1. Exam question (from last lesson)
Begin 2nd lesson of session with this?

2.  = ln 2
t1/2
 = 1.3×10-5
the probability per second (1)
of the decay of a (single) nucleus/atom (1)

3. Remember this beautiful thing?
dN = -N dt

4. dN = -N dt Remember this beautiful thing? Number of nuclei left
Rate of decay
ALSO CALLED ACTIVITY dt
This constant is called the decay constant. It is measured in s-1 . It is negative because this is exponential decay (going down). It is the probability of a nuclei decaying each second
Differential equation

5. How does this fit into “real life”?
You already know that…..
The rate that radioactive nuclei are decaying (the gradient on the graph) is called the ACTIVITY (measured in s-1 or Bequerels, Bq). It is a pretty good indication of how dangerous a radioactive isotope is right now

6. What would this look like on a graph?
If we wanted to work out how much of the radioactive substance was remaining, or measure the activity, we could use the graph…or an iterative model….
Complete the table using our differential equation for an isotope with a half life of 1.16s. A’s use t = 1s, B’s use t = 0.25s
t (s) N ΔN
Number remaining
1000 1 2 3
What would this look like on a graph?
Sketch these graphs onto the whiteboards and compare.
What’s wrong with this?

7. Make the time interval you measure over as small as possible!
This would be the value if measured from 0 to 50 experimentally.
This is a problem too with measuring the activity from experimental data.
If you use a long time interval the activity will have changed dramatically over that time interval
Solution????
Make the time interval you measure over as small as possible!

8. We’ve actually already been using the idea of infinitely small amounts of time…
The ds mean a little bitty small change – as small as you can get
dN = -N dt
You sometimes see it written down as slightly bigger changes (like in the previous slide)
N = -N t
We need to solve this equation so that we end up with a useful equation which we can use to tell us how many nuclei there are at any one time?

9. N = N0e-t The Exponential Decay Equation for Radioactivity
Decay constant in s-1
Time in s
N = N0e-t
Nuclei left undecayed (no unit)
Minus because it is decay
Nuclei at start (t=0) (no unit)
Want to see the derivation from dN/dt = -λN?
Want to try the derivation yourself???
Start here:
And integrate w.r.t t

10. N = N0e-t Worked example
Suppose there is 10 million atoms with a decay constant of 0.5 s-1 How many are left after 30 seconds?
N = N0e-t
N0 =
N = ?
= 0.5 s-1
t = 30 s
N= × e – 0.5 ×30
N = 3
The try one where a period of time has to be calculated. Take numbers from the students.

11. What to do… E C A Grade Outcome Task I’m the don!!!
Please note you DO NOT chose how far down this list you go, you complete all tasks, but this gives you an idea of their level.
Grade
Outcome
Task E
I can use the exponential equation in simple substitutions
Use the lucky dip to choose 3 examples to complete. C
I can use the exponential equation in more complex substitutions
Go back to task 2 and complete Q1(iii) and Q2c (iii) A
I can apply ideas and relate them to simple decay curves.
1) Work out the decay constant for it.
2) Write the differential equation for this experiment showing the decay constant as a number. Use your graph to work out two values one second apart. Check the differential equation works
I’m the don!!!
I can apply complex ideas to derive relationships.
Starting with the differential equation for radioactive decay and using your knowledge of half life, attempt to prove the equation linking half life and decay constant.

12. To finish…

13. I can show my understanding of effects ideas and relationships…
Plenary
Learning Review – use the tick sheet to grade yourself – today we did the statements:
I can show my understanding of effects ideas and relationships…
Radioactive decay modelled as an exponential relationship between the number of undecayed atoms, with a fixed probability of random decay per unit time
I can use the following words and phrases accurately…
For radioactivity: half-life, decay constant, random, probability
relationships of the form dx/dy = –kx , i.e. where a rate of change is proportional to the amount present
I can sketch, plot and interpret graphs of:
Radioactive decay against time
I can make calculations and estimates making use of:
iterative numerical or graphical methods to solve a model of a decay equation