The half life of a radioactive nucleus is 50 days. The time interval (t2 – t1) between the time t2when 2/3 of itshas decayed and the time t1 when 1/3of it had decayed is : [2012]a)30 daysb)50 daysc)60 daysd)15 daysCorrect answer is option 'B'. Can you explain this answer?

Question

The half life of a radioactive nucleus is 50 days. The time interval (t2 &ndash; t1) between the time t2when 2/3 of itshas decayed and the time t1 when 1/3of it had decayed is :   [2012]a)30 daysb)50 daysc)60 daysd)15 daysCorrect answer is option 'B'. Can you explain this answer?

Answer

-t1) required for the number of radioactive nuclei to decrease from 1000 to 250 is:

First, we need to find the decay constant (λ) using the formula:

λ = ln(2) / half-life

λ = ln(2) / 50 days
λ = 0.0139 per day

Next, we can use the formula for radioactive decay:

N(t) = N0 * e^(-λt)

Where:
N(t) = number of radioactive nuclei at time t
N0 = initial number of radioactive nuclei
e = Euler's number (approximately 2.718)
λ = decay constant
t = time elapsed

We can use this formula to find the time (t1) it takes for the number of radioactive nuclei to decrease from 1000 to 500:

500 = 1000 * e^(-0.0139t1)

e^(-0.0139t1) = 0.5

-0.0139t1 = ln(0.5)

t1 = ln(0.5) / -0.0139
t1 = 49.8 days (rounded to one decimal place)

Similarly, we can find the time (t2) it takes for the number of radioactive nuclei to decrease from 1000 to 250:

250 = 1000 * e^(-0.0139t2)

e^(-0.0139t2) = 0.25

-0.0139t2 = ln(0.25)

t2 = ln(0.25) / -0.0139
t2 = 99.5 days (rounded to one decimal place)

Therefore, the time interval required for the number of radioactive nuclei to decrease from 1000 to 250 is:

t2 - t1 = 99.5 - 49.8
t2 - t1 = 49.7 days (rounded to one decimal place)

Answer

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