Nuclear Radius

Question 1

Describe how the distance of closest approach is used to estimate the radius of an atom

Answer 1

• An alpha particle fired at a gold nucleus will have an initial kinetic energy which can be measured
• As it moves towards the positively charged nucleus it will experience an electrostatic force of repulsion and slow down as its kinetic energy is converted to electric potential energy
• The point at which the particle stops and has no kinetic energy is its distance of closest approach
• Its electrical potential energy is equal to its initial kinetic energy due to conservation of energy
• Therefore the alpha particle’s kinetic energy can be equated to electrical potential energy in order to find the radius

Question 2

Describe the Distance of Closest Approach Equation

Answer 2

E_k = Q /4π x ε0 x r

Kinetic Energy of Alpha Particle = Charge / 4π x Permetivity of Free Space x Distance

Question 3

Why is Electron Diffraction more accurate the Distance of Closest Approach

Answer 3

• Electrons are leptons meaning they will not interact with nucleons in the nucleus through the strong nuclear force as an alpha particle would
• Alpha Particles will give an overestimate for the radius
• Electron diffraction gives a far more accurate estimate of nuclear radius

Question 4

Describe how Electron Diffraction gives an estimate for the radius of an atom

Answer 4

• The electrons are accelerated to very high speeds so that their De Broglie wavelength is around 10^-15 m
• They are directed a very thin film of material in front of a screen causing them to diffract through the gaps between nuclei and form a diffraction pattern
• The diffraction pattern formed with a central bright spot, which get dimmer as you move away from the centre, using this pattern you can plot a graph of intensity against diffraction angle from which you can find the diffraction angle of the first minimum
• Using this measurement you can find an estimate of nuclear radius by using the following formula: sin θ = 0.61λ/ R

Question 5

What is the Approximate Radius of a Nucleus

Answer 5

1 x 10^-15 m

Question 6

Describe the Graph of Intensity against angle for electron diffraction by a nucleus

Answer 6

sin θ = 0.61λ / R

Question 7

Describe the Graph of Nuclear Radius against Nucleon Number

Answer 7

Question 8

Describe the Graph of the Natural Log of the Nuclear Radius against the Nucleon Number

Answer 8

R = kAⁿ

ln R = ln (k) + n ln (A)

The gradient = 1/3

Y-intercept = ln (k)

Question 9

What is the Nuclear Radius Equation

Answer 9

R = R₀ x A^1/3

Radius of the Nucleus = The constant R₀ x Nucleon Number ^1/3

Question 10

What is the value of R₀

Answer 10

1.4 fm

(It will normally be given in the question)

Question 11

Describe how the Nuclear Radius Equation shows that Nuclear Density is Constant

Answer 11

Density = Mass / Volume

Density = A x mass _{nucleon /} 4/3 π R³

Density = A x mass_nucleon / 4/3 π x (R₀ x A^1/3) ³

Density = A x mass_nucleon / 4/3 π x R₀³ x A

Density = mass_nucleon / 4/3 x R₀³

Density is a Constant Value

Question 12

What is the Nuclear Density of a Nucleus

Answer 12

Density = mass_nucleon / 4/3 x x π R₀³

Density = 1.45 x 10¹⁷ kg m^-3

Question 13

What does the value of the Nuclear Density tell you about any nucleus

Answer 13

The nuclear density is much larger than the density of an atom, suggesting an atom is mostly empty space with most of its mass concentrated in its centre

Question 14

What is meant by the random nature of radioactive decay

Answer 14

• There is equal probability of any nucleus decaying
• It cannot be known which particular nucleus will decay next

Question 15

What is radioactive half life

Answer 15

Half-life is time for the number of nuclei to halve for a particular isotope

Question 16

What is the Radioactive Decay Constant

Answer 16

The probability of a nucleus decaying per unit time

Question 17

What are the equations for radioactive decay

Answer 17

Δ N / Δ t = - λ N

Change in the number of Nucli / Change in Time = - Constant Decay Probability x Number of Nucli

N = N₀ x e

Question 18

Write the radioactive decay equation in the form y = mx + c

Answer 18

• N = N₀ x e^-λt
• ln (N) = ln(N₀ e ^-λt )
• ln(N) = ln(N₀) +ln(e ^-λt )
• ln (N) = ln(N₀) - λt
• ln(N) = - λt+ ln(N₀)
• y = mx + c

Question 19

What is the Activity of a radioactive sample

Answer 19

The number of nuclei that decay per second

Question 20

What is Activity measursed in

Answer 20

Bq (decays per second)

Question 21

What are the Equations for Activity

Answer 21

A = Nλ

Activity = Number of Nucli x Decay Constant Probability

A = A₀ x e^-λt

Acivity = Initial Activity x e ^{- Decay Constant Probability x Time}

Question 22

What is the Equation for Half Life

Answer 22

T _1/2 = ln 2 / λ

Half Life = ln 2 / Decay Constant Probability

Question 23

What are the Applications of knowing a Nucleus’ Half Life

Answer 23

• Dating of objects - nuclei with a long half-life such as carbon-14, which has a half-life of 5730 years can be used to date organic objects, such as those found in archaeological sites. This is done by measuring the current amount of carbon-14 and comparing it to the initial amount
• Medical diagnosis - nuclei with relatively short half-lives are used as radioactive tracers in medical diagnosis
• How to store nuclei - nuclei with an extremely long half-life will have to be suitably stored, for example in steel casks underground, to prevent these nuclei from damaging the environment and the people

Question 24

Describe the Graph of Number of Neutrons against Number of Protons for a Stable Nucli and Non-Stable Nucli

Answer 24

Question 25

What are the 4 reasons why a nucleus may become unstable

Answer 25

• It has too many neutrons - decays through beta-minus emission or sometimes by neutron emission
• t has too many protons - decays through beta-plus emission or electron capture
• It has too many nucleons - decays through alpha emission
• It has too much energy - decays through gamma emission (this usually occurs after a different type of decay, such as alpha or beta decay because the nucleus becomes excited and has excess energy)

Question 26

Describe the energy level diagram for a nucleus undergoing alpha decay

Answer 26

Question 27

Describe the energy level diagram for a nucleus undergoing beta minus decay

Answer 27

Question 28

Explain why Technetium-99m is used in medical diagnosis

Answer 28

• Technetium-99m is a pure gamma emmited
• Gamma Radiation is weakly ionising and is easily detectable
• It only has a half life of 6 hours making it short enough to have limited exposure but long enough for tests to be carried out