Radioactive Decay Formula: Radioactive decay refers to the spontaneous disintegration of an atomic nucleus within a radioactive substance, leading to the release of radiation from the nucleus. In this process, the radioactive material initially present is known as the parent nuclide, while the newly formed material is called the daughter nuclide. The rate at which nuclei undergo decay in a radioactive substance is directly proportional to the total number of nuclei within the sample material. This overall decay rate is also referred to as the activity of the sample and is typically measured in becquerels (Bq).

Also Check – Vapor Pressure Formula

Radioactive Decay Formula

The mathematical representation for radioactive decay is expressed as:

Where:

N represents the remaining quantity of the substance that has not yet decayed.

N0 is the initial amount of the sample.

e stands for Euler’s number, approximately 2.71828.

λ denotes the radioactive decay constant or disintegration constant.

t signifies the total time elapsed in the decay process.

The half-life of an isotope refers to the duration required for half of its initial nuclei to decay. It is defined as the time at which both the rate of decay and the number of nuclei decrease to half of their initial values. This half-life is denoted by the symbol t1/2 and is mathematically related to the decay constant as follows:

Here, λ still represents the radioactive decay constant or disintegration constant.

Also Check – Gibbs Free Energy Formula

Radioactive Decay Formula Solved Examples

Example 1: Let’s begin by calculating the remaining amount of a sample with an initial quantity of 100 grams, a decay constant of 0.322, and a total time of 5 seconds.

Solution:

Given values:

Initial quantity (N0) = 100 grams

Decay constant (λ) = 0.322

Total time (t) = 5 seconds

Using the decay formula

N(t) = N0 * e^(-λt), we find:

N(t) = 100 * (2.71828)^(-0.322 * 5)

N(t) = 100 * (2.71828)^(-1.61)

N(t) ≈ 100 / 5

N(t) ≈ 20 grams

Also Check – Theoretical Yield Formula

Example 2: Now, let’s calculate the remaining amount of a sample with an initial quantity of 200 grams, a decay constant of 0.139, and a total time of 10 seconds.

Solution:

Given values:

Initial quantity (N0) = 200 grams

Decay constant (λ) = 0.139

Total time (t) = 10 seconds

Using the same decay formula:

N(t) = 200 * (2.71828)^(-0.139 * 10)

N(t) = 200 * (2.71828)^(-1.39)

N(t) ≈ 200 / 4

N(t) ≈ 50 grams

Also Check – Percentage Yield Formula

Example 3: Next, we’ll calculate the decay constant when the initial amount of a sample was 50 grams, the final amount is 5 grams, and the total time is 6 seconds.

Solution:

Given values:

Final quantity (N) = 5 grams

Initial quantity (N0) = 50 grams

Total time (t) = 6 seconds

Using the decay formula, we can rearrange it to solve for the decay constant (λ):

N(t) = N0 * e^(-λt)

log(N0/N) = λt

6λ = log(50/5)

6λ = log(10)

6λ ≈ 3.32

λ ≈ 0.553 (rounded to three decimal places)

So, the decay constant (λ) is approximately 0.553.

Also Check – Partial Pressure Formula

Example 4: Determine the remaining amount of a sample with an initial quantity of 80 grams, a decay constant of 0.225, after a total time of 3 seconds.

Solution:

Given values:

Initial quantity (N0) = 80 grams

Decay constant (λ) = 0.225

Total time (t) = 3 seconds

Using the decay formula

N(t) = N0 * e^(-λt):

N(t) = 80 * (2.71828)^(-0.225 * 3)

N(t) = 80 * (2.71828)^(-0.675)

N(t) ≈ 80 / 2.002

N(t) ≈ 39.97 grams (rounded to two decimal places)

Example 5: Calculate the remaining amount of a sample with an initial quantity of 150 grams, a decay constant of 0.042, after a total time of 20 seconds.

Solution:

Given values:

Initial quantity (N0) = 150 grams

Decay constant (λ) = 0.042

Total time (t) = 20 seconds

Using the same decay formula:

N(t) = 150 * (2.71828)^(-0.042 * 20)

N(t) = 150 * (2.71828)^(-0.84)

N(t) ≈ 150 / 2.315

N(t) ≈ 64.77 grams (rounded to two decimal places)

Example 6: Calculate the decay constant for a sample with an initial amount of 200 grams, a final amount of 25 grams, and a total time of 8 seconds.

Solution:

Given values:

Final quantity (N) = 25 grams

Initial quantity (N0) = 200 grams

Total time (t) = 8 seconds

Using the decay formula and rearranging to solve for λ:

N(t) = N0 * e^(-λt)

log(N0/N) = λt

λ = log(N0/N) / t

λ = log(200/25) / 8

λ = log(8) / 8

λ ≈ 0.169 (rounded to three decimal places)

So, the decay constant (λ) is approximately 0.169.

Example 7: Determine the remaining amount of a sample with an initial quantity of 60 grams, a decay constant of 0.05, after a total time of 15 seconds.

Solution:

Given values:

Initial quantity (N0) = 60 grams

Decay constant (λ) = 0.05

Total time (t) = 15 seconds

Using the decay formula:

N(t) = 60 * (2.71828)^(-0.05 * 15)

N(t) = 60 * (2.71828)^(-0.75)

N(t) ≈ 60 / 2.117

N(t) ≈ 28.34 grams (rounded to two decimal places)

These examples illustrate calculations involving radioactive decay and the determination of remaining quantities or decay constants for different scenarios.