A radioactive isotope has a half-life of T years. How long will it take the activity to reduce to a) 3.125%, b) 1% of its original value?

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Potassium-40 - Wikipedia
Potassium-40 (40K) is a radioactive isotope of potassium which has a long half- life of 1.251×109 years. It makes up 0.012% (120 ppm) of the total amount of ...
For more information, see Potassium-40 - Wikipedia
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Law of Radioactive Decay, Decay Rate, Half-Mean Life, Q&A
In this topic, we will learn about the Laws of Radioactive Decay. ... Now, integrating both the sides of the above equation, we get, ... This rate gives us the number of nuclei decaying per unit time. ... Next, let's find the relation between the mean life τ and the disintegration constant λ. ... Download NCERT Notes and Solutions.
For more information, see Law of Radioactive Decay, Decay Rate, Half-Mean Life, Q&A
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What percentage of radioactive isotopes are left at each half-life ...
After 1 half-life there will be 50% of the original isotope, and 50% of the decay product. ... Answered 4 years ago · Author has 5.6K answers and 3.6M answer views ... In the half life of a radioactive isotope, why does only half of the nuclei decay ... undergo radioactive decay, decay until they finally become a stable isotope of ...
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Required formulas:
Radioactive Decay Law is a law that regulates the decay of radioactive materials
After period t, the total number of radioactive nuclei left is N.
N = N_0 e^{- \lambda t}, where, N _0 - original amount of the substance, \lambda - decay constant and t - time
Half life of the radioactive substance T_{\frac{1}{2}} = \frac{\ln 2}{\lambda}
Given that
Half - life of the radio active isotope T_{\frac{1}{2}} = T years
(a) Activity reduced to 3.125%
Step 1: Get an expression for time period in terms of its decay constant
Number of radioactive nuclei left after decay N = 3.125 % of N_0
\frac{N}{N_0} = 3.125 % = \frac{3.125}{100}
We know that N=N_0e^{-\lambda t}\ \Rightarrow\ \frac{N}{N_0}\ =\ e^{-\lambda t}
\frac{3.125}{100} = e^{- \lambda t}
\frac{1}{32} = e^{- \lambda t}
Applying the log on both sides
\log 1 - \log 32 = - \lambda t
- 3.47 = - \lambda t
t = \frac{3.47}{\lambda} ---------------------(1)
Step 2: Calculating the time required to decay
Half life T_{\frac{1}{2}} = \frac{\ln 2}{\lambda}
\lambda=\frac{\ln2}{T}
Substituting \lambda value in equation (1)
t = \frac{3.47}{\frac{\ln 2}{T}}
t = \frac{3.47}{0.693} T
t = 5T
Hence, time required to decay 3.125%, t = 5T yeras
Similarly,
(b) Activity reduced to 1%
Step 1: expression for time period in terms of its decay constant
\frac{N}{N_0} = \frac{1}{100}
\frac{1}{100} = e^{-\lambda t}
t = \frac{4.61}{\lambda}
Step 2: Calculating the time required to decay
t = \frac{4.61}{\frac{\ln 2}{T}}
t = 6.65 T years
Hence, time required to decay 1%, t = 6.65 T years