7.1 - Atomic structure - questions

35 - In the reaction B → C + β + ν the total binding energy of B is 79 MeV and the total binding energy of C is 92 MeV.
What is the energy of the beta particle ?

36 - What is the reaction when ₁₃²⁷Al decays with alpha, beta and gamma radiation ?

37 - What is ionising radiation ?

38 - What is a Geiger-Muller detector ?

39 - What is the properties of different types of radiation ?

7.7 - Half-life - questions

40 - What does it mean that radioactive decays are a random process ?

41 - What does radioactive half-life mean ?

42 - What does a decay curve looks like for different decay constants ?

43 - What does the decay equation that give the decay curve look like ?

44 - What is the relationship between the decay constant and the half-life ?

45 - How can you determine the half-life from a decay curve ?

46 - What does the activity equation look like ?

47 - What is the difference between A and N ?

48 - How does Potassium-argon dating work ?

49 - You have 10¹¹ K-40 nuclei.
After 10¹¹ years only 58% is left.
What is the decay constant of K-40 ?

50 - You have 10¹¹ K-40 nuclei.
55 nuclei decay in one year.
What is the decay constant of K-40 ?

51 - How does Carbon dating work ?

7.8 - Nuclear reactions - questions

52 - What is transmutation ?

53 - Give an example of a transmutation ?

54 - What is the process called that gives energy to the sun ?

55 - What is the process called that gives energy in a nuclear power plant ?

56 - Explain what fusion is ?

57 - Explain what fission is ?

58 - Explain with the help of a binding energy curve why two H-2 atoms wants to undergo fusion to one He-4 atom.

59 - Explain with the help of a binding energy curve why one U-236 atom wants to undergo fission to one Ba-141 atom and one Kr-92 atom.

7.1 - Atomic structure

Rurherford let alpha particles hit a gold foil. Instead of all going through more or less without change of direction, some had a large change in direction because they hit the nucleus of a gold atom. This was the first evidence for atoms having a nucleus.

In the Bohr model the electrons move around the nucleus (like the planets around the sun) in fixed orbits. The force that keeps the electrons in their orbit is the electrical attraction between the charge of the electrons and the charge of the nucleus (like gravity that keeps the planets going around the sun).

In the Bohr model the electrons should loose energy when they move around the nucleus and therefore get closer and closer to the nucleus.

Electrons in an atom can only have some particular discrete energy values. The electrons can jump between the energy levels by sending out or absorbing photons (electromagnetic carrier particles).

If one put hydrogen in a tube and then send an electric current through it one has created a hydrogen lamp. If one let the light from the hydrogen lamp go through a prism one does not get a continuous spectrum (rainbow) as with a normal lamp. Instead one get several thin lines that comes from electrons jumping between different energy levels in the hydrogen atom. If one uses other atoms than hydrogen one gets different lines.

7.5 - Nuclear structure

Z = the number of protons in the nucleus.

N = the number of neutrons in the nucleus.

A = Z + N = the number of protons and neutrons in the nucleus.

A nuclide is a bunch of atoms with a well-defined nucleus. With other words they have a certain number of protons (Z) and a certain number of neutrons (N). So atoms that belong to the same nuclide all have the same Z and N numbers. All atoms belonging to the same nuclide has exactly the same nucleus.

Atoms that have the same value of Z (protons) but different values of N (neutrons) have the same chemical characteristics and are called isotopes (isotopes are a sub-set of nuclides). They differ in mass (A) because of the different number of neutrons but react the same. C-12 (Z=6,N=6) and C=13 (Z=6,N=7) are two isotopes of carbon.

Assume that you have:

Two atoms with the same Z and N → They then belong to the same nuclide and the same isotope.

Two atoms with the same Z but different N → They then belong to different nuclides and different isotopes.

Two atoms with different Z but the same N → They then belong to different nuclides (and one does not use the word isotopes for atoms that have different Z) .

Nucleons is another word for Protons and Neutrons. So instead of saying "the nucleus has 2 protons and 1 neutron" one can say "the nucleus has 3 nucleons". Remember that nucleon, proton and neutron all ends with "on".

An ion is an atom that do not have the same number of electrons and protons. It might have lost electrons and become positively charged or taken up electrons and become negatively charged.

A unified atomic mass unit (u) is defined as the mass of 1/12 of the nucleus of a carbon-12 isotope.

m_p = 1.0078 u = 1.672 x 10^-27 kg

m_n = 1.0086 u = 1.675 x 10^-27 kg

m_e = 0.0005 u = 9.110 x 10^-31 kg

So a proton and a neutron has almost the same mass and it is almost equal to 1 u. While an electron has a much smaller mass.

Look at the data booklet:

An elementary charge unit (e) is defined as the (opposite sign) charge of one electron.

Q_p = +1 e = +1.60 x 10^-19 C

Q_n = 0 e = 0 C

Q_e = -1 e = -1.60 x 10^-19 C

So a proton and an electron have exactly the same charge equal to 1 e. While a neutron has zero charge.

Look at the data booklet:

An Aluminium nucleus has Z=13 so the charge is Q_Al = +13 e = 13 x 1.60 x 10^-19 C

An Aluminium atom has zero charge because it has an equal number of protons and electrons and their charge cancels out.

The nuclear force is the force that glues the protons and neutrons together in the nucleus. It is much stronger than the electromagnetic force because otherwise the positively charged protons would fly apart. It works the same on protons and neutrons (it is always attractive) and only works over a short distance (the electromagnetic force between charges can work over large distances).

An electronvolt is the kinetic energy that an electron gets if it is accelerated by a field of 1 Volts. A mega-electronvolt (MeV) is one million electronvolts.

The binding energy is defined as the amount of work needed to pull apart the protons and the neutrons in the nucleus.

Look at the data booklet:

E: Energy in J
m: mass in kg
c: speed of light m/s (a constant = 300000 km/s)

The formula tells us that energy is equivalent to mass.

When protons and neutrons are put together in a nucleus some of their mass is converted into binding energy. The difference between the mass of the nucleus of an atom and the mass of the individual protons and neutrons is called the mass defect.

Look at the data booklet:

The mass of a He-3 nucleus is 3.0160 u. What is the binding energy per nucleon in MeV ?

A He-3 nucleus consists of two protons with mass = 2 x 1.0078 u and one neutron with mass = 1.0086 u which gives a total mass of the particles of 3.0242 u.

The total binding energy = mass defect = 3.0242 u - 3.0160 u = 0.0082 u.

Change the units to MeV: 0.0082 x 931.5 = 7.638 MeV.

Calculate the binding energy per nucleon by dividing with the number of nucleons (3): 7.638/3 = 2.55 MeV

For small A the binding energy is small which means the nucleus is not so stable. There is maximum around Fe-56 which means that Iron has the most stable nucleus. At large A the binding energy is going down because many protons means that there are repulsive forces that makes the nucleus less stable.

7.6 - Radioactive decay

Natural radioactive decays is when a nucleus lose energy by emitting radiation. This happens when the nucleus can change to a more stable nucleus (higher binding energy per nucleon) by sending out the radiation. So a nucleus will decay if there is a set of particles with lower total mass that can be reached by any radioactive decay process.

Alpha radiation consists of Helium nuclei i.e. two protons and two neutrons. They are sent out by a parent nucleus which then loses two protons and two neutrons and becomes a new lighter atom (the daughter atom).

A Ra atom with mass 226.005 u decays to an alpha particle with mass 4 u and a Rn atom with mass 222 u. What is the kinetic energy of the alpha particle ?

Beta radiation consists of electrons (or its anti-particle the positron). They are sent out by a parent nucleus when a neutron changes to a proton, electron and anti-neutrino (or to a proton, positron and neutrino). The new proton stays in the nucleus but the electron (or positron) is sent out as Beta radiation and the neutrino disappears as neutrino radiation.

n → p + e + ν

Gamma radiation consists of photons which are electromagnetic radiation. They are sent out by a parent nucleus which then lose energy. But the nucleus is not changed as for beta and alpha radiation.

In the reaction B → C + β + ν the total binding energy of B is 79 MeV and the total binding energy of C is 92 MeV.

Kinetic energy of both the beta and the neutrino is 92-79 = 13 MeV.

That means that the kinetic energy of the beta can be between 0 and 13 MeV.

₁₃²⁷Al   →  ₂⁴He + ₁₁²³Na

₁₃²⁷Al   →  e^- + ν + ₁₄²⁷Si

₁₃²⁷Al   →  γ + ₁₃²⁷Al

Ionising radiation is alpha, beta and gamma radiation that has enough energy to kick out electrons from atoms and turn them into ions. Gamma radiation is photons (like light or radio waves) but with a higher energy (shorter wave length) which means that they can create ions. This makes them dangerous because ionising radiation can produce hydrogen ions in the body and these can give cancer.

A Geiger-Muller detector is used to measure radiation. It has a cylinder filled with gas and with a wire in the middle. A high voltage is put between the wire and the cylinder. When the radiation goes into the cylinder and hits the gas atoms some electrons are knocked out (this process is called ionisation) and they travel to the wire where they give a signal.

7.7 - Half-life

If one have a sample of a radioactive material it is impossible to say when a particular atom will decay. But one can calculate the probability for a decay and how many atoms will be left after a particular time.

The half-life is the time it takes for half of a sample of unstable nuclei to decay.

A decay curve shows how many nuclei in a sample is left as a function of time. The larger the decays constant (λ), the faster the material decay.

Look at the data booklet:

N₀: the number of radioactive nuclei at time = 0 (when the measurement starts).
N: the number of radioactive nuclei at time = t.
N/N₀: the fraction of nuclei left after a time = t.
λ: the decay constant (can be calculated from the half-life).

Look at the data booklet:

T_1/2: Half-life in s.
λ: the decay constant in s^-1.
ln(2) = 0.693....

Look at which time 50% of the original sample is left:

Look at the data booklet:

N₀: the number of radioactive nuclei at time = 0 (when the measurement starts).
N: the number of radioactive nuclei at time = t.
λ: the decay constant (can be calculated from the half-life).

Another way of writing this equation is A = A₀ e^-λt

A₀=λN₀: the activity at a time = 0. Unit: Bq = Becquerel = decay / s
A: the activity at a time = t. Unit: Bq = Becquerel = decay / s
A/A₀: the fraction of the activity left after a time = t.

When K-40 decays to Ar-40 in rock the argon is trapped inside the rock.

If one melts the rock one can get the argon out and see how much there is and from this one can measure the age of the rock. If there is a lot of argon the rock is old.

You have 10¹¹ K-40 nuclei.
After 10¹¹ years only 58% is left.
what is the decay constant of K-40 ?

Look at the data booklet:

N/N₀ = 0.58
t = 10¹¹ years

λ = -ln(N/N₀) / t = 5.5•10¹¹ years^-1

You have 10¹¹ K-40 nuclei.
55 nuclei decay in one year.
what is the decay constant of K-40 ?

Look at the data booklet:

55 decays of 10¹¹ means that N and N₀ are almost the same numbers.

A = λN

λ= A / N = 55 / 10¹¹ = 5.5•10¹¹ years^-1

Most carbon is C-12 which is not radioactive.

There is, however, a small amount of C-14 in air that plants take up.

When the plants die the C-14 in them starts to decay.

By measuring the amount of C-14 one can determine the age of the plant material (or the age of bones from animals).

One can measure the age of stuff which is not older than 60,000 years.

7.8 - nuclear reactions

Transmutation is when one change one atom to another atom i.e. when one change the Z-number.
This can happen due to different nuclear processes:
1. Radioactive decay.
2. Bombardment of protons from an accelerator.
3. Bombardment of neutrons from a reactor.

Example of transmutation: N-14 + n → C-14 + p

A nitrogen atom absorbs a neutron, becomes a carbon atom and give out a proton.

Fusion

Fission

Nuclear fusion is when two small nuclei is joining together to form a larger one. This can happen if the pressure is very high and the temperature is millions of degrees because then the nuclei have enough kinetic energy to overcome the repulsion due to the positive charge of the protons.

When they join together some of the original mass is converted into energy.

In the plot below two hydrogen atoms undergo fusion and form a helium nuclei and a neutron.

Fission is when very large nuclei is split into two smaller nuclei + a couple of neutrons. This is normally done by hitting the large nuclei with slow neutrons.

In the plot below a U-235 atom is split into one Kr-91 and one Ba-142 atom + 3 neutrons. The neutrons goes on and split other U-235 atoms. One has a chain reaction. Every time a U-235 atom is split a small amount of mass is turned into energy.

The total binding energy of two H-2 atoms is 4•1 = 4 MeV.

The total binding energy of one He-4 atoms is 4•7 = 28 MeV.

Increase in total binding energy = 28 - 4 = 24 MeV.

The higher binding energy per nucleon for He-4 means that it is more stable.

The change of total binding energy means that 24 MeV of energy has to be released in form of gamma radiation.

If one added up the masses before and after the fusion one would see that a mass corresponding to 24 MeV had disappeared.

The total binding energy of Ba-141 and Kr-92 is 141•8 + 92•8.2 = 1882.4 MeV.

The total binding energy of one U-236 atom is 236•7.6 = 1793.6 MeV.

Increase in total binding energy = 1882.4 - 1793.6 = 88.8 MeV.

The higher binding energy per nucleon for Ba-141 and Kr-92 means that they are more stable than U-236.

The change of total binding energy means that 88.8 MeV of energy has to be released in form of neutron radiation and kinetic energy.

If one added up the masses before and after the fission one would see that a mass corresponding to 88.8 MeV had disappeared.