Identify the element with 35 protons and write the symbols for its isotopes with 44 and 46 neutrons.</p>
<p class="para" id="averill_1.0-ch01_s06_p17"><strong class="emphasis bold">Answer: </strong><span class="inlineequation"><math xml:id="averill_1.0-ch01_m015" display="inline"><semantics><mrow><mmultiscripts><mtext>B</mtext><mprescripts></mprescripts><mrow><mn>35</mn></mrow><mrow><mn>79</mn></mrow></mmultiscripts><mtext>r</mtext></mrow></semantics></math></span> and <span class="inlineequation"><math xml:id="averill_1.0-ch01_m016" display="inline"><semantics><mrow><mmultiscripts><mtext>B</mtext><mprescripts></mprescripts><mrow><mn>35</mn></mrow><mrow><mn>81</mn></mrow></mmultiscripts><mtext>r</mtext></mrow></semantics></math></span> or, more commonly, <sup class="superscript">79</sup>Br and <sup class="superscript">81</sup>Br.</p>
</div>
<p class="para editable block" id="averill_1.0-ch01_s06_p18">Although the masses of the electron, the proton, and the neutron are known to a high degree of precision (<a class="xref" href="averill_1.0-ch01_s05#averill_1.0-ch01_s05_t01">Table 1.3 "Properties of Subatomic Particles*"</a>), 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 <sup class="superscript">1</sup>H (hydrogen) and <sup class="superscript">2</sup>H (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 (<a class="xref" href="averill_1.0-ch20#averill_1.0-ch20">Chapter 20 "Nuclear Chemistry"</a>).</p>
<p class="para editable block" id="averill_1.0-ch01_s06_p19">Because atoms are much too small to measure individually and do not have a charge, there is no convenient way to accurately measure <em class="emphasis">absolute</em> atomic masses. Scientists can measure <em class="emphasis">relative</em> atomic masses very accurately, however, using an instrument called a <em class="emphasis">mass spectrometer</em>. 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 <span class="margin_term"><a class="glossterm">ions</a><span class="glossdef">A charged particle produced when one or more electrons is removed from or added to an atom or molecule.</span></span>. 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 (<a class="xref" href="#averill_1.0-ch01_s06_f02">Figure 1.25 "Determining Relative Atomic Masses Using a Mass Spectrometer"</a>). 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 <em class="emphasis">absolute</em> atomic masses cannot be measured, but <em class="emphasis">relative</em> 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.</p>
<div class="figure large editable block" id="averill_1.0-ch01_s06_f02">
<p class="title"><span class="title-prefix">Figure 1.25</span> Determining Relative Atomic Masses Using a Mass Spectrometer</p>
<img src="section_05/ce4e100228e41595807a2be40259746f.jpg">
<p class="para">Chlorine consists of two isotopes, <sup class="superscript">35</sup>Cl and <sup class="superscript">37</sup>Cl, 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 Cl<sub class="subscript">2</sub> molecules into chlorine atoms and convert the chlorine atoms to Cl<sup class="superscript">+</sup> 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 <sup class="superscript">35</sup>Cl<sup class="superscript">+</sup> ions are deflected more than the heavier <sup class="superscript">37</sup>Cl<sup class="superscript">+</sup> 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.</p>
</div>
<p class="para block" id="averill_1.0-ch01_s06_p20">The arbitrary standard that has been established for describing atomic mass is the <span class="margin_term"><a class="glossterm">atomic mass unit (amu)</a><span class="glossdef">One-twelfth of the mass of one atom of <span class="inlineequation"><math xml:id="averill_1.0-ch01_m017" display="inline"><semantics><mrow><mmultiscripts><mtext>C</mtext><mprescripts></mprescripts><none></none><mrow><mtext>12</mtext></mrow></mmultiscripts></mrow></semantics></math></span>; <span class="inlineequation"><math xml:id="averill_1.0-ch01_m018" display="inline"><semantics><mrow><mtext>1 amu </mtext><mo>=</mo><mtext> 1</mtext><mo>.</mo><mtext>66 </mtext><mo>×</mo><mtext> 1</mtext><msup><mn>0</mn><mrow><mo>−</mo><mtext>24</mtext></mrow></msup><mtext>g</mtext></mrow></semantics></math></span>.</span></span>, defined as one-twelfth of the mass of one atom of <sup class="superscript">12</sup>C. Because the masses of all other atoms are calculated relative to the <sup class="superscript">12</sup>C standard, <sup class="superscript">12</sup>C is the only atom listed in <a class="xref" href="#averill_1.0-ch01_s06_t02">Table 1.5 "Properties of Selected Isotopes"</a> whose exact atomic mass is equal to the mass number. Experiments have shown that 1 amu = 1.66 × 10<sup class="superscript">−24</sup> g.</p>
<p class="para editable block" id="averill_1.0-ch01_s06_p21">Mass spectrometric experiments give a value of 0.167842 for the ratio of the mass of <sup class="superscript">2</sup>H to the mass of <sup class="superscript">12</sup>C, so the absolute mass of <sup class="superscript">2</sup>H is</p>
<span class="informalequation block">
<math xml:id="averill_1.0-ch01_m019" display="block">
<semantics>
<mrow>
<mfrac>
<mrow>
<mtext>mass of </mtext>
<mmultiscripts>
<mtext>H</mtext>
<mprescripts></mprescripts>
<none></none>
<mtext>2</mtext>
</mmultiscripts>
</mrow>
<mrow>
<menclose notation="updiagonalstrike">
<mrow>
<mtext>mass of </mtext>
<mmultiscripts>
<mtext>C</mtext>
<mprescripts></mprescripts>
<none></none>
<mrow>
<mtext>12</mtext>
</mrow>
</mmultiscripts>
</mrow>
</menclose>
</mrow>
</mfrac>
<mo>×</mo>
<menclose notation="updiagonalstrike">
<mrow>
<mtext>mass of </mtext>
<mmultiscripts>
<mtext>C</mtext>
<mprescripts></mprescripts>
<none></none>
<mrow>
<mtext>12</mtext>
</mrow>
</mmultiscripts>
</mrow>
</menclose>
<mo>=</mo>
<mtext> 0</mtext>
<mtext>.167842 </mtext>
<mo>×</mo>
<mtext> 12 amu </mtext>
<mo>=</mo>
<mtext> 2</mtext>
<mtext>.104104 amu</mtext>
</mrow>
</semantics>
</math>
</span>
<p class="para editable block" id="averill_1.0-ch01_s06_p22">The masses of the other elements are determined in a similar way.</p>
<p class="para editable block" id="averill_1.0-ch01_s06_p23">The periodic table (see <a class="xref" href="s36-appendix-h-periodic-table-of-e.html">Chapter 32 "Appendix H: Periodic Table of Elements"</a>) lists the atomic masses of all the elements. If you compare these values with those given for some of the isotopes in <a class="xref" href="#averill_1.0-ch01_s06_t02">Table 1.5 "Properties of Selected Isotopes"</a>, 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 <strong class="emphasis bold">atomic mass</strong> 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% <sup class="superscript">12</sup>C (mass = 12 amu by definition) and 1.11% <sup class="superscript">13</sup>C (mass = 13.003355 amu). The percent abundance of <sup class="superscript">14</sup>C is so low that it can be ignored in this calculation. The <em class="emphasis">average</em> atomic mass of carbon is then calculated as</p>
<span class="informalequation block">
<span class="mathphrase">(0.9889 × 12 amu) + (0.0111 × 13.003355 amu) = 12.01 amu</span>
</span>
<p class="para editable block" id="averill_1.0-ch01_s06_p24">Carbon is predominantly <sup class="superscript">12</sup>C, so its average atomic mass should be close to 12 amu, which is in agreement with our calculation.</p>
<p class="para editable block" id="averill_1.0-ch01_s06_p25">The value of 12.01 is shown under the symbol for C in the periodic table (see <a class="xref" href="s36-appendix-h-periodic-table-of-e.html">Chapter 32 "Appendix H: Periodic Table of Elements"</a>), although without the abbreviation <em class="emphasis">amu</em>, which is customarily omitted. Thus the tabulated <em class="emphasis">atomic mass</em> of carbon or any other element is the weighted average of the masses of the naturally occurring isotopes.</p>
<div class="exercises block" id="averill_1.0-ch01_s06_n03">
<h3 class="title">Example 6</h3>
<p class="para" id="averill_1.0-ch01_s06_p26">Naturally occurring bromine consists of the two isotopes listed in the following table:</p>
<div class="informaltable" frame="all">
<table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center">Isotope</th>
<th align="center">Exact Mass (amu)</th>
<th align="center">Percent Abundance (%)</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<sup class="superscript">79</sup>Br</td>
<td align="right">78.9183</td>
<td align="right">50.69</td>
</tr>
<tr>
<td>
<sup class="superscript">81</sup>Br</td>
<td align="right">80.9163</td>
<td align="right">49.31</td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="averill_1.0-ch01_s06_p27">Calculate the atomic mass of bromine.</p>
<p class="para" id="averill_1.0-ch01_s06_p28"><strong class="emphasis bold">Given: </strong>exact mass and percent abundance</p>
<p class="para" id="averill_1.0-ch01_s06_p29"><strong class="emphasis bold">Asked for: </strong>atomic mass</p>
<p class="para" id="averill_1.0-ch01_s06_p30">
<strong class="emphasis bold">Strategy:</strong>
</p>
<p class="para" id="averill_1.0-ch01_s06_p31"><strong class="emphasis bold">A</strong> Convert the percent abundances to decimal form to obtain the mass fraction of each isotope.</p>
<p class="para" id="averill_1.0-ch01_s06_p32"><strong class="emphasis bold">B</strong> Multiply the exact mass of each isotope by its corresponding mass fraction (percent abundance ÷ 100) to obtain its weighted mass.</p>
<p class="para" id="averill_1.0-ch01_s06_p33"><strong class="emphasis bold">C</strong> Add together the weighted masses to obtain the atomic mass of the element.</p>
<p class="para" id="averill_1.0-ch01_s06_p34"><strong class="emphasis bold">D</strong> Check to make sure that your answer makes sense.</p>
<p class="para" id="averill_1.0-ch01_s06_p35">
<strong class="emphasis bold">Solution:</strong>
</p>
<p class="para" id="averill_1.0-ch01_s06_p36"><strong class="emphasis bold">A</strong> The atomic mass is the weighted average of the masses of the isotopes. In general, we can write</p>
<span class="informalequation">
<span class="mathphrase">atomic mass of element = [(mass of isotope 1 in amu) (mass fraction of isotope 1)] + [(mass of isotope 2) (mass fraction of isotope 2)] + …</span>
</span>
<p class="para" id="averill_1.0-ch01_s06_p37">Bromine has only two isotopes. Converting the percent abundances to mass fractions gives</p>
<span class="informalequation">
<math xml:id="averill_1.0-ch01_m020" display="block">
<semantics>
<mtable columnalign="left">
<mtr>
<mtd>
<mmultiscripts>
<mtext>B</mtext>
<mprescripts></mprescripts>
<none></none>
<mrow>
<mn>79</mn>
</mrow>
</mmultiscripts>
<mtext>r: </mtext>
<mfrac>
<mrow>
<mn>50.69</mn>
</mrow>
<mrow>
<mn>100</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.5069</mn>
<mtext> </mtext>
</mtd>
</mtr>
<mtr>
<mtd>
<mmultiscripts>
<mtext>B</mtext>
<mprescripts></mprescripts>
<none></none>
<mrow>
<mn>81</mn>
</mrow>
</mmultiscripts>
<mtext>r: </mtext>
<mfrac>
<mrow>
<mn>49.31</mn>
</mrow>
<mrow>
<mn>100</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.4931</mn>
</mtd>
</mtr>
</mtable>
</semantics>
</math>
</span>
<p class="para" id="averill_1.0-ch01_s06_p38"><strong class="emphasis bold">B</strong> Multiplying the exact mass of each isotope by the corresponding mass fraction gives the isotope’s weighted mass:</p>
<span class="informalequation">
<span class="mathphrase"><sup class="superscript">79</sup>Br: 79.9183 amu × 0.5069 = 40.00 amu</span>
</span>
<span class="informalequation">
<span class="mathphrase"><sup class="superscript">81</sup>Br: 80.9163 amu × 0.4931 = 39.90 amu</span>
</span>
<p class="para" id="averill_1.0-ch01_s06_p39"><strong class="emphasis bold">C</strong> The sum of the weighted masses is the atomic mass of bromine is</p>
<span class="informalequation">
<span class="mathphrase">40.00 amu + 39.90 amu = 79.90 amu</span>
</span>
<p class="para" id="averill_1.0-ch01_s06_p40"><strong class="emphasis bold">D</strong> This value is about halfway between the masses of the two isotopes, which is expected because the percent abundance of each is approximately 50%.</p>
<p class="simpara">Exercise</p>
<p class="para" id="averill_1.0-ch01_s06_p41">Magnesium has the three isotopes listed in the following table:</p>
<div class="informaltable" frame="all">
<table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center">Isotope</th>
<th align="center">Exact Mass (amu)</th>
<th align="center">Percent Abundance (%)</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<sup class="superscript">24</sup>Mg</td>
<td align="right">23.98504</td>
<td align="right">78.70</td>
</tr>
<tr>
<td>
<sup class="superscript">25</sup>Mg</td>
<td align="right">24.98584</td>
<td align="right">10.13</td>
</tr>
<tr>
<td>
<sup class="superscript">26</sup>Mg</td>
<td align="right">25.98259</td>
<td align="right">11.17</td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="averill_1.0-ch01_s06_p42">Use these data to calculate the atomic mass of magnesium.</p>
<p class="para" id="averill_1.0-ch01_s06_p43"><strong class="emphasis bold">Answer: </strong>24.31 amu</p>
</div>
<div class="callout editable block" id="averill_1.0-ch01_s06_n04">
<h3 class="title">Summary</h3>
<p class="para" id="averill_1.0-ch01_s06_p44">Each atom of an element contains the same number of protons, which is the <strong class="emphasis bold">atomic number</strong> (<em class="emphasis bolditalic">Z</em>). Neutral atoms have the same number of electrons and protons. Atoms of an element that contain different numbers of neutrons are called <strong class="emphasis bold">isotopes</strong>. Each isotope of a given element has the same atomic number but a different <strong class="emphasis bold">mass number</strong> (<em class="emphasis bolditalic">A</em>), which is the sum of the numbers of protons and neutrons. The relative masses of atoms are reported using the <strong class="emphasis bold">atomic mass unit</strong> (<strong class="emphasis bold">amu</strong>), which is defined as one-twelfth of the mass of one atom of carbon-12, with 6 protons, 6 neutrons, and 6 electrons. The <strong class="emphasis bold">atomic mass</strong> 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 <strong class="emphasis bold">ion</strong> is produced, whose charge is indicated by a superscript after the symbol.</p>
</div>
<div class="key_takeaways editable block" id="averill_1.0-ch01_s06_n05">
<h3 class="title">Key Takeaway</h3>
<ul class="itemizedlist" id="averill_1.0-ch01_s06_l02">
<li>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.</li>
</ul>
</div>
<div class="qandaset block" id="averill_1.0-ch01_s06_qs01" defaultlabel="number">
<h3 class="title">Conceptual Problems</h3>
<ol class="qandadiv" id="averill_1.0-ch01_s06_qs01_qd01">
<li class="qandaentry" id="averill_1.0-ch01_s06_qs01_qd01_qa01">
<div class="question">
<p class="para">Complete the following table for the missing elements, symbols, and numbers of electrons.</p>
<div class="informaltable" frame="all">
<table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center">Element</th>
<th align="center">Symbol</th>
<th align="center">Number of Electrons</th>
</tr>
</thead>
<tbody>
<tr>
<td>molybdenum</td>
<td></td>
<td align="right"></td>
</tr>
<tr>
<td></td>
<td></td>
<td align="right">19</td>
</tr>
<tr>
<td>titanium</td>
<td></td>
<td align="right"></td>
</tr>
<tr>
<td></td>
<td>B</td>
<td align="right"></td>
</tr>
<tr>
<td></td>
<td></td>
<td align="right">53</td>
</tr>
<tr>
<td></td>
<td>Sm</td>
<td align="right"></td>
</tr>
<tr>
<td>helium</td>
<td></td>
<td align="right"></td>
</tr>
<tr>
<td></td>
<td></td>
<td align="right">14</td>
</tr>
</tbody>
</table>
</div>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch01_s06_qs01_qd01_qa02">
<div class="question">
<p class="para">Complete the following table for the missing elements, symbols, and numbers of electrons.</p>
<div class="informaltable" frame="all">
<table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center">Element</th>
<th align="center">Symbol</th>
<th align="center">Number of Electrons</th>
</tr>
</thead>
<tbody>
<tr>
<td>lanthanum</td>
<td></td>
<td align="right"></td>
</tr>
<tr>
<td></td>
<td>Ir</td>
<td align="right"></td>
</tr>
<tr>
<td>aluminum</td>
<td></td>
<td align="right"></td>
</tr>
<tr>
<td></td>
<td></td>
<td align="right">80</td>
</tr>
<tr>
<td>sodium</td>
<td></td>
<td align="right"></td>
</tr>
<tr>
<td></td>
<td>Si</td>
<td align="right"></td>
</tr>
<tr>
<td></td>
<td></td>
<td align="right">9</td>
</tr>
<tr>
<td></td>
<td>Be</td>
<td align="right"></td>
</tr>
</tbody>
</table>
</div>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch01_s06_qs01_qd01_qa03">
<div class="question">
<p class="para">Is the mass of an ion the same as the mass of its parent atom? Explain your answer.</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch01_s06_qs01_qd01_qa04">
<div class="question">
<p class="para">What isotopic standard is used for determining the mass of an atom?</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch01_s06_qs01_qd01_qa05">
<div class="question">
<p class="para">Give the symbol <span class="inlineequation"><math xml:id="averill_1.0-ch01_m021" display="inline"><semantics><mrow><mmultiscripts><mtext>X</mtext><mprescripts></mprescripts><mi>Z</mi><mi>A</mi></mmultiscripts></mrow></semantics></math></span> for these elements, all of which exist as a single isotope.</p>
<ol class="orderedlist" numeration="loweralpha">
<li>beryllium</li>
<li>ruthenium</li>
<li>phosphorus</li>
<li>aluminum</li>
<li>cesium</li>
<li>praseodymium</li>
<li>cobalt</li>
<li>yttrium</li>
<li>arsenic</li>
</ol>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch01_s06_qs01_qd01_qa06">
<div class="question">
<p class="para">Give the symbol <span class="inlineequation"><math xml:id="averill_1.0-ch01_m022" display="inline"><semantics><mrow><mmultiscripts><mtext>X</mtext><mprescripts></mprescripts><mi>Z</mi><mi>A</mi></mmultiscripts></mrow></semantics></math></span> for these elements, all of which exist as a single isotope.</p>
<ol class="orderedlist" numeration="loweralpha">
<li>fluorine</li>
<li>helium</li>
<li>terbium</li>
<li>iodine</li>
<li>gold</li>
<li>scandium</li>
<li>sodium</li>
<li>niobium</li>
<li>manganese</li>
</ol>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch01_s06_qs01_qd01_qa07">
<div class="question">
<p class="para">Identify each element, represented by X, that have the given symbols.</p>
<ol class="orderedlist" numeration="loweralpha">
<li>
<span class="inlineequation">
<math xml:id="averill_1.0-ch01_m023" display="inline">
<semantics>
<mrow>
<mmultiscripts>
<mtext>X</mtext>
<mprescripts></mprescripts>
<mrow>
<mn>26</mn>
</mrow>
<mrow>
<mn>55</mn>
</mrow>
</mmultiscripts>
</mrow>
</semantics>
</math>
</span>
</li>
<li>
<span class="inlineequation">
<math xml:id="averill_1.0-ch01_m024" display="inline">
<semantics>
<mrow>
<mmultiscripts>
<mtext>X</mtext>
<mprescripts></mprescripts>
<mrow>
<mn>33</mn>
</mrow>
<mrow>
<mn>74</mn>
</mrow>
</mmultiscripts>
</mrow>
</semantics>
</math>
</span>
</li>
<li>
<span class="inlineequation">
<math xml:id="averill_1.0-ch01_m025" display="inline">
<semantics>
<mrow>
<mmultiscripts>
<mtext>X</mtext>
<mprescripts></mprescripts>
<mrow>
<mn>12</mn>
</mrow>
<mrow>
<mn>24</mn>
</mrow>
</mmultiscripts>
</mrow>
</semantics>
</math>
</span>
</li>
<li>
<span class="inlineequation">
<math xml:id="averill_1.0-ch01_m026" display="inline">
<semantics>
<mrow>
<mmultiscripts>
<mtext>X</mtext>
<mprescripts></mprescripts>
<mrow>
<mn>53</mn>
</mrow>
<mrow>
<mn>127</mn>
</mrow>
</mmultiscripts>
</mrow>
</semantics>
</math>
</span>
</li>
<li>
<span class="inlineequation">
<math xml:id="averill_1.0-ch01_m027" display="inline">
<semantics>
<mrow>
<mmultiscripts>
<mtext>X</mtext>
<mprescripts></mprescripts>
<mrow>
<mn>18</mn>
</mrow>
<mrow>
<mn>40</mn>
</mrow>
</mmultiscripts>
</mrow>
</semantics>
</math>
</span>
</li>
<li>
<span class="inlineequation">
<math xml:id="averill_1.0-ch01_m028" display="inline">
<semantics>
<mrow>
<mmultiscripts>
<mtext>X</mtext>
<mprescripts></mprescripts>
<mrow>
<mn>63</mn>
</mrow>
<mrow>
<mn>152</mn>
</mrow>
</mmultiscripts>
</mrow>
</semantics>
</math>
</span>
</li>
</ol>
</div>
</li>
</ol>
</div>
<div class="qandaset block" id="averill_1.0-ch01_s06_qs02" defaultlabel="number">
<h3 class="title">Numerical Problems</h3>
<ol class="qandadiv" id="averill_1.0-ch01_s06_qs02_qd01">
<p class="para">
<em class="emphasis">Please be sure you are familiar with the topics discussed in Essential Skills 1 (<a class="xref" href="averill_1.0-ch01_s09#averill_1.0-ch01_s09">Section 1.9 "Essential Skills 1"</a>) before proceeding to the Numerical Problems.</em>
</p>
<li class="qandaentry" id="averill_1.0-ch01_s06_qs02_qd01_qa02">
<div class="question">
<p class="para">The isotopes <sup class="superscript">131</sup>I and <sup class="superscript">60</sup>Co are commonly used in medicine. Determine the number of neutrons, protons, and electrons in a neutral atom of each.</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch01_s06_qs02_qd01_qa01">
<div class="question">
<p class="para">Determine the number of protons, neutrons, and electrons in a neutral atom of each isotope:</p>
<ol class="orderedlist" numeration="loweralpha">
<li>
<sup class="superscript">97</sup>Tc</li>
<li>
<sup class="superscript">113</sup>In</li>
<li>
<sup class="superscript">63</sup>Ni</li>
<li>
<sup class="superscript">55</sup>Fe</li>
</ol>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch01_s06_qs02_qd01_qa04">
<div class="question">
<p class="para">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.</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch01_s06_qs02_qd01_qa03">
<div class="question">
<p class="para">The following isotopes are important in archaeological research. How many protons, neutrons, and electrons does a neutral atom of each contain?</p>
<ol class="orderedlist" numeration="loweralpha">
<li>
<sup class="superscript">207</sup>Pb</li>
<li>
<sup class="superscript">16</sup>O</li>
<li>
<sup class="superscript">40</sup>K</li>
<li>
<sup class="superscript">137</sup>Cs</li>
<li>
<sup class="superscript">40</sup>Ar</li>
</ol>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch01_s06_qs02_qd01_qa06">
<div class="question">
<p class="para">Copper, an excellent conductor of heat, has two isotopes: <sup class="superscript">63</sup>Cu and <sup class="superscript">65</sup>Cu. Use the following information to calculate the average atomic mass of copper:</p>
<div class="informaltable" frame="all">
<table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center">Isotope</th>
<th align="center">Percent Abundance (%)</th>
<th align="center">Atomic Mass (amu)</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<sup class="superscript">63</sup>Cu</td>
<td align="right">69.09</td>
<td align="right">62.9298</td>
</tr>
<tr>
<td>
<sup class="superscript">65</sup>Cu</td>
<td align="right">30.92</td>
<td align="right">64.9278</td>
</tr>
</tbody>
</table>
</div>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch01_s06_qs02_qd01_qa05">
<div class="question">
<p class="para">Silicon consists of three isotopes with the following percent abundances:</p>
<div class="informaltable" frame="all">
<table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center">Isotope</th>
<th align="center">Percent Abundance (%)</th>
<th align="center">Atomic Mass (amu)</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<sup class="superscript">28</sup>Si</td>
<td align="right">92.18</td>
<td align="right">27.976926</td>
</tr>
<tr>
<td>
<sup class="superscript">29</sup>Si</td>
<td align="right">4.71</td>
<td align="right">28.976495</td>
</tr>
<tr>
<td>
<sup class="superscript">30</sup>Si</td>
<td align="right">3.12</td>
<td align="right">29.973770</td>
</tr>
</tbody>
</table>
</div>
<p class="para">Calculate the average atomic mass of silicon.</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch01_s06_qs02_qd01_qa07">
<div class="question">
<p class="para">Complete the following table for neon. The average atomic mass of neon is 20.1797 amu.</p>
<div class="informaltable" frame="all">
<table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center">Isotope</th>
<th align="center">Percent Abundance (%)</th>
<th align="center">Atomic Mass (amu)</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<sup class="superscript">20</sup>Ne</td>
<td align="right">90.92</td>
<td align="right">19.99244</td>
</tr>
<tr>
<td>
<sup class="superscript">21</sup>Ne</td>
<td align="right">0.257</td>
<td align="right">20.99395</td>
</tr>
<tr>
<td>
<sup class="superscript">22</sup>Ne</td>
<td align="right"></td>
<td align="right"></td>
</tr>
</tbody>
</table>
</div>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch01_s06_qs02_qd01_qa08">
<div class="question">
<p class="para">Are <span class="inlineequation"><math xml:id="averill_1.0-ch01_m029" display="inline"><semantics><mrow><mmultiscripts><mtext>X</mtext><mprescripts></mprescripts><mrow><mn>28</mn></mrow><mrow><mn>63</mn></mrow></mmultiscripts></mrow></semantics></math></span> and <span class="inlineequation"><math xml:id="averill_1.0-ch01_m030" display="inline"><semantics><mrow><mmultiscripts><mtext>X</mtext><mprescripts></mprescripts><mrow><mn>29</mn></mrow><mrow><mn>62</mn></mrow></mmultiscripts></mrow></semantics></math></span> isotopes of the same element? Explain your answer.</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch01_s06_qs02_qd01_qa10">
<div class="question">
<p class="para">Complete the following table:</p>
<div class="informaltable" frame="all">
<table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center">Isotope</th>
<th align="center">Number of Protons</th>
<th align="center">Number of Neutrons</th>
<th align="center">Number of Electrons</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<sup class="superscript">238</sup>X</td>
<td align="right"></td>
<td align="right"></td>
<td align="right">95</td>
</tr>
<tr>
<td>
<sup class="superscript">238</sup>U</td>
<td align="right"></td>