Radioactive Decay Formula

A parent nucleus splits into two or more daughter nuclei to reach the stage of stability. So the unstable nucleus is considered radioactive and while splitting, it loses energy in the form of radiation.

So the whole process initiating from subdivision to loss of energy is the radioactive decay.

The three most common types of decays are:

• Alpha decay

• Beta-decay (β - decay)

• Gamma decay (𝛾 - decay)

There are certain radioactive equations for three of these, say, for the gamma decay process, we have the gamma decay formula, proceeding with this, we have the activity of radioactive substance formula. 

Also, the formula for half-life decay helps us determine the time needed for half of the original population of radioactive atoms to decay, which we will understand with the help of the radioactive half-life formula.

Radioactive Decay Equation

As per the activity of radioactive substance formula, the average number of radioactive decays per unit time or the change in the number of radioactive nuclei present is given as:

                                A   =  -  dN/dt

Here,

• A is the total activity

• N is the number of particles

• T = time taken for the whole activity to complete

Also, we understand the following key points from the above radioactivity equation:

• The total activity relies entirely on the number of nuclei present, as we can see A dN....(a)

• During the radioactive decay, A decreases with time, as we can see that A ∝ 1/dt…..(b)

Below is the graph representing (a) and (b) definitions:

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Now, let us have a look at alpha, beta, and gamma radioactive equations.

History of Radioactivity

In the years 1899 and 1900, a British Ernest Rutherford (working at McGill University in Montreal, Canada) and the French Physicist named Paul Villard (working in Paris) did experimental investigations on electromagnetic radiation and separated them into three kinds.

Further, Rutherford named them alpha, beta, and gamma rays depending on the penetration of matter and deflection by a magnetic field.

Here, we will talk about the following three radioactive equations:

• Alpha decay formula

• Beta decay formula

• Gamma decay formula

Alpha Decay Formula

Alpha decay results in the emission of α-particles from the radioactive nucleus.

For example, the alpha decay of  92U238 into   90Th234  is as follows:

In the above radioactive decay formula, we notice the following things:

1. The Uranium nucleus emits an α-particle, and therefore, its mass and charge reduced, shown in the following equation:

                 Mass number:  238 -  4 = 234 

                 Charge number: 92 - 2 = 90        

Following this, a new element formed is Thorium (Th).

In general, the radioactivity equation (1) can be represented as:

zXA   ⇾   z - ₂YA-2  +  ₂He⁴ + Q  

Also, we see that after a spontaneous α-decay process, the total mass of  90Th234 and  2He 4 was less than 92U238.          

This means, the total mass-energy (Q) also decreases, equivalent to the difference between the Initial mass-energy and the final mass-energy, stated as:

                             Q = (mx - m - mHe)     

                              Q = (mx - my - mHe). c2   

In this equation, Q is the disintegration energy, which is shared by the daughter nucleus ‘y’ and an alpha particle He.

So, the alpha particle emission occurs in the following manner:

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Beta Decay Formula

Beta decay of Thorium, i.e.,  90Th234 emits a β-particle, where the mass number of the

daughter nucleus remains invariant, while the charge increments by 1, therefore, a new element Palladium 91Pa234  forms in the following way:

90Th234 ⇾  91Pa234 + -1e0 (β-particle)

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Here,

2. The mass number of Palladium is:

               234 - 0 = 234

Its charge number becomes 91 (90+1).

The general radioactivity equation for the beta decay process is:

Z XA ⇾  z+1YA + -1e0 + Q 

 Q = the energy released in β-decay.

Gamma Decay Formula

We know that gamma rays are emitted during the decay of radioactive atomic nuclei and definite subatomic particles. 

These powerful rays are produced by the hottest and most energetic objects in the universe and are present in the electromagnetic spectrum.

 Below is the gamma decay equation of Technetium-99m to Technetium-99:

 43Tc99m      →      43Tc99      +   0γ0   (Gamma radiation) ….(2)

Another example that initiates from the β-decay of 27Co60 turning into an exciting 28Ni60 nucleus, the radioactive decay equation for the same with the energy released is as follows:

27Co60  ⇾ 28Ni60**   + -1e0……(3)

So, when this exciting nucleus reaches the ground state, as a result, gamma rays are emitted with a release of energy in Mega electron Volts.

  28Ni60** ⇾ 28Ni60* + Eγ ( = 1.17 MeV)……(4)

  28Ni60* ⇾ 28Ni + Eγ (= 1.33 MeV)……(5)

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Radioactive Half Life Formula

The formula for half-life decay is:

\[N(t)=N_{0}(\frac{1}{2})^{\frac{t}{t_{1/2}}}\]……(6)

Here,

• N (t) is a function of time, which shows the amount of substance remaining after the decay in a given time.

• N is the initial quantity of the substance

• t  is the time elapsed, and

• t1/2  is the half-life of the decaying component

Definition of the Half-Life:

When half of the radioactive atom undergoes the decay process, the time needed for a quantity to reduce to half of its initial value is the half-life. When talking about the decay of half of the radioactive atoms, the time taken is the radioactive half-life. 

For this, we have a radioactive half-life formula:

\[t_{1/2}=\frac{0.693}{\lambda }\]

Here, λ is the decay constant. 

Now, let us understand the decay constant formula:

Let’s suppose that ‘N’ is the size of a population of radioactive atoms at a given time ‘t,’ and dN is the amount by which the population of the radioactive atom decreases in time dt; therefore,  the rate of change is given by the following equation:                          

dN/dt     = - λ N,  (λ  = decay constant)

Radioactive Half-Life Table

Radioactive Half-Life Graph

The graph of the above table is as follows:

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Radioactive Half-Life Formula Derivation

Let’s describe equation (6) in an exponential form:

\[N(t)=N_{0}(\frac{1}{2})^{\frac{t}{t_{1/2}}}\]

\[N(t)=N_{0}e^{\frac{-t}{\tau}}\]

Here,

τ =  A mean lifetime of the decaying quantity, which is positive 

λ is also positive

The three parameters (t1/2, τ , and λ are all directly related to each other in the following way:

\[t_{1/2}=\frac{ln(2)}{\lambda }\]  =   (ln (2). τ

We know that the value of the natural logarithmic of 2 is 0.693, so rewriting equation (7):

\[t_{1/2}=\frac{0.693}{\lambda }\]

This is the required radioactivity formula for radioactive half-life.

Conclusion

A phenomenon in which a ‌heavy‌ ‌unstable element‌ ‌disintegrates‌ ‌itself‌ into two or more daughter nuclei ‌without‌ ‌being‌ forced‌ ‌by‌ ‌any‌ ‌external‌ ‌agent‌ ‌to‌ ‌do‌ ‌so.‌ The process involves the emission of the following particles:

• α particle

• β particle

• 𝛾 particle