Nuclei: common doubts

Question 1

What is one atomic mass unit?

Accepted Answer

One atomic mass unit is one-twelfth of the mass of a carbon-12 atom: 1 u = 1.660539 × 10⁻²⁷ kg. Kilograms are clumsy for such tiny masses, so atomic and nuclear masses are given in u, where protons, neutrons and most atoms come out close to whole numbers. Its energy equivalent, 931.5 MeV, makes mass-defect calculations quick.

Question 2

Why is the atomic mass of chlorine 35.5 u and not a whole number?

Accepted Answer

Because natural chlorine is a mixture of two isotopes, and the quoted atomic mass is their weighted average. About three-quarters of chlorine atoms have a mass close to 35 u and about one-quarter close to 37 u, which averages to about 35.5 u. Each individual isotope has a mass near a whole number; only the mixture gives a fractional value.

Question 3

What is the difference between isotopes, isobars and isotones?

Accepted Answer

Isotopes have the same atomic number Z but different neutron numbers, so they share chemistry but differ in mass, like the three forms of hydrogen. Isobars have the same mass number A but different Z, like ³₁H and ³₂He. Isotones have the same neutron number N but different Z, like ¹⁹⁸₈₀Hg and ¹⁹⁷₇₉Au, which both have 118 neutrons. Keep A = Z + N in mind.

Question 4

Why was the neutron discovered much later than the proton and electron?

Accepted Answer

Because it carries no charge, so electric and magnetic fields do not deflect it and it is hard to detect directly. Chadwick found it by bombarding beryllium with alpha-particles: a neutral radiation came out that knocked protons out of helium, carbon and nitrogen. Photons could not explain the energies involved, but a neutral particle about as heavy as a proton could.

Question 5

How does the radius of a nucleus change with its mass number?

Accepted Answer

It grows as the cube root of the mass number: R = R₀A^(1/3), with R₀ = 1.2 fm. So a nucleus with eight times as many nucleons is only twice as wide. For A = 27 the radius is about 3.6 fm, and for A = 125 about 6 fm. Ratio questions are quickest done by taking cube roots of the mass numbers.

Question 6

Why do all nuclei have nearly the same density?

Accepted Answer

Because the radius grows as R = R₀A^(1/3), so the volume, which goes as R³, is proportional to A. The mass is also roughly proportional to A, so mass divided by volume comes out nearly the same for every nucleus, about 2.3 × 10¹⁷ kg m⁻³. Heavier nuclei are bigger, not denser. This density is vastly greater than that of any ordinary material.

Question 7

Does mass really get converted into energy in nuclear reactions?

Accepted Answer

Yes, in the sense that mass and energy are equivalent through E = mc², and their combined total is conserved. In a nuclear reaction the products have slightly less mass than the starting particles, and that missing mass appears as kinetic energy or radiation. Chemical reactions do the same, but their mass change is about a million times smaller, far too small to measure.

Question 8

What is the energy equivalent of 1 u, and how is it used?

Accepted Answer

One atomic mass unit is equivalent to 931.5 MeV. Putting 1 u = 1.66 × 10⁻²⁷ kg and c = 3 × 10⁸ m/s into E = mc² gives about 1.5 × 10⁻¹⁰ J, which equals 931.5 MeV. So a mass defect written in u, multiplied by 931.5, gives the binding energy in MeV straight away, with no need to convert to kilograms and joules.

Question 9

Why is the mass of a nucleus less than the total mass of its protons and neutrons?

Accepted Answer

Because energy is released when nucleons bind together, and that energy carries mass away with it. The shortfall, ΔM = [Zmp + (A − Z)mn] − M, is the mass defect, and ΔMc² is the binding energy, the energy needed to pull the nucleus apart into free protons and neutrons. The larger the binding energy, the larger the mass defect.

Question 10

How should I handle electron masses when using atomic masses to find a mass defect?

Accepted Answer

Either subtract the masses of Z electrons from the atomic mass to get the nuclear mass, or use the mass of a hydrogen atom in place of the proton mass, so the electrons cancel. Tables give atomic masses, which include the electrons. Using an atomic mass for the nucleus but bare proton masses leaves Z electron masses unbalanced and gives the wrong binding energy.

Question 11

Is a nucleus with a larger total binding energy always more stable?

Accepted Answer

No. Stability depends on binding energy per nucleon, Ebn = Eb/A, not on the total. Uranium has a far larger total binding energy than iron simply because it has many more nucleons, but each nucleon in iron is held more tightly. The graph of Ebn against A peaks near A = 56, so nuclei in that region are the most tightly bound.

Question 12

How does the binding energy curve explain energy release in both fission and fusion?

Accepted Answer

Energy is released whenever nucleons end up more tightly bound. A heavy nucleus near A = 240 has Ebn of about 7.6 MeV, while middle-sized nuclei have about 8.5 MeV, so splitting a heavy nucleus releases energy: fission. Very light nuclei have a low Ebn, so joining them into a heavier nucleus also releases energy: fusion. Both move nuclei towards the peak of the curve.

Question 13

Why is the binding energy per nucleon nearly constant for medium-mass nuclei?

Accepted Answer

Because the nuclear force has a short range and saturates. Each nucleon interacts only with its nearest neighbours, so once a nucleus is big enough for a nucleon to be fully surrounded, adding more nucleons does not bind each one any harder. That keeps Ebn close to 8 MeV for A between about 30 and 170. Very light nuclei have a larger share of nucleons at the surface, with fewer neighbours.

Question 14

What are the main properties of the nuclear force?

Accepted Answer

It is much stronger than the Coulomb force at nuclear distances, which is how it holds protons together despite their repulsion. It acts only over a few femtometres. It is attractive beyond about 0.8 fm but strongly repulsive at closer distances, which stops nucleons from collapsing together. It is also nearly the same between two protons, two neutrons, or a proton and a neutron.

Question 15

What happens in alpha, beta and gamma decay?

Accepted Answer

They are the three kinds of natural radioactive decay, by which an unstable nucleus gives off particles or energy. In alpha decay it emits a helium nucleus, ⁴₂He, and so loses two protons and two neutrons. In beta decay it emits an electron or a positron. In gamma decay it emits a high-energy photon, of hundreds of keV or more, giving up energy without changing its numbers of protons or neutrons.

Question 16

Why are some nuclei radioactive while others are stable?

Accepted Answer

Because their balance of neutrons and protons lies too far from the stable range. Light stable nuclei have roughly equal numbers of each, while heavy stable nuclei need extra neutrons, up to about 3 neutrons for every 2 protons, to offset the growing repulsion between protons. Nuclei too far from these ratios are unstable and decay, giving off particles or radiation.

Question 17

What is a positron?

Accepted Answer

A positron is the antiparticle of the electron: it has exactly the same mass but a positive charge of the same size. Positrons are given off in some beta decays and in the first step of the fusion reactions in the sun. When a positron meets an electron, the two annihilate, and their energy appears as gamma-ray photons, about 1.02 MeV in total.

Question 18

How much energy is released in the fission of one uranium nucleus?

Accepted Answer

About 200 MeV. A quick estimate: a nucleus with A = 240 has Ebn of about 7.6 MeV, while its two fragments, each near A = 120, have about 8.5 MeV, a gain of roughly 0.9 MeV per nucleon. Multiplying by 240 nucleons gives over 200 MeV. It appears first as kinetic energy of the fragments and neutrons and ends up as heat.

Question 19

Are the numbers of protons and neutrons conserved in nuclear fission?

Accepted Answer

Yes. The total number of protons and the total number of neutrons are the same before and after fission; they are only shared out among the fragments and the free neutrons. The energy does not come from destroying nucleons. It comes from the fragments being more tightly bound, so the products have slightly less total mass, and that difference is released as energy.

Question 20

Why does nuclear fuel give so much more energy than coal?

Accepted Answer

Because nuclear reactions involve energies of millions of electron volts per nucleus, while burning involves only a few electron volts per molecule. For the same mass of fuel the gap is enormous: fission of 1 kg of uranium gives about 10¹⁴ J, while burning 1 kg of coal gives about 10⁷ J. That is why a small amount of nuclear fuel can run a power station for a long time.

Question 21

Why does nuclear fusion need extremely high temperatures?

Accepted Answer

Because both nuclei are positively charged and repel, yet they must get close enough for the short-range nuclear force to take over. To get over this Coulomb barrier, about 400 keV for two protons, the nuclei need very large kinetic energies, which means temperatures of many millions of kelvin. Fusion driven by such heat is called thermonuclear fusion, and it powers the stars.

Question 22

How does the sun produce its energy?

Accepted Answer

By fusing hydrogen into helium in its core, mainly through the proton–proton cycle. In a chain of reactions four hydrogen nuclei become one helium nucleus, with positrons, neutrinos and gamma rays produced along the way, and about 26.7 MeV is released for each helium nucleus formed. The core is at about 1.5 × 10⁷ K, so only the fastest protons manage to fuse.

Question 23

What is the difference between nuclear fission and nuclear fusion?

Accepted Answer

Fission splits a heavy nucleus into two middle-sized ones, while fusion joins very light nuclei into a heavier one. In fission, uranium-235 absorbs a neutron and breaks into two fragments plus a few neutrons, releasing about 200 MeV. In fusion, light nuclei such as hydrogen isotopes combine, which needs temperatures of millions of kelvin to overcome their repulsion. Fission runs nuclear reactors; fusion powers the sun and other stars.

Nuclei: common doubts, answered

About the chapter

What is the difference between nuclear fission and nuclear fusion?

Atomic mass unit and isotopes