Activity: Nucleons and BE

Aims

Define atomic mass and the unified mass unit.
Define nuclear charge.
State that a nucleus is made of neutrons and protons.
Describe the nature of the nuclear force.
Define the BE in terms of mass defect.
Calculate BE.
Plot the BE/nucleon vs nucleon number curve.

Mass and charge of the nucleus

First some history

1805 John Dalton: matter is made of atoms of different mass.

1811 Amedeo Avogadro: equal volumes of gas at the same pressure and temperature have the same number of molecules

1871 Dmitri Mendelev: ordered elements into the periodic table.

1911 Ernest Rutherford: shows that atoms have a small positive nucleus.

1913 Henry Moseley: ordered elements according to charge.

So by 1913 a lot was known about the size, mass and charge of the nucleus but not what it was made out of. Here is a small selection.

Note the relative atomic masses are approximate.

  • Does this data support the hypothesis that nuclei are all made up of hydrogen nuclei?

If the charge of every nucleus is due to hydrogen nuclei plus a certain number of other particles:

  • What is the mass of that particle?
  • what is it's charge?

Protons and Neutrons

It fits the data if we assume that all nuclei are made of two particles of almost the same mass, one has positive charge (and is the same as the hydrogen nucleus) the other no charge.

  • According to the table above, how many protons and neutrons would Carbon have?

Unified mass unit (u)

The relative mass of all nuclei could be expressed in terms of the smallest one hydrogen but it isn't. Instead 1/12 of the mass carbon is used as the standard.

1u = 1/12 mass of a carbon 12 atom
(1.66054 x 10-27 kg)

Using the table above express the following atomic masses in u

  • Carbon
  • Hydrogen
  • Oxygen

As mentioned the relative atomic masses given here are rounded off, for example, the actual atomic mass of hydrogen is 1.00782 u.

  • The atomic mass of carbon 12 is exactly 12u, why is this exact?
Proton1.00728 u
Neutron1.00866 u

The different nuclei are classified in terms of the number of particles.

A nucleon number = protons + neutrons (approximately equal to the mass)
Z proton number = protons (Ze equals the charge in Coulombs)
N neutron number = neutrons ( A - Z)

Elements

The elements are defined by their chemical properties which depend on the charge of the nucleus, all atoms of a given element therefore have the same Z but can have different N.

Isotopes

Isotopes of an element have the same number of protons but different number of neutrons. Isotopes have the same chemical properties but different physical properties (e.g boiling point).

Nuclear symbol

A nucleus is represented in the following way

C presubscript Z presuperscript A

  • Why don't you really need to write the number Z?
  • How many neutrons does C presubscript 6 presuperscript 13 have?

Ions

A neutral atom has the same number of electrons as protons. If the number is different the atom will have a net charge. A charged atom is called an ion.

You can build atoms with this simulation

  • Exercises 16 to 19 page 296

Nuclear radius

The size of a nucleus is defined by how close you can get to it. If an alpha particle is scattered at 180° it will have come close to the nucleus, stopped and travelled back again as shown below.

(HL) If the mass of the alpha is m and its velocity v show that the closest it will approach a nucleus of proton number Z is (An alpha particle is actually a He nucleus with a charge of +2e)

r equals fraction numerator 4 k Z e squared over denominator m v squared end fraction

From this the radius is found to be in the order of 10-15 m (the atomic radius is about 10-10 m)

  • (HL) Calculate the distance of closest approach for the simulation where:

mass of the projectile = 1kg (mass of target is 200 kg)
velocity of projectile = 10 ms-1
Both spheres have charge = 1 x 10-4 C
k = 9 x 109 Nm2C-2

  • (HL) Given that the radius of the big sphere = 0.7 m, is your answer reasonable?
  • (HL) Why would this method only work for nuclei that were heavier than the alpha particle?

Experiments of this type lead to the conclusion the radius3 is proportional to the mass.

  • Why does this imply that all nuclei have the same density?

Nuclear force

Nucleons are held together by a strong attractive force.

  • Why isn't this force electric?
  • Why isn't the force gravitation?

We know the force must be strong because it's difficult to break the nucleus apart. Think of the alpha scattering experiment. When high energy alphas hit gold atoms they knock off lots of electrons but no nucleons. This means the nuclear force is much stronger than the electric force holding the electrons in position.

In Algodoo you can vary the force holding particles together by setting the attraction in materials

Apart from being strong the nuclear force is also very short range. If it had an infinite range like electromagnetic and gravity nuclei would attract each other over long distances.

  • Why is the density of the inside of a star greater than the outside layers?

Algodoo has the option of making objects sponge like. See what happens when the attraction between a group on spongey balls is increased.

Summary of fundamental forces
ForceStrengthrange
Nuclear110-15 m
Electromagnetic10-2infinite
Gravity10-38infinite

The nuclear force is so strong it is called the strong force.

Binding Energy

The nucleons are held together by the strong force so to pull them apart work has to be done, this means that when the nucleus was formed energy must be released, this is not very intuitive so let's consider a group of perfectly elastic balls pulled together by an attractive force.

Answer the following before looking at the hidden images.

  • What happens when the balls collide with each other?
  • Will the balls stay together?

  • What if the balls were not elastic and lost energy when the collided, would they stay together now?

  • Where did this energy go?

So to stay together the balls need to lose energy. This is the same with nucleons, however nucleons can't get rid of energy as heat, what happens is that when the nucleus comes together some nucleons and photons are ejected, these particles take away the energy. You can see that this sort of happened in the first animation.

Mass energy relationship

The annihilation of matter and antimatter led Einstein to his famous mass - energy formula:

E = mc2

c = the speed of light in a vacuum (3 x 108 ms-1)

This means that when a body is given energy the mass of the body increases. However, when considering cars and balls the increase in mass is so small compared to the mass of the body that we don't consider it.

  • Calculate the increase in mass of a 1 kg body accelerated from rest to a velocity of 10 ms-1.

We have seen that the strong force is, strong (well named then) which means that the energy required to take a nucleon out of the nucleus is in the order of MeV. (to remove an electron from an atom requires about 1 eV).

  • Calculate the increase in mass of a proton if it requires 5 MeV to pull it out of a nucleus. (1 eV = 1.6 x 10-19 J)
  • If 1u = 1.6605 x 10-27 kg calculate the energy equivalent of 1u in MeV.

Binding energy and mass defect

If a nucleus were pulled apart into its individual parts work would be done, so energy would be transferred to the parts. The parts would therefore have a greater mass than the original nucleus if we measure the change in mass we can calculate the BE.

Atomic masses

You can find atomic masses of different isotopes in tables such as this - hyperphysics scroll down to see nuclear data.

Note that although this is nuclear data the header says "Atomic mass". This means that electrons are included.

  • How many electrons does a neutral atom of 4He have?

So if we want to know the nuclear mass we would have to subtract the electrons. To calculate the BE the equation would be:

{(mass of 2no + 2p+) - (atomic mass of 4He - 2e-)} x 931.5 MeV

(The 931.5 MeV is to convert u into MeV)

This can be re-arranged to give {(mass of 2no + 2p+ + 2e-) - (atomic mass of 4He)} x 931.5 MeV

But a proton plus an electron is a hydrogen atom so this is the same as

{(mass of 2no + 2 x atomic mass 1H) - (atomic mass of 4He)} x 931.5 MeV

So, if we simply use the mass of a hydrogen atom instead of the mass of a proton we can perform the calculation in atomic mass.

Don't worry if you didn't understand that last bit, the point is that if you use the mass of a H nucleus instead of a proton then you can use the atomic mass directly from the table without having to subtract the electrons.

  • Use the data in the tables to show that the BE of 4He is 28.3 MeV.
  • By looking up data in the hyperphysics database show that the BE of 54Fe is 471.7 MeV

BE/nucleon vs nucleon number curve

There is a very interesting and important relationship between the BE/nucleon and the number of nucleons. Note that these are atomic masses of particular isotopes, if you look up the atomic mass of an element you will get the average mass, taking into account the relative abundance of the different isotopes. For example carbon has an atomic mass of 12.011 as there are several different isotopes of carbon.

  • Using the data in this spreadsheet (Atomic masses) calculate the binding energy for each nucleus and plot a graph of BE/nucleon vs nucleon number.

(old excel version)

We will be referring to this curve a lot in the next couple of activities.

Binding energy multiple choice quiz

Binding Energy problems

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