10: Modelling decay

Question 1

What makes nucleus unstable?

Answer 1

Too many neutrons, not enough neutrons, too many nucleons in total, or too much energy

Question 2

What happens to unstable nuclei?

Answer 2

They break down by releasing energy and/or particles, until they reach a stable form – this process is called radioactive decay

Question 3

Is radioactive decay a random process? Why?

Answer 3

Yes – you can’t tell when anyone nucleus will decay, or which nucleus in a sample will be the next to decay

Question 4

Describe how radioactive decay can be modelled

Answer 4

Based on a very large number of undecayed nuclei

Modelled by exponential decay

Question 5

What is the activity of a radioactive sample?

Answer 5

The number of unstable nuclei that decay each second

Question 6

What is the activity of a sample proportional to? Why?

Answer 6

The size – as nuclei decay, the sample size gets smaller, so the activity falls

Question 7

What does the decay constant measure?

Answer 7

Measures how quickly an isotope will decay – it’s the probability of a given nucleus decaying in a certain time. The bigger the value of the decay constant, the more likely a decay is, so the faster the rate of decay.

Question 8

What is activity measured in? What does the unit mean?

Answer 8

Becquerels (Bq)

An activity of 1 Bq means that 1 nucleus decays every second

Question 9

What is the half life of an isotope?

Answer 9

The average time it takes for the number of undecayed nuclei and a halve

Question 10

The longer the half life of an isotope, the [ ] a sample of a given initial size will remain radioactive

Answer 10

Longer

Question 11

How do you find the half life on a graph of number of undecayed nuclei against time?

Answer 11

Find the value of undecayed nuclei when t=0
Go to half the original value of N
Work out the corresponding time
(You can check this by finding the quarter of N)

Question 12

What is the difference between activity and count rate? What is their relationship?

Answer 12

Activity is the number of decays per second and count rate is the number of decays detected per second.
As detectors generally only detect radiation that’s emitted in one direction, but sources emit radiation in all directions, count rate will be proportional to, but less than, the source’s activity

Question 13

Which isotope do you generally use when investigating decay? How do you source it? How do measure the decay?

Answer 13

Protactinium-234

It’s formed when uranium decays. You can measure the decay rate using a protactinium generator

Question 14

What is a protactinium generator?

Answer 14

A bottle containing a uranium salt, the decay products of uranium and two solvents, which separate out into layers

Question 15

Explain how to investigate the decay of protactinium using a protactinium generator

Answer 15

Shake the bottle to mix the solvents together, then add it to the equipment.
Wait for the liquids to separate. The protactinium will be in solution in the top layer, and the uranium salt will stay in the bottom layer. Then you can point the Geiger-Müller tube at the top layer to measure the activity of the protactinium
As soon as the liquids separate, record the count rate. Remeasure the count rate at sensible intervals.
Subtract the background count rate from your measured count rates, then plot a graph of count rate against time. You can then find the half life.

Question 16

Protactinium experiment: how do you measure the background count rate? What is background radiation?

Answer 16

Once you’ve collected your data, leave the bottle to stand for a least 10 minutes, this should be long enough for all the protactinium in the top layer to decay, then take the count rate again. This is the background count rate corresponding to background radiation – the low-level radiation you get everywhere

Question 17

Briefly describe the equipment used in the investigation of count rate

Answer 17

A Geiger counter attached to the Geiger-Müller tube which is on a clamp and stand. A protactinium generator in front of the tube.

Question 18

Define capacitance

Answer 18

The amount of charge stored per volt

Question 19

What is a capacitor? What is it made of?

Answer 19

Capacitors are made of two metal plates separated by an air gap or an insulator, so no charge can flow between them. These plates store electrical charge – a bit like a charge bucket.
Capacitors also store energy

Question 20

Describe how capacitors store energy (using the circuit with battery, capacitor and bulb)

Answer 20

When the switch is flicked (so that the battery and capacitor are connected), charge builds up on the plates of the capacitor. Energy provided by the battery is stored, in the capacitor, in the form of electrical potential energy
When the switch is flipped the other way, the energy stored on the plates full discharge through the bulb, converting electrical potential energy into light and heat

Question 21

Describe the setup of a capacitor in a circuit, to explain how capacitors store energy

Answer 21

Set up a circuit. A battery in parallel with a capacitor in parallel with a bulb. With a switch at the junction between the battery capacitor and bulb.

Question 22

Describe what work is done on a capacitor charging

Answer 22

Work is done removing charge from one plate and depositing charge onto the other one. The energy for this must come from the energy stored in the battery

Question 23

What does the graph of potential difference against charge look like? What is the area?

Answer 23

Straight line through the origin, area = energy

Question 24

Describe the set up of the investigation as to what happens when you charge a capacitor

Answer 24

In series, a high resistance resistor, a DC power supply, a switch, an ammeter, and a capacitor. In parallel with the capacitor, a voltmeter. In parallel with the voltmeter and ammeter, a data logger

Question 25

Describe the investigation of what happens when you charge a capacitor

Answer 25

Close the switch to connect to the uncharged capacitor to the power supply.
Let the capacitor charge whilst the data logger records both the potential difference and the current.
When the current through the ammeter is zero, the capacitor is fully charged. Plot the data

Question 26

Investigation of what happen when charging a capacitor:

Describe the charge in the capacitor, when it is charging

Answer 26

As soon as the switch closes, current starts to flow. The electrons flow onto the plate connected to the negative terminal of the power supply, so a negative charge builds up.
This buildup of negative charge repels electrons off the plate connected to the positive terminal of the power supply, making that plate positive. These electrons are attracted to the positive terminal of the power supply.
And equal but opposite charge builds up on each plate, causing a potential difference between the plates. No charge can flow between the plates because they are separated by an insulator

Question 27

Describe the current when a capacitor is charging

Investigation of what happens when a capacitor is charging

Answer 27

As soon as the switch closes, current starts to flow. Initially, the current through the circuit is high. But, as charge builds up on the plate, electrostatic repulsion makes it harder and harder for more electrons to be deposited. When the PD across the capacitor is equal to the PD across the power supply, the current falls to zero

Question 28

Capacitor charging:

Describe the graph of current against time

Answer 28

Current starts high when t=0 and exponentially decrease to 0 as time increases

Question 29

Capacitor charging:

Describe the graph of voltage against time

Answer 29

Same as charge against time

Voltage starts at zero, initially increases rapidly as time increases, then voltage increases slowly as time increases

Question 30

Capacitor charging:

Describe the graph of charge against time

Answer 30

Same as voltage against time

Charge starts at zero, initially increases rapidly as time increases, then charge increases slowly as time increases

Question 31

Investigation of what happens when you charge a capacitor:

How do you discharge the capacitor?

Answer 31

Disconnect the power from the test circuit and close the switch to let the capacitor discharge whilst the data logger records the potential difference in current over time. When the current through the ammeter and the potential difference across the plates is zero, the capacitor is fully discharged. Plot the results.

Question 32

Describe the current of the discharging capacitor.

Investigation of what happens when you charge/discharge a capacitor

Answer 32

The current flows in the opposite direction to the charging current. As the potential difference decreases, the current decreases as well

Question 33

Investigation of what happens when you charge/discharge a capacitor:

Describe the charge and p.d of the discharging capacitor.

Answer 33

The amount of charge on and potential difference between the plates falls exponentially with time. That means it always takes the same length of time for the charge or potential difference to halve, no matter what value it starts at. The same is true for the amount of current flowing through the circuit

Question 34

Describe the graph of current/voltage/charge against time for a discharging capacitor.

Answer 34

All the same graph shape

All start of high when t=0. All done exponentially decrease as time increases

Question 35

What does the time it takes to charge or discharge a capacitor depend on?

Answer 35

The capacitance of the capacitor. This affects the amount of charge that can be transferred for given potential difference

The resistance of the circuit. This affects the current in a circuit

Question 36

Capacitor: what is the relationship between discharge rate and charge remaining

Answer 36

Proportional

Question 37

The proportion of charge that is transferred from a discharging capacitor in a given time interval is….

Answer 37

Constant

Question 38

What is the time constant?

Capacitors

Answer 38

The time taken for the charge, potential difference or current of a discharging capacitor to fall to 37% of its initial value
The time taken for the charger or potential difference of a charging capacitor to rise to 63% of its value when fully charged

Question 39

What is the relationship between the resistance in series with the capacitor and the time it takes a capacitor to charge or discharge?

Answer 39

The larger the resistance in series with the capacitor, the longer the capacitor takes to charge or discharge

Question 40

What is the time taken for a capacitor to charge or discharge fully?
(Roughly)

Answer 40

5RC

Question 41

What is the quantity that decays in discharging capacitors and radioactive isotopes?

Answer 41

Q, the amount of charge left on the plates of the capacitor

N, the number of undecayed nuclei remaining

Question 42

What method can be used to solve differential equations?

Answer 42

Iterative numerical methods

Question 43

Why do you iterative methods only give you approximate answers?

Answer 43

They assume that dx/dt doesn’t change over the time interval you’re using. In fact, dx/dt is changing all the time

Question 44

What makes an iterative of method more accurate?

Answer 44

Use a smaller value for the time interval

Question 45

What is the disadvantage of using a smaller time interval, when doing iterative methods?

Answer 45

You need to do more iterations