Radioactive Decay Formula

Radioactive decay is the spontaneous breakup of an atomic nucleus of a radioactive material that results in the emission of radiation from the nucleus. The nuclide that decays in a radioactive process is referred to as a parent nuclide, and the nuclide that is created in the radioactive process is referred to as a daughter nuclide. The number of nuclei decaying per unit time in a radioactive substance is proportional to the total number of nuclei in the sample material. The total decay rate of a sample is also known as the sample’s activity. Its standard unit of measurement is the becquerel (Bq).

Formula

N(t) = N₀ e^{– λt}

where,

N is the quantity still remained and not yet decayed,

N₀ is the initial amount of sample, is the half-life of the decaying quantity,

e is the Euler’s number with a value of 2.71828,

λ is the radioactive decay constant or disintegration constant,

t is the total time of decay rate.

An isotope’s half-life is the amount of time it takes for its nucleus to decay to half of its initial number. It is defined as the time it takes for the rate of decay and the number of nuclei to be lowered to half of their starting levels. It is denoted by the symbol t_1/2.

 t_1/2 = 0.693/λ

where, λ is the radioactive decay constant or disintegration constant.

Sample Problems

Problem 1. Calculate the remaining amount of sample if its initial amount was 100 g, decay constant is 0.322 for a total time of 5 s.

Solution:

We have,

N₀ = 100

λ = 0.322

t = 5

Using the formula we get,

N(t) = N₀ e^–λt

= 100 (2.71828)^{-0.322 × 5}

= 100/5

= 20 g

Problem 2. Calculate the remaining amount of sample if its initial amount was 200 g, decay constant is 0.139 for a total time of 10 s.

Solution:

We have,

N₀ = 200

λ = 0.139

t = 10

Using the formula we get,

N(t) = N₀ e^–λt

= 200 (2.71828)^{-0.139 × 10}

= 200/4

= 50 g

Problem 3. Calculate the decay constant if the initial amount of sample was 50 g, the final amount is 5 g for a total time of 6 s.

Solution:

We have,

N = 5

N₀ = 50

t = 6

Using the formula we get,

N(t) = N₀ e^–λt

=> log (N₀/N) = λt

=> 6λ = log (50/5)

=> 6λ = 3.32

=> λ = 0.383

Problem 4. Calculate the decay constant if the initial amount of sample was 300 g, the final amount is 100 g for a total time of 8 s.

Solution:

We have,

N = 100

N₀ = 300

t = 8

Using the formula we get,

N(t) = N₀ e^–λt

=> log (N₀/N) = λt

=> 8λ = log (300/100)

=> 8λ = 1.58 

=> λ = 0.1375

Problem 5. Calculate the initial amount of sample if the decay constant is 0.141, the final amount is 30 g for a total time of 15 s.

Solution:

We have,

N = 30

t = 15

λ = 0.141

Using the formula we get,

N(t) = N₀ e^–λt

N₀ = N e^λt

= 30 (2.71828)^{0.141 × 15}

= 250 g

Problem 6. Calculate the time taken for a sample to decay if the decay constant is 0.233, the initial amount is 200 g and the final amount is 40 g.

Solution:

We have,

N₀ = 200

N = 40

λ = 0.233

Using the formula we get,

N(t) = N₀ e^–λt

=> log (N₀/N) = λt

=> log (200/40) = 0.233t

=> t = 2.322/0.233

=> t = 7 s

Problem 7. Calculate the decay constant for a sample if its half-life is 3.5s.

Solution:

We have,

t_1/2 = 3.5

Using the formula we get,

 t_1/2 = 0.693/ λ

=> λ = 0.693/3.5

=> λ = 0.19805