ln(No/N) = kt
No = 80 mg
N = 10 mg
t = 30 min
solve for k, then
k = 0.693/t1/2
Substitute k from above and solve for t1/2
10 min is correct.
10 minute is correct
The person asking the question is asking why ? s/he did not challenge the answer 10min. But most respond just show how to get the answer and did not answer the why .. So here is my attempt.
I think first off, you need to re-affirm your understanding of the definition of half-life .
Half-life is the time taken for the element to decay to half of its original mass. so if it started off as 80gm at the start ,it reaches it first half-life when the mass is 40gm.
Since it is a MCQ multiple choice question . Let just say we find the answer by elimination . We can start from an answer we might think is most likely wrong .. For me I will try (c).
Consider (c. 30min). If this answer is true, the sample ought to reduce from 80gm to 40gm after 30min. but the question says that it is already left with 10gm after only 30 min. So (c) 30min, is quickly recognizable as a wrong answer.
if you tried to do (d) and (b) you should get similar conclusion .
Now I skipped to (a) to try.
Consider (a. 10min). if it this half-life. The sample, ought to reduce from 80gm to 40gm after 10min. And then from 40gm to 20gm in the next 10min. And then from 20gm to 10gm after another 10min. So all in all, from 80gm to 10gm in 30min. This seems consistent the original description of how it was observed to have behaved.
Note that if you have got the definition of half-life wrong, you would not recognize (a) as the correct answer.
This question is not so much a test of mathematics skill. More a test of the concept and definition.
If they really want to be confusing they can set the questions as...
"IF 80.8991234gm of a radioactive element decays to 10.12389gm in 30.1284938 minutes, the half-life of this element is ".... . But they didn't. They made the mathematics very very very simple.