11.1 Nuclear structure and stability (Page 4/17)

Ut austin - principles of Page 4 / 17

A graph is shown where the x-axis is labeled “Number of neutrons, open parenthesis, n, close parenthesis” and has values of 0 to 180 in increments of 10. The y-axis is labeled “Number of protons, open parenthesis, Z, close parenthesis” and has values of 0 to 120 in increments of 10. A green shaded band of varying width, labeled “Radioactive,” extends from point 0 on both axes to 178 on the y-axis and 118 on the x-axis in a linear manner. The width of this band varies from 8 to 18 units in width according to the x-axis measurements. A blue line in a roughly zig-zag pattern runs through the middle of the shaded band and stops at 128 on the y-axis and 82 on the x-axis. This line is labeled “Nonradioactive.” An unlabeled, black, solid line extends from point 0, 0 to 120, 120 in a linear manner. — This plot shows the nuclides that are known to exist and those that are stable. The stable nuclides are indicated in blue, and the unstable nuclides are indicated in green. Note that all isotopes of elements with atomic numbers greater than 83 are unstable. The solid line is the line where n = Z.

The nuclei that are to the left or to the right of the band of stability are unstable and exhibit radioactivity . They change spontaneously (decay) into other nuclei that are either in, or closer to, the band of stability. These nuclear decay reactions convert one unstable isotope (or radioisotope ) into another, more stable, isotope. We will discuss the nature and products of this radioactive decay in subsequent sections of this chapter.

Several observations may be made regarding the relationship between the stability of a nucleus and its structure. Nuclei with even numbers of protons, neutrons, or both are more likely to be stable (see [link] ). Nuclei with certain numbers of nucleons, known as magic numbers , are stable against nuclear decay. These numbers of protons or neutrons (2, 8, 20, 28, 50, 82, and 126) make complete shells in the nucleus. These are similar in concept to the stable electron shells observed for the noble gases. Nuclei that have magic numbers of both protons and neutrons, such as $_{2}^{4} He,$ $_{8}^{16} O,$ $_{20}^{40} Ca,$ and $_{82}^{208} Pb,$ are called “double magic” and are particularly stable. These trends in nuclear stability may be rationalized by considering a quantum mechanical model of nuclear energy states analogous to that used to describe electronic states earlier in this textbook. The details of this model are beyond the scope of this chapter.

Stable Nuclear Isotopes
Number of Stable Isotopes	Proton Number	Neutron Number
157	even	even
53	even	odd
50	odd	even
5	odd	odd

The relative stability of a nucleus is correlated with its binding energy per nucleon , the total binding energy for the nucleus divided by the number or nucleons in the nucleus. For instance, we saw in [link] that the binding energy for a $_{2}^{4} He$ nucleus is 28.4 MeV. The binding energy per nucleon for a $_{2}^{4} He$ nucleus is therefore:

\frac{28.4 MeV}{4 nucleons} = 7.10 MeV/nucleon

In [link] , we learn how to calculate the binding energy per nucleon of a nuclide on the curve shown in [link] .

A graph is shown where the x-axis is labeled “binding energy per nucleon, open parenthesis, M e V, close parenthesis” and has values of 0 to 10 in increments of 1. The y-axis is labeled “Mass number” and has values of 0 to 260 in increments of 20. A line of best fit beginning at point 0, 0 is drawn through points “8, 5.5; 9, 7.3; 18, 7.1; 20, 7.5; 19, 7.9; 27, 7.8; 21, 8.1; 25, 8.4; 37, 8.6; 43, 8.8; 57, 8.6; 60, 8.9; 70, 9; 88, 8.8; 102, 8.9; 108, 8.5; 126, 8.7; 133, 8.8; 143, 8.2; 157, 8.1; 167, 8.2; 195, 7.9; 205, 7.9; 241, 7.3 and 255, 75. An upward-facing arrow near the bottom left of the graph is labeled “Fusion” while a left-facing arrow near the top right is labeled “Fission.” — The binding energy per nucleon is largest for nuclides with mass number of approximately 56.

Calculation of binding energy per nucleon

The iron nuclide

_{26}^{56} Fe

lies near the top of the binding energy curve ( [link] ) and is one of the most stable nuclides. What is the binding energy per nucleon (in MeV) for the nuclide

_{26}^{56} Fe

(atomic mass of 55.9349 amu)?

Solution

As in [link] , we first determine the mass defect of the nuclide, which is the difference between the mass of 26 protons, 30 neutrons, and 26 electrons, and the observed mass of an

_{26}^{56} Fe

atom:

\begin{array}{l} Mass defect = [(26 \times 1.0073 amu) + (30 \times 1.0087 amu) + (26 \times 0.00055 amu)] - 55.9349 amu \\ = 56.4651 amu - 55.9349 amu \\ = 0.5302 amu \end{array}

We next calculate the binding energy for one nucleus from the mass defect using the mass-energy equivalence equation:

\begin{array}{l} E = m c^{2} = 0.5302 amu \times \frac{1.6605 \times 10^{−27} kg}{1 amu} \times {(2.998 \times 10^{8} m/s)}^{2} \\ = 7.913 \times 10^{−11} kg ⋅ {m/s}^{2} \\ = 7.913 \times 10^{−11} J \end{array}

We then convert the binding energy in joules per nucleus into units of MeV per nuclide:

7.913 \times 10^{−11} J \times \frac{1 MeV}{1.602 \times 10^{−13} J} = 493.9 MeV

Finally, we determine the binding energy per nucleon by dividing the total nuclear binding energy by the number of nucleons in the atom:

Binding energy per nucleon = \frac{493.9 MeV}{56} = 8.820 MeV/nucleon

Note that this is almost 25% larger than the binding energy per nucleon for $_{2}^{4} He .$

(Note also that this is the same process as in [link] , but with the additional step of dividing the total nuclear binding energy by the number of nucleons.)

Check your learning

What is the binding energy per nucleon in

_{9}^{19} F

(atomic mass, 18.9984 amu)?

Answer:

7.810 MeV/nucleon

Got questions? Get instant answers now!

Key concepts and summary

An atomic nucleus consists of protons and neutrons, collectively called nucleons. Although protons repel each other, the nucleus is held tightly together by a short-range, but very strong, force called the strong nuclear force. A nucleus has less mass than the total mass of its constituent nucleons. This “missing” mass is the mass defect, which has been converted into the binding energy that holds the nucleus together according to Einstein’s mass-energy equivalence equation, E = mc ² . Of the many nuclides that exist, only a small number are stable. Nuclides with even numbers of protons or neutrons, or those with magic numbers of nucleons, are especially likely to be stable. These stable nuclides occupy a narrow band of stability on a graph of number of protons versus number of neutrons. The binding energy per nucleon is largest for the elements with mass numbers near 56; these are the most stable nuclei.

Key equations

E = mc ²

Chemistry end of chapter exercises

Write the following isotopes in hyphenated form (e.g., “carbon-14”)

(a) $_{11}^{24} Na$

(b) $_{13}^{29} Al$

(d) $_{77}^{194} Ir$

(a) sodium-24; (b) aluminum-29; (c) krypton-73; (d) iridium-194

Got questions? Get instant answers now!

Write the following isotopes in nuclide notation (e.g., $"_{6}^{14} C ")$

(a) oxygen-14

(b) copper-70

(d) francium-217

Got questions? Get instant answers now!

For the following isotopes that have missing information, fill in the missing information to complete the notation

(a) $_{14}^{34} X$

(b) $_{X}^{36} P$

(d) $_{56}^{121} X$

(a) $_{14}^{34} Si;$ (b) $_{15}^{36} P;$ (c) $_{25}^{57} Mn;$ (d) $_{56}^{121} Ba$

Got questions? Get instant answers now!

For each of the isotopes in [link] , determine the numbers of protons, neutrons, and electrons in a neutral atom of the isotope.

Got questions? Get instant answers now!

Write the nuclide notation, including charge if applicable, for atoms with the following characteristics:

(a) 25 protons, 20 neutrons, 24 electrons

(b) 45 protons, 24 neutrons, 43 electrons

(d) 97 protons, 146 neutrons, 97 electrons

(a) $_{25}^{45} {Mn}^{+1};$ (b) $_{45}^{69} {Rh}^{+2};$ (c) $_{53}^{142} I^{−1};$ (d) $_{97}^{243} Bk$

Got questions? Get instant answers now!

Calculate the density of the $_{12}^{24} Mg$ nucleus in g/mL, assuming that it has the typical nuclear diameter of 1 $\times$ 10 ^–13 cm and is spherical in shape.

Got questions? Get instant answers now!

What are the two principal differences between nuclear reactions and ordinary chemical changes?

Nuclear reactions usually change one type of nucleus into another; chemical changes rearrange atoms. Nuclear reactions involve much larger energies than chemical reactions and have measureable mass changes.

Got questions? Get instant answers now!

The mass of the atom $_{11}^{23} Na$ is 22.9898 amu.

(a) Calculate its binding energy per atom in millions of electron volts.

(b) Calculate its binding energy per nucleon.

Got questions? Get instant answers now!

Which of the following nuclei lie within the band of stability shown in [link] ?

(a) chlorine-37

(b) calcium-40

(d) ⁵⁶ Fe

(e) ²⁰⁶ Pb

(f) ²¹¹ Pb

(g) ²²² Rn

(h) carbon-14

(a), (b), (c), (d), and (e)

Got questions? Get instant answers now!

Which of the following nuclei lie within the band of stability shown in [link] ?

(a) argon-40

(b) oxygen-16

(d) ⁵⁸ Ni

(e) ²⁰⁵ Tl

(f) ²¹⁰ Tl

(g) ²²⁶ Ra

(h) magnesium-24

Got questions? Get instant answers now!

<< Chapter < Page Page > Chapter >>

Practice Key Terms 13

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Ut austin - principles of chemistry. OpenStax CNX. Mar 31, 2016 Download for free at http://legacy.cnx.org/content/col11830/1.13

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Ut austin - principles of chemistry' conversation and receive update notifications?

Ask

	Anthropology Life Cycle By Richley Crapo Start Assignment
©flickr: Aaron	Computer Literacy Exam 1 By Lakeima Roberts Start Quiz
	15 Dr. Amberg Pharm quiz By Brooke Delaney Start Exam
	19 AP Key Terms 19 Cardiovascular System Heart By OpenStax Start Key Terms
	NCE Ch 05 Theories in Counseling Helping... By Anh Dao Start Exam
©flickr:	Word Roots and Prefixes By Ellie Banfield Start Quiz
	4 Practice Neuroanatomy MCQ Exam By David Corey Start Exam
	11 Clin Sci I - GI Twedt quiz 2 By Brooke Delaney Start Exam
	Cardiac Electrophysiology Basic 2 By Mistry Bhavesh Start Test
	7 AP 07 Axial Skeleton MCQ By OpenStax Start Quiz