Expert: Ahmed Salami - 11/1/2009

Question
QUESTION: Data below is of radioactive decay of a material x Time t min y Count rate N cps ( counts per second )

Time t min      10   20  30   40   50  60  70  80  90
Count Rate cps 340  236  165  112  77  53  36  25  17

It is believed the data fits a graph with an equation of the form

N=Noe-at The a should be alpha

Where No and a are constants. I have already plotted this graph.

The exponential equation can be transformed to give a linear equation lnN=lnNo-at. Prepare a table of values for plotting a straight line graph (I dont know how to do that)Plot the line and find values for No and a

ANSWER: Hi Stephen,
Well, the new table of values is just one that contains values of lnN and Time t(just as before). In case you're wondering, lnN refers to the natural logarithm of N and you can get all the values by using a calculator to compute lnN from N.
The graph of InN against t would be a straight line and can be used to find the values of the unknown constants.
Referring to the general equation of a straight line,
y = mx + c
where m is the slope of the line and c is the intercept on the y axis.
From your graph of lnN against t,
lnN = lnNo - αt
comparing with y = mx + c, we can see that
m = -α and c = lnNo
which ultimately means that the slope of your graph would be -α and it'll intercept the vertical axis at a value of lnNo
So now, all you need do is draw the graph and find the slope m(which i assume you can) and of course you should see the vertical intercept c. And then,
α = -m
and
No = e^c

You can always get back to me.

Regards

---------- FOLLOW-UP ----------

QUESTION: I ahve drawn the straight line graph and got thew values for No and a. From the graph how do I determine the half life of the material in decay?

Thanks for the help in the first place

Steve

Answer
Hi Steve,
To get the half-life you simply use the equation
N = No.e^(-αt)
If T is the half-life, then t = T when N = No/2, i.e
No/2 = No.e^(-αT)
1/2 = e^(-αT)
Taking the natural logarithms of both sides,
ln(1/2) = -αT
-0.693 = -αT
T = 0.693/α
And of course you have the value of α from your graph.

Regards