How Does the Mean Life of a Radioactive Element Affect Its Activity Over Time?

[utkarshakash]

Homework Statement

The mean life of a certain radioactive element is 6 hrs. By what fraction will its initial activity decrease over a time period of 5 hrs.

Homework Equations

t_{avg}=\frac{1}{k}
Initial Activity = kN_{0}
Final Activity = kN_{t}

The Attempt at a Solution

From eqn 1 I obtain the value of k = 0.166 hr^{-1}
Since I have to calculate the fraction of initial and final activity
Dividing eqn 2 by 3
\frac{Initial Activity}{Final Activity}=\frac{N_{0}}{N_{t}}
So this means now I have to calculate \frac{N_{0}}{N_{t}} which will be my answer

So using first order kinetics equation

k=\frac{2.303}{5}log\frac{N_{0}}{N_{t}}

Plugging the known values I get \frac{N_{0}}{N_{t}}=2.3006
which is unfortunately incorrect. But how?

 

[JohnRC]

What happened to the 0.166 /hr which you went to so much trouble to calculate?

And remember that the factor 2.303 only applies for logs with base 10; for natural logs and exponentials it disappears.

And finally, the question "by what fraction will it decrease?" is ambiguous. The wording might mean "what fraction of the initial activity is the final activity?" or "by what fraction of the initial activity has the final activity decreased?" To give an example, if the final activity was found to be 25% of the initial activity, was the questioner really expecting the answer 25% or 75%? But in either case, an answer greater than 1 does not make sense.

 

[utkarshakash]

What happened to the 0.166 /hr which you went to so much trouble to calculate?

And remember that the factor 2.303 only applies for logs with base 10; for natural logs and exponentials it disappears.

And finally, the question "by what fraction will it decrease?" is ambiguous. The wording might mean "what fraction of the initial activity is the final activity?" or "by what fraction of the initial activity has the final activity decreased?" To give an example, if the final activity was found to be 25% of the initial activity, was the questioner really expecting the answer 25% or 75%? But in either case, an answer greater than 1 does not make sense.

Yes I also agree with you but the correct answer is itself greater than 1.

 

[JohnRC]

Yes I also agree with you but the correct answer is itself greater than 1.

OK, so the ambiguity has a third and a fourth string:

Interpretation 1: (final activity)/(initial activity)
Interpretation 2: (Initial activity – final activity)/(Initial activity)
Interpretation 3: (Initial activity – final activity)/(final activity)
Interpretation 4: (Initial activity)/(final activity)

The problem setter must surely have been seeking an award for the most ambiguous problem wording! Any one of those four is a reasonable interpretation of the actual wording!

 

[utkarshakash]

OK, so the ambiguity has a third and a fourth string:

Interpretation 1: (final activity)/(initial activity)
Interpretation 2: (Initial activity – final activity)/(Initial activity)
Interpretation 3: (Initial activity – final activity)/(final activity)
Interpretation 4: (Initial activity)/(final activity)

The problem setter must surely have been seeking an award for the most ambiguous problem wording! Any one of those four is a reasonable interpretation of the actual wording!

I have already tried 1 and 4 and they did not work. Give me some hints for 2 and 3.

 

[JohnRC]

OK I have had another look at the detail of your work.

Plugging the known values I get N0Nt=2.3006
which is unfortunately incorrect. But how?

Your answer is quite correct. The other three possible answers according to my last posting would be (1) 0.43 (2) 0.57 or (3) 1.30. The 2.30 you have quoted is the correct value for option (4).

To understand how to arrive at (2) and (3) simply separate each of the two terms in the fraction in my formulae for these interpretations.

 

[utkarshakash]

OK I have had another look at the detail of your work.


Your answer is quite correct. The other three possible answers according to my last posting would be (1) 0.43 (2) 0.57 or (3) 1.30. The 2.30 you have quoted is the correct value for option (4).

To understand how to arrive at (2) and (3) simply separate each of the two terms in the fraction in my formulae for these interpretations.

OK the correct answer is 1.0565. But none of the interpretations match with the answer. Only the 3rd interpretation is somewhat close to the answer. So should I go with the third interpretation of yours?