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You are here: Experts > Science > Mathematics > Advanced Math > exponential equation to linear equation
Advanced Math - exponential equation to linear equation
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
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