Mass defect and binding energy

The atomic mass of any other atom is less than the sum of the

masses of the protons, neutrons and electrons present.

Mass defect is a measure of the binding energy of the protons

and neutrons in the nucleus.

The loss in mass and liberation of energy are related by Einstein’s

equation

!E = !mc2

where !E = energy liberated,

!m = loss of mass, and

c = speed of light in a vacuum = 2.998 x 108 m/s

Worked example: Nuclear binding energy

Assuming that the mass defect originates solely from the

interaction of protons and neutrons in the nucleus, estimate the

nuclear binding energy of 73Li given the following data:

Observed atomic mass of 73Li = 7.01600 u

1u = 1.66054 x 10-27 kg

Electron rest mass = 9.10939 x 1031 kg

Proton rest mass = 1.67262 x 1027 kg

Neutron rest mass = 1.67493 x 1027 kg

c = 2.998 x 108 m/s

The actual mass of a 73Li atom

= 7.01600 u x 1.66054 x 1027 kg/u

=1.16503 x 1026 kg

The sum of the masses of the protons, neutrons and electrons in

a 73Li atom

= (3 x 9.10939 x 10-31) + (3 x 1.67262 x 10-27)

+ (4 x 1.67493 x 10-27)

= 1.17203 x 10-26 kg

The difference in mass = !m

= (1.17203 x 10-26 ) - (1.16503 x 10-26 )

= 0.00700 x 10-26 kg

The difference !E = !mc2

= (0.00700 x 10-26 ) x (2.998 x 108)2 kg m2 s-2

= 6.29160 x 10-12 J per atom (J = kg m2 s-2)

This corresponds to 3.79 x 1012 J or 3.79 x 109 kJ per mole of nuclei

3. Estimate the nuclear binding energy of 168O in J per atom,

given that the observed atomic mass is 15.99491 u. Other

data you require are given above.

Exercises:

1. Estimate the nuclear binding energy of 42He given that its

observed atomic mass is 4.002 60 u; other necessary data are

given above.

[Ans. 4:53 x 10-12 J per atom]

2. If the nuclear binding energy of an atom of 94Be is 9.3182 x 10-12 J per atom, calculate the atomic mass of 9

4Be; other

necessary data are given above.

[Ans. 9.012 18 u]

[Ans. 2.045 x 10-11 J per atom]

The average binding energy per nucleon

The average binding energy per nucleon is BE per atom divided

by the sum of protons and neutrons.

For 73Li, binding energy per nucleon

!

=6.29x10

"12

7= 8.98x10

"13J

BE per nucleon values represent the energy released per

nucleon upon the formation of the nucleus from its fundamental

particles.

These values can be used to give a measure of the relative

stabilities of nuclei with respect to decomposition into those

particles.

The nucleus with the greatest

binding energy is 5626Fe and this is

therefore the most stable nucleus.

Nuclei with Z around 60 have the highest

average BE per nucleon.

Variation in average binding energy per nucleon as a function of A.

- stable Fe and Ni are believed to constitute

the bulk of the Earth’s core

Nuclei with A = 4, 12 and 16 have

relatively high BE per nucleon.

Variation in average binding energy per nucleon as a function of A.

- these are used as projectiles in the synthesis

of heavy nuclei

The BE per nucleon decreases

remarkably for A > 100.

Magic Numbers for Nuclear Stability

No. of Protons: 2 8 20 28 50 82 114

No. of Neutrons: 2 8 20 28 50 82 126 184

Distribution of Naturally Occurring Stable Nuclides

Combination Number of Nuclides

Z even - N even 163Z even - N odd 55

Z odd - N even 50

Z odd - N odd 4

Neutron-to-proton ratio and the stability of nuclides

Possible heavy nuclides of high stability

(long radioactive half-lives)

Naturally occurring stable nuclides

(ranging from H to Bi)

Naturally occurring and

artificially made radioactive

nuclides

BE values are important for the application of nuclear reactions as

energy sources.

A reaction involving nuclei will be exothermic if:

. a heavy nucleus is divided into two nuclei of medium mass (so-

called nuclear fission), or

. two light nuclei are combined to give one nucleus of medium

mass (so-called nuclear fusion).

Radioactivity

Nuclear emissions

Nuclear transformations are usually very slow, but even so

spontaneous changes of many heavy nuclides occur.

e.g. 23892U and 232

90Th

3 Types of emission recognized by Rutherford:

1.. Alpha (") decay - involves emission of alpha particles, 42" or

(42He2+)

!

92

238U"

90

234Th+

2

4He + #e.g.

# " decay results in a decrease in the atomic number by 2 units and the

mass number by 4 units.

2. Beta ($#)decay - involves emission of electrons (0-1e or 0

-1!) from

the nucleus (called beta particles).

# " easily captures two electrons; it can be represented as 42He.

e.g.

!

6

14C"

7

14N+#1

0$

$# decay results in an increase in the atomic number by one and leaves

the mass number unchanged.

# a neutron is converted into a proton

3. Gamma (%) emission (high energy X-rays) usually accompanies

other types of nuclear emission.

% emission leaves the atomic number and mass number unchanged.

The decay of some nuclei also involves the emission of other types

of particle:

4. Positron ($+) decay - involves emission of positron (like electron

except that its charge is +)

!

8

15O"

7

15N++1

0#e.g.

$+ decay results in a decrease in the atomic number by one and leaves

the mass number unchanged.

5. Neutrino (&e)- an “electron neutrino” is associated with electron

but is uncharged, possesses near zero mass, accompany

positron emission.

Muon Neutrino (&µ) - associated with muon w/c is a heavier

version (200x in mass) of electron

Two other types of neutrino:

Tau Neutrino (&') - associated with tau w/c is a heavier version

(2500x mass of e-)of electron and muon

The electron, muon and tau, and their corresponding neutrinos

constitute the 6 leptons, which are seen as elementary particles.

The other elementary particles are the 6 “flavors” of quarks:

u (up), d (down), c (charm), s (strange), t (top), b (bottom)

Quarks occur only in combinations of

two quarks : mesons (are bosons; have integer spin;

three quarks: baryons (are fermions; have half-integral

spin)(e.g. proton(uud) and neutron (udd))

five quarks: pentaquarks

6. Antineutrinos - the corresponding antimatter of the neutrinos.

A comparison of the penetrating powers of "-particles, $-

particles, %-radiation and neutrons. Neutrons are especially

penetrating and their use in a nuclear reactor calls for

concrete-wall shields of 2m in thickness.

Nuclear transformations

Many nuclear reactions, as opposed to ordinary chemical

reactions, change the identity of (transmute) the starting

element.

Steps involving the loss of an "- or $-particle may be part of a

decay series.!

92

238U"

90

234Th+

2

4He+

0

0#Examples

!

6

14C"

7

14N+#1

0$

The decay series from 23892U to

20682Pb. Only the last nuclide in

the series, 20682Pb, is stable with

respect to further decay.

Exercise: Three of the nuclides are

not labeled. What are their

identities?