Calculating the Energy of a Nuclear Reaction with the Mass–Energy Relationship What is the energy, in joules and in megaelectronvolts, associated with the α decay of [latex]^{238}U[/latex]? [latex]^{238}_{92}U → ^{234}_{90}Th + ^{4}_{2}He[/latex] The nuclidic (atomic) masses in atomic mass units (u) are from Table D.5 in Appendix D: [latex]^{238}_{92}U[/latex] = 238.0508 u [latex]^{234}_{90}Th[/latex] = 234.0437 u [latex]^{4}_{2}He[/latex] = 4.0026 u

Analyze The key concept here is the fact that, during a nuclear reaction, a loss or gain of mass is balanced by a gain or loss of energy. We need to determine the loss or gain of mass and then use the conversion factors (25.21) and (25.22) to convert the loss or gain of mass to the corresponding amount of energy. 1 atomic mass unit (u) = 1.4924 × [latex]10^{−10}[/latex] J (25.21) 1 atomic mass unit (u) = 1.4924 × [latex]10^{−10} J × \frac{1 MeV}{1.6022 × 10^{−13} J}[/latex] = 931.5 MeV (25.22) Solve The net change in mass that accompanies the decay of a single nucleus of [latex]^{238}U[/latex] is shown below. Note that the masses of the extranuclear electrons do not enter into the calculation of the net change in mass. change in mass = nuclear mass of [latex]^{234}_{90}Th[/latex] + nuclear mass of [latex]^{4}_{2}He[/latex] − nuclear mass of [latex]^{238}_{92}U[/latex] = [234.0437 u − (90 × mass [latex]e^{−}[/latex])] + [4.0026 u − (2 × mass [latex]e^{−}[/latex])] − [238.0508 u − (92 × mass [latex]e^{−}[/latex])] = 234.0437 u + 4.0026 u − 238.0508 u − 92 × mass [latex]e^{−}[/latex] + 92 × mass [latex]e^{−}[/latex] = −0.0045 u We can use this loss of mass and conversion factors (25.21) and (25.22) to write E = −0.0045 u × [latex]\frac{1.49 × 10^{−10} J}{u} = −6.7 × 10^{−13}[/latex] J or E = −0.0045 u × [latex](\frac{931.5 MeV}{u})[/latex] = −4.2 MeV Assess The negative sign denotes that energy is lost in the nuclear reaction. This is the kinetic energy of the departing α particle. Note that you can use either nuclear or nuclidic (atomic) masses in calculations based on a nuclear equation. The change in mass will be the same in either case because the masses of the electrons will cancel out.

Question 25.5: Calculating the Energy of a Nuclear Reaction with the Mass–E......

General Chemistry Principles and Modern Applications [1096733]

Calculating the Energy of a Nuclear Reaction with the Mass–Energy Relationship

What is the energy, in joules and in megaelectronvolts, associated with the α decay of $^{238}U$ ?

$^{238}_{92}U → ^{234}_{90}Th + ^{4}_{2}He$

The nuclidic (atomic) masses in atomic mass units (u) are from Table D.5 in Appendix D:

$^{238}_{92}U$ = 238.0508 u $^{234}_{90}Th$ = 234.0437 u $^{4}_{2}He$ = 4.0026 u

$\begin{aligned} &\text { TABLE D.5 Isotopic Masses and Their Abundance }{ }^a\\ &\begin{array}{|c|c|c|c|c|} \hline z & \text { Name } & \text { Symbol } & \text { Mass of Atom, } u & \% \text { Abundance } \\ \hline {1} & \text { Hydrogen } & { }^1 \mathrm{H} & 1.007825 & 99.9885 \\ \hline & \text { Deuterium } & { }^2 \mathrm{H} & 2.014102 & 0.0115 \\ \hline & \text { Tritium } & { }^3 \mathrm{H} & 3.016049 & – \\ \hline {2} & \text { Helium } & { }^3 \mathrm{He} & 3.016029 & 0.000137 \\ \hline & & { }^4 \mathrm{He} & 4.002603 & 99.999863 \\ \hline {3} & \text { Lithium } & { }^6 \mathrm{Li} & 6.015122 & 7.59 \\ \hline & & { }^7 \mathrm{Li} & 7.016004 & 92.41 \\ \hline 4 & \text { Beryllium } & { }^9 \mathrm{Be} & 9.012182 & 100 \\ \hline {5} & \text { Boron } & { }^{10} \mathrm{~B} & 10.012937 & 19.9 \\ \hline & & { }^{11} \mathrm{~B} & 11.009305 & 80.1 \\ \hline {6} & \text { Carbon } & { }^{12} \mathrm{C} & 12.000000 & 98.93 \\ \hline & & { }^{13} \mathrm{C} & 13.003355 & 1.07 \\ \hline & & { }^{14} \mathrm{C} & 14.003242 & – \\ \hline {7} & \text { Nitrogen } & { }^{14} \mathrm{~N} & 14.003074 & 99.632 \\ \hline & & { }^{15} \mathrm{~N} & 15.000109 & 0.368 \\ \hline {8} & \text { Oxygen } & { }^{16} \mathrm{O} & 15.994915 & 99.757 \\ \hline & & { }^{17} \mathrm{O} & 16.999132 & 0.038 \\ \hline & & { }^{18} \mathrm{O} & 17.999160 & 0.205 \\ \hline 9 & \text { Fluorine } & { }^{19} \mathrm{~F} & 18.998403 & 100 \\ \hline {10} & \text { Neon } & { }^{20} \mathrm{Ne} & 19.992440 & 90.48 \\ \hline & & { }^{21} \mathrm{Ne} & 20.993847 & 0.27 \\ \hline & & { }^{22} \mathrm{Ne} & 21.991386 & 9.25 \\ \hline 11 & \text { Sodium } & { }^{23} \mathrm{Na} & 22.989770 & 100 \\ \hline {12} & \text { Magnesium } & { }^{24} \mathrm{Mg} & 23.985042 & 78.99 \\ \hline & & { }^{25} \mathrm{Mg} & 24.985837 & 10.00 \\ \hline & & { }^{26} \mathrm{Mg} & 25.982593 & 11.01 \\ \hline 13 & \text { Aluminum } & { }^{27} {\mathrm{Al}} & 26.981538 & 100 \\ \hline{14} & \text { Silicon } & ^{28} \mathrm{Si} & 27.976927 & 92.2297 \\ \hline & & ^{29} \mathrm{Si} & 28.976495 & 4.6832 \\ \hline & & ^{30} \mathrm{Si} & 29.973770 & 3.0872 \\ \hline 15 & \text { Phosphorus } & { }^{31} \mathrm{p} & 30.973762 & 100 \\ \hline {16} & \text { Sulfur } & { }^{32} S & 31.972071 & 94.93 \\ \hline & & { }^{33} \mathrm{~S} & 32.971458 & 0.76 \\ \hline & & ^{34} S & 33.967867 & 4.29 \\ \hline & & ^{36} \mathrm{~S} & 35.967081 & 0.02 \\ \hline{17} & \text { Chlorine } & { }^{35} \mathrm{Cl} & 34.968853 & 75.78 \\ \hline & & { }^{37} \mathrm{Cl} & 36.965903 & 24.22 \\ \hline{18} & \text { Argon } & { }^{36} \mathrm{Ar} & 35.967546 & 0.3365 \\ \hline & & { }^{38} \mathrm{Ar} & 37.962732 & 0.0632 \\ \hline & & { }^{40} \mathrm{Ar} & 39.962383 & 99.6003 \\ \hline {19} & \text { Potassium } & { }^{39} \mathrm{~K} & 38.963707 & 93.2581 \\ \hline & & { }^{40} \mathrm{~K} & 39.963999 & 0.0117 \\ \hline & & { }^{41} \mathrm{~K} & 40.961826 & 6.7302 \\ \hline{20} & \text { Calcium } & { }^{40} \mathrm{Ca} & 39.962591 & 96.941 \\ \hline & & { }^{42} \mathrm{Ca} & 41.958618 & 0.647 \\ \hline & & { }^{43} \mathrm{Ca} & 42.958767 & 0.135 \\ \hline & & { }^{44} \mathrm{Ca} & 43.955481 & 2.086 \\ \hline & & { }^{46} \mathrm{Ca} & 45.953693 & 0.004 \\ \hline & & { }^{48} \mathrm{Ca} & 47.952534 & 0.187 \\ \hline \end{array} \end{aligned}$

$^{a}$ The isotopic mass data are from G. Audi and A. H. Wapstra, and M. Dedieu, Nuclear Physics A, volume 565, pages 1-65 (1993) and G. Audi and A. H. Wapstra, Nuclear Physics A, volume 595, pages 409-480 (1995). The percent natural abundance data are from K.J.R. Rosman and P.D.P. Taylor, Pure and Applied Chemistry, volume 70, pages 217-235 (1998).

$\begin{array}{|c|c|c|c|c|} \hline \text { Z } & \text { Name } & \text { Symbol } & \text { Mass of Atom, } u & \text { \% Abundance } \\ \hline 21 & \text { Scandium } & { }^{45} \mathrm{Sc} & 44.955910 & 100 \\ \hline {22} & \text { Titanium } & { }^{46} \mathrm{Ti} & 45.952629 & 8.25 \\ \hline & & { }^{47} \mathrm{Ti} & 46.951764 & 7.44 \\ \hline & & { }^{48} \mathrm{Ti} & 47.947947 & 73.72 \\ \hline & & { }^{49} \mathrm{Ti} & 48.947871 & 5.41 \\ \hline & & { }^{50} \mathrm{Ti} & 49.944792 & 5.18 \\ \hline {23} & \text { Vanadium } & { }^{50} \mathrm{~V} & 49.947163 & 0.250 \\ \hline & & { }^{51} \mathrm{~V} & 50.943964 & 99.750 \\ \hline {24} & \text { Chromium } & { }^{50} \mathrm{Cr} & 49.946050 & 4.345 \\ \hline & & { }^{52} \mathrm{Cr} & 51.940512 & 83.789 \\ \hline & & { }^{53} \mathrm{Cr} & 52.940654 & 9.501 \\ \hline & & { }^{54} \mathrm{Cr} & 53.938885 & 2.365 \\ \hline 25 & \text { Manganese } & { }^{58} \mathrm{Mn} & 54.938050 & 100 \\ \hline 26& \text { Iron } & { }^{54} \mathrm{Fe} & 53.939615 & 5.845 \\ \hline & & { }^{56} \mathrm{Fe} & 55.934942 & 91.754 \\ \hline & & { }^{57} \mathrm{Fe} & 56.935399 & 2.119 \\ \hline & & { }^{58} \mathrm{Fe} & 57.933280 & 0.282 \\ \hline 27 & \text { Cobalt } & { }^{59} \mathrm{Co} & 58.933200 & 100 \\ \hline {28} & \text { Nickel } & { }^{58} \mathrm{Ni} & 57.935348 & 68.0769 \\ \hline & & { }^{60} \mathrm{Ni} & 59.930791 & 26.2231 \\ \hline & & { }^{61} \mathrm{Ni} & 60.931060 & 1.1399 \\ \hline & & { }^{62} \mathrm{Ni} & 61.928349 & 3.6345 \\ \hline & & { }^{64} \mathrm{Ni} & 63.927970 & 0.9256 \\ \hline {29} & \text { Copper } & { }^{63} \mathrm{Cu} & 62.929601 & 69.17 \\ \hline & & { }^{65} \mathrm{Cu} & 64.927794 & 30.83 \\ \hline {30} & \text { Zinc } & { }^{64} \mathrm{Zn} & 63.929147 & 48.63 \\ \hline & & { }^{66} \mathrm{Zn} & 65.926037 & 27.90 \\ \hline & & { }^{67} \mathrm{Zn} & 66.927131 & 4.10 \\ \hline & & { }^{68} \mathrm{Zn} & 67.924848 & 18.75 \\ \hline & & { }^{30} \mathrm{Zn} & 69.925325 & 0.62 \\ \hline {31} & \text { Gallium } & { }^{69} \mathrm{Ga} & 68.925581 & 60.108 \\ \hline & & { }^{71} \mathrm{Ga} & 70.924705 & 39.892 \\ \hline {32} & \text { Germanium } & { }^{70} \mathrm{Ge} & 69.924250 & 20.84 \\ \hline && { }^{72} \mathrm{Ge} & 71.922076 & 27.54 \\ \hline & & { }^{73} \mathrm{Ge} & 72.923459 & 7.73 \\ \hline & & { }^{74} \mathrm{Ge} & 73.921178 & 36.28 \\ \hline & & { }^{76} \mathrm{Ge} & 75.921403 & 7.61 \\ \hline 33 & \text { Arsenic } & { }^{75} \mathrm{As} & 74.921596 & 100 \\ \hline {34} & \text { Selenium } & { }^{74} \mathrm{Se} & 73.922477 & 0.89 \\ \hline & &^{76} \mathrm{Se} & 75.919214 & 9.37 \\ \hline & & { }^{77} \mathrm{Se} & 76.919915 & 7.63 \\ \hline & & { }^{78} \mathrm{Se} & 77.917310 & 23.77 \\ \hline & & { }^{80} \mathrm{Se} & 79.916522 & 49.61 \\ \hline & & { }^{82} \mathrm{Se} & 81.916700 & 8.73 \\ \hline {35} & \text { Bromine } & { }^{79} \mathrm{Br} & 78.918338 & 50.69 \\ \hline & & { }^{81} \mathrm{Br} & 80.916291 & 49.31 \\ \hline {36} & \text { Krypton } & { }^{78} \mathrm{Kr} & 77.920386 & 0.35 \\ \hline & & { }^{80} \mathrm{Kr} & 79.916378 & 2.28 \\ \hline & & { }^{82} \mathrm{Kr} & 81.913485 & 11.58 \\ \hline & & { }^{83} \mathrm{Kr} & 82.914136 & 11.49 \\ \hline & & { }^{84} \mathrm{Kr} & 83.911507 & 57.00 \\ \hline & & { }^{86} \mathrm{Kr} & 85.910610 & 17.30 \\ \hline \end{array}$

$\begin{array}{|c|c|c|c|c|} \hline Z & \text { Name } & \text { Symbol } & \text { Mass of Atom, } u & \% \text { Abundance } \\ \hline {37} & \text { Rubidium } & { }^{85} \mathrm{Rb} & 84.911789 & 72.17 \\ \hline & & { }^{87} \mathrm{Rb} & 86.909183 & 27.83 \\ \hline {38} & \text { Strontium } & ^{84} \mathrm{Sr} & 83.913425 & 0.56 \\ \hline & & ^{86} \mathrm{Sr} & 85.909262 & 9.86 \\ \hline & & { }^{87} \mathrm{Sr} & 86.908879 & 7.00 \\ \hline & & { }^{88} \mathrm{Sr} & 87.905614 & 82.58 \\ \hline 39 & \text { Yttrium } & { }^{89} \mathrm{Y} & 88.905848 & 100 \\ \hline {40} & \text { Zirconium } & ^{90} \mathrm{Zr} & 89.904704 & 51.45 \\ \hline & & ^{91} \mathrm{Zr} & 90.905645 & 11.22 \\ \hline & & { }^{92} \mathrm{Zr} & 91.905040 & 17.15 \\ \hline & & ^{94} \mathrm{Zr} & 93.906316 & 17.38 \\ \hline & & ^{96} \mathrm{Zr} & 95.908276 & 2.80 \\ \hline 41 & \text { Niobium } & { }^{93} \mathrm{Nb} & 92.906378 & 100 \\ \hline {42} & \text { Molybdenum } & { }^{92} \mathrm{Mo} & 91.906810 & 14.84 \\ \hline & & { }^{94} \mathrm{Mo} & 93.905088 & 9.25 \\ \hline & & { }^{95} \mathrm{Mo} & 94.905841 & 15.92 \\ \hline & & { }^{96} \mathrm{Mo} & 95.904679 & 16.68 \\ \hline & & { }^{97} \mathrm{Mo} & 96.906021 & 9.55 \\ \hline & & { }^{98} \mathrm{Mo} & 97.905408 & 24.13 \\ \hline & & { }^{100} \mathrm{Mo} & 99.907477 & 9.63 \\ \hline 43 & \text { Technetium } & { }^{98} \mathrm{Tc} & 97.907216 & – \\ \hline {44} & \text { Ruthenium } & { }^{96} \mathrm{Ru} & 95.907598 & 5.54 \\ \hline & & { }^{98} \mathrm{Ru} & 97.905287 & 1.87 \\ \hline & & { }^{99} \mathrm{Ru} & 98.905939 & 12.76 \\ \hline & & { }^{100} \mathrm{Ru} & 99.904220 & 12.60 \\ \hline & & { }^{101} \mathrm{Ru} & 100.905582 & 17.06 \\ \hline & & { }^{102} \mathrm{Ru} & 101.904350 & 31.55 \\ \hline & & { }^{104} \mathrm{Ru} & 103.905430 & 18.62 \\ \hline 45 & \text { Rhodium } & { }^{103} \mathrm{Rh} & 102.905504 & 100 \\ \hline {46} & \text { Palladium } & { }^{102} \mathrm{Pd} & 101.905608 & 1.02 \\ \hline & & { }^{104} \mathrm{Pd} & 103.904035 & 11.14 \\ \hline & & { }^{105} \mathrm{Pd} & 104.905084 & 22.33 \\ \hline & & { }^{106} \mathrm{Pd} & 105.903483 & 27.33 \\ \hline & & { }^{105 \mathrm{Pd}} & 107.903894 & 26.46 \\ \hline & & { }^{110} \mathrm{Pd} & 109.905152 & 11.72 \\ \hline {47} & \text { Silver } & { }^{100} \mathrm{Ag} & 106.905093 & 51.839 \\ \hline & & { }^{109} \mathrm{Ag} & 108.904756 & 48.161 \\ \hline {48} & \text { Cadmium } & { }^{106} \mathrm{Cd} & 105.906458 & 1.25 \\ \hline & & { }_{108} \mathrm{Cd} & 107.904183 & 0.89 \\ \hline & & { }^{110} \mathrm{Cd} & 109.903006 & 12.49 \\ \hline & & { }^{111} \mathrm{Cd} & 110.904182 & 12.80 \\ \hline & & { }^{112} \mathrm{Cd} & 111.902757 & 24.13 \\ \hline & & { }^{113} \mathrm{Cd} & 112.904401 & 12.22 \\ \hline & & { }^{114} \mathrm{Cd} & 113.903358 & 28.73 \\ \hline & & { }^{116} \mathrm{Cd} & 115.904755 & 7.49 \\ \hline {49} & \text { Indium } & { }^{113} \text { In } & 112.904061 & 4.29 \\ \hline & & { }^{115} \text { In } & 114.903878 & 95.71 \\ \hline {50} & \text { Tin } & { }^{112} \mathrm{Sn} & 111.904821 & 0.97 \\ \hline & & { }^{114} \mathrm{Sn} & 113.902782 & 0.66 \\ \hline & & { }^{115} \mathrm{Sn} & 114.903346 & 0.34 \\ \hline \end{array}$

$\begin{array}{|c|c|c|c|c|} \hline Z & \text { Name } & \text { Symbol } & \text { Mass of Atom, } u & \text { \% Abundance } \\ \hline {50} & \text { Tin (continued) } & { }^{116} \mathrm{Sn} & 115.901744 & 14.54 \\ \hline & & { }^{117} \mathrm{Sn} & 116.902954 & 7.68 \\ \hline & & { }^{118} \mathrm{Sn} & 117.901606 & 24.22 \\ \hline & & { }^{119} \mathrm{Sn} & 118.903309 & 8.59 \\ \hline & & { }^{120} \mathrm{Sn} & 119.902197 & 32.58 \\ \hline & & { }^{122} \mathrm{Sn} & 121.903440 & 4.63 \\ \hline & & { }^{124} \mathrm{Sn} & 123.905275 & 5.79 \\ \hline {51} & \text { Antimony } & { }^{121} \mathrm{Sb} & 120.903818 & 57.21 \\ \hline & & { }^{123} \mathrm{Sb} & 122.904216 & 42.79 \\ \hline {52} & \text { Tellurium } & { }^{120} \mathrm{Te} & 119.904020 & 0.09 \\ \hline & & { }^{122} \mathrm{Te} & 121.903047 & 2.55 \\ \hline & & { }^{123} \mathrm{Te} & 122.904273 & 0.89 \\ \hline & & { }^{124} \mathrm{Te} & 123.902819 & 4.74 \\ \hline & & { }^{125} \mathrm{Te} & 124.904425 & 7.07 \\ \hline & & { }^{126} \mathrm{Te} & 125.903306 & 18.84 \\ \hline & & { }^{128} \mathrm{Te} & 127.904461 & 31.74 \\ \hline & & { }^{130} \mathrm{Te} & 129.906223 & 34.08 \\ \hline 53 & \text { Iodine } & { }^{127} \mathrm{I} & 126.904468 & 100 \\ \hline {54} & \text { Xenon } & { }^{124} \mathrm{Xe} & 123.905896 & 0.09 \\ \hline & & { }^{126} \mathrm{Xe} & 125.904269 & 0.09 \\ \hline & & { }^{128} \mathrm{Xe} & 127.903530 & 1.92 \\ \hline & & { }^{129} \mathrm{Xe} & 128.904779 & 26.44 \\ \hline & & { }^{130} \mathrm{Xe} & 129.903508 & 4.08 \\ \hline & & { }^{131} \mathrm{Xe} & 130.905082 & 21.18 \\ \hline & & { }^{132} \mathrm{Xe} & 131.904154 & 26.89 \\ \hline & & { }^{134} \mathrm{Xe} & 133.905395 & 10.44 \\ \hline & & { }^{136} \mathrm{Xe} & 135.907220 & 8.87 \\ \hline 55 & \text { Cesium } & { }^{133} \mathrm{Cs}_s & 132.905447 & 100 \\ \hline {56} & \text { Barium } & { }^{130} \mathrm{Ba} & 129.906310 & 0.106 \\ \hline & & { }^{132} \mathrm{Ba} & 131.905056 & 0.101 \\ \hline & & { }^{134} \mathrm{Ba} & 133.904503 & 2.417 \\ \hline & & { }^{135} \mathrm{Ba} & 134.905683 & 6.592 \\ \hline & & { }^{136} \mathrm{Ba} & 135.904570 & 7.854 \\ \hline & & { }^{137} \mathrm{Ba} & 136.905821 & 11.232 \\ \hline & & { }^{138} \mathrm{Ba} & 137.905241 & 71.698 \\ \hline{57} & \text { Lanthanum } & { }^{138} \mathrm{La} & 137.907107 & 0.090 \\ \hline & & { }^{139} \mathrm{La} & 138.906348 & 99.910 \\ \hline {58} & \text { Cerium } & { }^{136} \mathrm{Ce} & 135.907144 & 0.185 \\ \hline & & { }^{138} \mathrm{Ce} & 137.905986 & 0.251 \\ \hline & & { }^{140} \mathrm{Ce} & 139.905434 & 88.450 \\ \hline & & { }^{142} \mathrm{Ce} & 141.909240 & 11.114 \\ \hline 59 & \text { Praseodymium } & { }^{141} \mathrm{Pr} & 140.907648 & 100 \\ \hline {60} & \text { Neodymium } & { }^{142} \mathrm{Nd} & 141.907719 & 27.2 \\ \hline & & { }^{143} \mathrm{Nd} & 142.909810 & 12.2 \\ \hline & & { }^{144} \mathrm{Nd} & 143.910083 & 23.8 \\ \hline & & { }^{145} \mathrm{Nd} & 144.912569 & 8.3 \\ \hline & & { }^{146} \mathrm{Nd} & 145.913112 & 17.2 \\ \hline & & { }^{148} \mathrm{Nd} & 147.916889 & 5.7 \\ \hline & & { }^{150} \mathrm{Nd} & 149.920887 & 5.6 \\ \hline 61 & \text { Promethium } & { }^{145} \mathrm{Pm} & 144.912744 & – \\ \hline \end{array}$

$\begin{array}{|c|c|c|c|c|} \hline \text { Z } & \text { Name } & \text { Symbol } & \text { Mass of Atom, } u & \text { \% Abundance } \\ \hline {62} & \text { Samarium } & { }^{144} \mathrm{Sm} & 143.911995 & 3.07 \\ \hline & & ^{146} \mathrm{Sm} & 145.913041 & \\ \hline & & { }^{147} \mathrm{Sm} & 146.914893 & 14.99 \\ \hline & & { }^{14} \mathrm{Sm} & 147.914818 & 11.24 \\ \hline & & { }^{199} \mathrm{Sm} & 148.917180 & 13.82 \\ \hline & & ^{150} \mathrm{Sm} & 149.917271 & 7.38 \\ \hline & & { }^{152} \mathrm{Sm} & 151.919728 & 26.75 \\ \hline & & { }^{154} \mathrm{Sm} & 153.922205 & 22.75 \\ \hline {63} & \text { Europium } & { }^{151} \mathrm{Eu} & 150.919846 & 47.81 \\ \hline & & { }^{153} \mathrm{Eu} & 152.921226 & 52.19 \\ \hline {64} & \text { Gadolinium } & { }^{152} \mathrm{Gd} & 151.919788 & 0.20 \\ \hline & & { }^{154} \mathrm{Gd} & 153.920862 & 2.18 \\ \hline & & { }^{155} \mathrm{Gd} & 154.922619 & 14.80 \\ \hline & & { }^{156} \mathrm{Gd} & 155.922120 & 20.47 \\ \hline & & { }^{157} \mathrm{Gd} & 156.923957 & 15.65 \\ \hline & & { }^{159} \mathrm{Gd} & 157.924101 & 24.84 \\ \hline & & { }^{169} \mathrm{Gd} & 159.927051 & 21.86 \\ \hline 65 & \text { Terbium } & { }^{159} \mathrm{~Tb} & 158.925343 & 100 \\ \hline {66} & \text { Dysprosium } & { }^{156} \mathrm{Dy} & 155.924278 & 0.06 \\ \hline & & { }^{158} \mathrm{Dy} & 157.924405 & 0.10 \\ \hline & & { }^{160} \mathrm{Dy} & 159.925194 & 2.34 \\ \hline & & { }^{161} \mathrm{Dy} & 160.926930 & 18.91 \\ \hline & & { }^{162} \mathrm{Dy} & 161.926795 & 25.51 \\ \hline & & { }^{163} \mathrm{Dy} & 162.928728 & 24.90 \\ \hline & & { }^{164} \mathrm{Dy} & 163.929171 & 28.18 \\ \hline 67 & \text { Holmium } & { }^{165} \mathrm{Ho} & 164.930319 & 100 \\ \hline {68} & \text { Erbium } & { }^{162} \mathrm{Er} & 161.928775 & 0.14 \\ \hline & & { }^{164} \mathrm{Er} & 163.929197 & 1.61 \\ \hline & & { }^{166} \mathrm{Er} & 165.930290 & 33.61 \\ \hline & & { }^{167} \mathrm{Er} & 166.932045 & 22.93 \\ \hline & & { }^{168} \mathrm{Er} & 167.932368 & 26.78 \\ \hline & & { }^{170} \mathrm{Er} & 169.935460 & 14.93 \\ \hline 69 & \text { Thulium } & { }^{169} \mathrm{Tm} & 168.934211 & 100 \\ \hline {70} & \text { Ytterbium } & { }^{168} \mathrm{Yb} & 167.933894 & 0.13 \\ \hline & & { }^{170} \mathrm{Yb} & 169.934759 & 3.04 \\ \hline & & { }^{171} \mathrm{Yb} & 170.936322 & 14.28 \\ \hline & & { }^{172} \mathrm{Yb} & 171.936378 & 21.83 \\ \hline & & { }^{173} \mathrm{Yb} & 172.938207 & 16.13 \\ \hline & & { }^{174} \mathrm{Yb} & 173.938858 & 31.83 \\ \hline & & { }^{176} \mathrm{Yb} & 175.942568 & 12.76 \\ \hline {71} & \text { Lutetium } & { }^{175} \mathrm{Lu} & 174.940768 & 97.41 \\ \hline & & { }^{176} \mathrm{Lu} & 175.942682 & 2.59 \\ \hline {72} & \text { Hafnium } & { }^{175} \mathrm{Hf} & 173.940040 & 0.16 \\ \hline & & { }^{176} \mathrm{Hf} & 175.941402 & 5.26 \\ \hline & & { }^{177} {\mathrm{Hf}} & 176.943220 & 18.60 \\ \hline & & { }^{178} \mathrm{Hf} & 177.943698 & 27.28 \\ \hline & & { }^{179} \mathrm{Hf} & 178.945815 & 13.62 \\ \hline & & { }^{180} \mathrm{Hf} & 179.946549 & 35.08 \\ \hline {73} & \text { Tantalum } & { }^{180} \mathrm{Ta} & 179.947466 & 0.012 \\ \hline & & { }^{181} \mathrm{Ta} & 180.947996 & 99.988 \\ \hline \end{array}$

$\begin{array}{|c|c|c|c|c|} \hline \text { Z } & \text { Name } & \text { Symbol } & \text { Mass of Atom, } u & \text { \% Abundance } \\ \hline {74} & \text { Tungsten } & { }^{180} \mathrm{~W} & 179.946706 & 0.12 \\ \hline & & { }^{182} \mathrm{~W} & 181.948206 & 26.50 \\ \hline & & { }^{183} \mathrm{~W} & 182.950224 & 14.31 \\ \hline & & { }^{184} \mathrm{~W} & 183.950933 & 30.64 \\ \hline & & { }^{186} \mathrm{~W} & 185.954362 & 28.43 \\ \hline {75} & \text { Rhenium } & { }^{185} \mathrm{Re} & 184.952956 & 37.40 \\ \hline & & { }^{187} \mathrm{Re} & 186.955751 & 62.60 \\ \hline {76} & \text { Osmium } & { }^{184} \mathrm{Os}_5 & 183.952491 & 0.02 \\ \hline & & ^{186} \mathrm{Os}_5 & 185.953838 & 1.59 \\ \hline & & { }^{187} \mathrm{Os} & 186.955748 & 1.96 \\ \hline & & { }^{188} \mathrm{Os}_5 & 187.955836 & 13.24 \\ \hline & & { }^{189} \mathrm{Os}_5 & 188.958145 & 16.15 \\ \hline & & { }^{190} \mathrm{Os} & 189.958445 & 26.26 \\ \hline & & { }^{192} \mathrm{Os} & 191.961479 & 40.78 \\ \hline {77} & \text { Iridium } & { }^{191} \mathrm{Ir} & 190.960591 & 37.3 \\ \hline & & { }^{193} \mathrm{Ir} & 192.962924 & 62.7 \\ \hline {78} & \text { Platinum } & { }^{190} \mathrm{Pt} & 189.959930 & 0.014 \\ \hline & & { }^{192} \mathrm{Pt} & 191.961035 & 0.782 \\ \hline & & { }^{194} \mathrm{Pt} & 193.962664 & 32.967 \\ \hline & & { }^{195} \mathrm{Pt} & 194.964774 & 33.832 \\ \hline & & { }^{196} \mathrm{Pt} & 195.964935 & 25.242 \\ \hline & & { }^{198} \mathrm{Pt} & 197.967876 & 7.163 \\ \hline 79 & \text { Gold } & { }^{197} \mathrm{Au} & 196.966552 & 100 \\ \hline {80} & \text { Mercury } & { }^{196} \mathrm{Hg} & 195.965815 & 0.15 \\ \hline & & { }^{198} \mathrm{Hg} & 197.966752 & 9.97 \\ \hline & & { }^{199} \mathrm{Hg} & 198.968262 & 16.87 \\ \hline & & { }^{200} \mathrm{Hg} & 199.968309 & 23.10 \\ \hline & & { }^{201} \mathrm{Hg} & 200.970285 & 13.18 \\ \hline & & { }^{202} \mathrm{Hg} & 201.970626 & 29.86 \\ \hline & & { }^{204} \mathrm{Hg} & 203.973476 & 6.87 \\ \hline {81} & \text { Thallium } & { }^{203} \mathrm{TI} & 202.972329 & 29.524 \\ \hline & & { }^{205} \mathrm{TI} & 204.974412 & 70.476 \\ \hline {82} & \text { Lead } & { }^{204} \mathrm{~Pb} & 203.973029 & 1.4 \\ \hline & & { }^{206} \mathrm{~Pb} & 205.974449 & 24.1 \\ \hline & & { }^{207} \mathrm{~Pb} & 206.975881 & 22.1 \\ \hline & & { }^{208} \mathrm{~Pb} & 207.976636 & 52.4 \\ \hline 83 & \text { Bismuth } & { }^{209} \mathrm{Bi} & 208.980383 & 100 \\ \hline 84 & \text { Polonium } & { }^{209} \mathrm{Po} & 208.982416 & – \\ \hline 85 & \text { Astatine } & { }^{210} \mathrm{At} & 209.987131 & – \\ \hline 86 & \text { Radon } & { }^m \mathrm{Rn} & 222.017570 & – \\ \hline 87 & \text { Francium } & { }^{223} \mathrm{Fr} & 223.019731 & – \\ \hline 88 & \text { Radium } & { }^{226} \mathrm{Ra} & 226.025403 & – \\ \hline 89 & \text { Actinium } & { }^{22} \mathrm{Ac} & 227.027747 & – \\ \hline {90} & \text { Thorium } & { }^{232} \mathrm{Th} & 232.038050 & 100 \\ \hline & & { }^{234} \mathrm{Th} & 234.043601 & \\ \hline 91 & \text { Protactinium } & { }^{231} \mathrm{~Pa} & 231.035879 & 100 \\ \hline {92} & \text { Uranium } & { }^{23} U & 234.040946 & 0.0055 \\ \hline & & { }^{235} U & 235.043923 & 0.7200 \\ \hline & & { }^{238} \mathrm{U} & 238.050783 & 99.2745 \\ \hline 93 & \text { Neptunium } & { }^{237} \mathrm{~Np} & 237.048167 & -\\ \hline 94 & \text { Plutonium } & { }^{244} \mathrm{Pu} & 244.064198 &-\\ \hline 95 & \text { Americium } & { }^{243} \mathrm{Am} & 243.061373 &- \\ \hline \end{array}$

$\begin{array}{lllcc} \hline Z & \text { Name } & \text { Symbol } & \text { Mass of Atom, u } & \text { \% Abundance } \\ \hline 96 & \text { Curium } & { }^{247} \mathrm{Cm} & 247.070347 & – \\ 97 & \text { Berkelium } & { }^{247} \mathrm{Bk} & 247.070299 & – \\ 98 & \text { Californium } & { }^{251} \mathrm{Cf} & 251.079580 & – \\ 99 & \text { Einsteinium } & { }^{252} \mathrm{Es} & 252.082972 & – \\ 100 & \text { Fermium } & { }^{257} \mathrm{Fm} & 257.095099 & – \\ 101 & \text { Mendelevium } & { }^{258} \mathrm{Md} & 258.098425 & – \\ 102 & \text { Nobelium } & { }^{259} \mathrm{No} & 259.101024 & – \\ 103 & \text { Lawrencium } & { }^{262} \mathrm{Lr} & 262.109692 & – \\ 104 & \text { Rutherfordium } & { }^{263} \mathrm{Rf} & 263.118313 & – \\ 105 & \text { Dubnium } & { }^{262} \mathrm{Db} & 262.011437 & – \\ 106 & \text { Seaborgium } & { }^{2665 \mathrm{Sg}} & 266.012238 & – \\ 107 & \text { Bohrium } & { }^{264} \mathrm{Bh} & 264.012496 & – \\ 108 & \text { Hassium } & { }^{269} \mathrm{Hs} & 269.001341 & – \\ 109 & \text { Meitnerium } & { }^{268} \mathrm{Mt} & 268.001388 & – \\ 110 & \text { Darmstadtium } & { }^{272} \mathrm{Ds} & 272.001463 & – \\ 111 & \text { Roentgenium } & { }^{272} \mathrm{Rg} & 272.001535 & – \\ 112 & \text { Copernicium } & { }^{277} \mathrm{Cn} & (277) & – \\ 114 & \text { Flerovium } & { }^{289} \mathrm{FI} & (289) & – \\ 116 & \text { Livermorium } & { }^{289} \mathrm{Lv} & (289) & – \\ 118 & \text { Ununoctium } & { }^{293} \mathrm{Uuo} & (293) & – \\ \hline \end{array}$

Step-by-Step

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Analyze
The key concept here is the fact that, during a nuclear reaction, a loss or gain of mass is balanced by a gain or loss of energy. We need to determine the loss or gain of mass and then use the conversion factors (25.21) and (25.22) to convert the loss or gain of mass to the corresponding amount of energy.

1 atomic mass unit (u) = 1.4924 × $10^{−10}$ J (25.21)

1 atomic mass unit (u) = 1.4924 × $10^{−10} J × \frac{1 MeV}{1.6022 × 10^{−13} J}$ = 931.5 MeV (25.22)

Solve
The net change in mass that accompanies the decay of a single nucleus of $^{238}U$ is shown below. Note that the masses of the extranuclear electrons do not enter into the calculation of the net change in mass.

change in mass

= nuclear mass of $^{234}_{90}Th$ + nuclear mass of $^{4}_{2}He$ − nuclear mass of $^{238}_{92}U$

= [234.0437 u − (90 × mass $e^{−}$ )] + [4.0026 u − (2 × mass $e^{−}$ )] − [238.0508 u − (92 × mass $e^{−}$ )]

= 234.0437 u + 4.0026 u − 238.0508 u − 92 × mass $e^{−}$ + 92 × mass $e^{−}$

= −0.0045 u

We can use this loss of mass and conversion factors (25.21) and (25.22) to write

E = −0.0045 u × $\frac{1.49 × 10^{−10} J}{u} = −6.7 × 10^{−13}$ J

E = −0.0045 u × $(\frac{931.5 MeV}{u})$ = −4.2 MeV

Assess
The negative sign denotes that energy is lost in the nuclear reaction. This is the kinetic energy of the departing α particle. Note that you can use either nuclear or nuclidic (atomic) masses in calculations based on a nuclear equation. The change in mass will be the same in either case because the masses of the electrons will cancel out.