Radioactive decay is the release of alpha, beta, and gamma particles from unbalanced atoms known as radionuclides. Some substances, like uranium, have no stable forms and are therefore always radioactive. Radioactive substances are referred to as radionuclides. Rate of Decay is calculated to tell the exact amount of radioactive material that is being radiated. In this article, we will learn about the rate of decay formula and its examples in detail.
Half-Life of Radioactive Substances
A radionuclide decays into a different atom known as a decay product. Until the atoms achieve a stable state and stop being radioactive, they continue to change into new decay products. Most radionuclides only undergo one decay before stabilizing. Series radionuclides are those that decay in more than one step. The decay chain is the collection of decay products produced to achieve this equilibrium.
Each radionuclide has a unique decay rate that is measured by its "half-life." The amount of time it takes for half of the radioactive atoms present to decay is known as the radioactive half-life. Others have half-lives of hundreds, millions, or billions of years. Some radionuclides have half-lives of just a few seconds.
The time it takes for the activity of a specific quantity of a radioactive substance to decay to half of its initial value is known as the half-life (t1/2).
t1/2 = ln(2) / λ = τ ln(2)
where,
t1/2 is the half-life of a radioactive substance.
λ is the mean lifetime of a radioactive substance.
τ is the average lifetime of a radioactive substance before decay.
Rate of Decay Formula
The breakdown of radioactive particles into new types of particles from the parent radionuclide is known as radioactive decay. There are three types of radioactive decay, alpha, beta, and gamma decay, and the half-lives of each type of decay have different values depending on the type of ionization. Due to the emission of many particle kinds, the decay rate differs for each type of decay.
It's been observed that radioactive disintegration only happens when the nucleus is unstable. Alpha, beta, and gamma radiations are released by the unstable nucleus along with other ionizing particles and radiations as it loses energy. A radioactive element is one that has a nucleus that is unstable. Usually, the radioactive substance splits into two parts.
The term "Parent nuclide" denotes to one component of a radioactive element. The other component disintegrates into a "Daughter nuclide," a modified atom that differs from the parent radionuclide as a result of the bombardment. The decay product is another name for the daughter nuclide. This is due to the fact that the parent nuclide's atoms continue to decay and transform into new decay products. The decay product stops decaying when it reaches a stable state where its radioactivity disappears.
A first-order decay process determines the rate of radioactive particle decay. This indicates that it exhibits an exponential decline pattern, which is simple to calculate.
Nt = N0 e-λt
where,
Nt is the amount of radioactive particles at time (t)
N0 is the amount of radioactive particles at time (0)
λ is the rate of decay constant
t is time
Since this decay rate is exponential, taking the natural log on both sides of the equation will result in:
ln (Nt /N0) = -λt
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Solved Examples on Rate of Decay Formula
Example 1: If U-238 has a half-life of 4.468 × 109 years, determine its rate of decay constant.
Solution:
The problem refers to half life of U-238, Half of the original sample has already decayed. Hence the ratio N0/Nt = 0.5.
ln (Nt /N0) = -λt
ln 0.5 = -λ × 4.468 × 109
λ = 1.55 x 10-10 years-1
Rate of decay constant (λ) is 8.38 x 10-11 years-1
Example 2: If U-238 has a 35% life of 5.142 × 109 years, determine its rate of decay constant.
Solution:
35% half life of U-238 has already decayed, Hence the ratio N0/Nt = 0.65 as 65 percent of original sample remains.
ln (Nt /N0) = -λt
ln 0.65 = -λ × 5.142 × 109
λ = 0.838 x 10-10 years-1
Rate of decay constant (λ) is 8.38 x 10-11 years-1
Example 3: Determine the amount of time it will take for 25% of a sample of U-238 to radioactively decay with a decay constant of 1.55 x 10-10 years-1.
Solution:
Since 75% of the original sample is still present, the ratio Nt/N0 = 0.75. Where 25% of the sample has undergone radioactive decay.
Rate of decay constant (λ) = 1.55 × 10-10 years-1
ln (Nt /N0) = -λt
ln 0.75 = -1.55 × 10-10 years-1 × t
t = 1.86 x 109 years
The amount of time for 25% radioactive U-238 decay is 1.86 x 109 years
Example 4: Determine the amount of time it will take for 45% of a sample of U-238 to radioactively decay with a decay constant of 1.55 x 10-10 years-1.
Solution:
Since 55% of the original sample is still present, the ratio Nt/N0 = 0.75. Where 45% of the sample has undergone radioactive decay.
Rate of decay constant (λ) = 1.55 × 10-10 years-1
ln (Nt /N0) = -λt
ln 0.55 = -1.55 × 10-10 years-1 × t
t = 3.86 x 109 years
The amount of time for 45% radioactive U-238 decay is 3.86 x 109 years
Example 5: The half-life of PD-100 is 3.6 days. How many atoms will remain after 20.0 days, if one has 6.02 x 1023 at the beginning?
Solution:
Time = 20 days
Half-life = 3.6 days
Initial atoms = 6.02 ×1023 atoms
Formula used to determine number of atoms after 20 days.
N = N0 × 1/2 × t/t1/2
N = 6.02 ×1023 × 1/2 × 20/3.6
= 1.28 × 1022
The number of atoms present is 1.28 × 1022