Radioactivity: Definition, Reason, Conservation Laws

Radioactivity: We all have heard about the nuclear bomb that was used against Japan in world war two or India’s nuclear test at Pokhran. Also, we all are aware of the nuclear power plants that generate household electricity, but how does these nuclear power plant function. What are the laws or principles that govern them? Is it the same principle behind the atom bomb and nuclear power plants? What are nuclear reactions? Why do they happen? How is it utilised to produce electricity? Let us read further to know more about radioactivity and the law related to it.

What is Radioactivity?

Radioactivity is the property of an element like uranium through which an unstable heavy nucleus gets converted into a stable nucleus emitting some bi-product, often an alpha-emission or gamma-emission. This process can occur through different types of reactions or decays depending on the bi-product of the reaction.
Example: Uranium \(238\) decays into thorium \(234\), and an alpha particle is emitted as a bi-product along with some amount of energy.
Law of conservation of momentum and law of conservation of mass-energy holds true during nuclear reactions.

Reason for Radioactivity

Radioactivity or nuclear reaction generally happens because the nucleus is unstable and it wants to attain stability and to attain stability, it undergoes a nuclear reaction.

Generally, for a lighter nucleus, when the number of protons and neutrons are equal, the given nucleus is stable, but in heavier atomic weight elements generally, the number of neutrons should be slightly more in order for the nucleus to be stable.

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If the number of protons is more than the required number, then it breaks into neutrons, but if the number of the neutron is more than the required number, then the neutron gets converted into a proton.

Alpha Decay

Alpha decay or alpha emission is a type of decay in which the unstable nuclei breaks into stable nuclei, and an alpha particle is released along with some amount of energy.
\(_{\text{Z}}^{\text{A}}{\text{X}} \to _{{\text{Z – 2}}}^{{\text{A – 4}}}{\text{Y + He + Q}}\)
\(Q\) is the value of energy released also known as \(Q\)-value.
It is the energy released due to the mass defect and the kinetic energy of the products. That is, some amount of mass gets converted into energy via Einstein’s mass-energy equivalence.
\(E=M c^{2}\)
\(Q\)-value is given by,
\(Q=\left(M_{x}-M_{y}-M_{H e}\right) c^{2}\)

Beta(-) Decay

Beta\((-)\) decay is a phenomenon in which the neutron breaks down into a proton and an electron.
\(n \rightarrow p+e+ {\bar v }\)
In the above equation, a particle known as antineutrino is also emitted.
\({}_{\text{Z}}^{\text{A}}{\text{X}} \to {}_{{\text{Z}} + 1}^{\text{A}}{\text{Y}} + {}_{ – 1}^0{\text{e}} + {\text{Q}}\)
\(Q\)-value is given by,
\(Q=\left(M_{x}-M_{y}-M_{e}\right) c^{2}\)

Beta(+) Decay

Beta\((+)\) decay or positron emission is a phenomenon in which a proton gets converted into a neutron with an emission of a positron.
\({\text{p}} \to {\text{n}} + {}_{ + 1}^0{\text{e}} + v\)
In the above equation, an additional particle known as neutrino is also emitted.
\(_{\text{Z}}^{\text{A}}{\text{X}} \to {}_{{\text{Z – 1}}}^{\text{A}}{\text{Y + }}{}_{{\text{ + 1}}}^{\text{0}}{\text{e + 0}}\)
\(Q\)-value of the equation is given by,
\(Q=\left(M_{x}-M_{y}-M_{e}\right) c^{2}\)
Gamma rays are generated sometimes because the resultant daughter nuclei may be in an excited state therefore, additional gamma photons are generated to bring it to a stable state.

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Conservation Laws in a Nuclear Reaction

Law of Radioactive Disintegration

The law of radioactive disintegration states that the rate of disintegration at any time is directly proportional to the amount of radioactive substance present at that time.
\(-\frac{d N}{d t} \propto N\)
\(\Rightarrow-\frac{d N}{d t}=\lambda N\)
Where,
\(\lambda\) is called the decay constant.
Decay constant depends on the nuclear reaction taken into consideration.
Let a nuclear reaction be,
\(X \rightarrow Y\)
At time \(t=0\), number of nuclei of \(X\) is \(N_{0}\) and number of nuclei of \(Y\) is zero.
At some time \(t=t\) the number of nuclei of \(X\) is \(N_{0}-N\) and number of nuclei of \(Y\) is \(N\)
We know that,
\(-\frac{d N}{d t}=\lambda N\)
\(\Rightarrow \int_{N_{0}}^{N}-\frac{d N}{N}=\int_{0}^{t} \lambda d t\)
\(\Rightarrow N=N_{0} e^{-\lambda t}\)

Half-life: It is the time after which the number of parent nuclei becomes half the initial amount.
\(t_{\frac{1}{2}}=\frac{\ln (2)}{\lambda}\)
Where,
\(\lambda\) is the decay constant.
Average life or mean life: It is the Average of life of all the nuclei in the sample.
\(T=\frac{\int_{0}^{\infty} N_{0} e^{-\lambda t} d t}{N_{0}}=\frac{1}{\lambda}\)

Successive Disintegration

When even after the first nuclear reaction, the product is not a stable nucleus, then the product itself undergoes a nuclear reaction again. This phenomenon is known as successive disintegration.
\(X \rightarrow Y \rightarrow Z\)
Here, since the \(Y\) is also unstable, therefore it further decays into \(Z\).
The law of radioactive disintegration is valid here.
At time \(t=0\), number of nuclei of \(X\) is \(N_{0}\), number of nuclei of \(Y\) is zero and number of nuclei of \(Z\) is zero.
At some time \(t=t\), the number of nuclei of \(X\) is \(N_{x} e^{-\lambda_{x} t}\), number of nuclei of \(Y\) is \(N_{y}\) and number of nuclei of \(Z\) is \(N_{0}-N_{x}-N_{y}\)
\(-\frac{d N_{y}}{d t}=\lambda_{x} N_{x}-\lambda_{y} N_{y}\)
And,
\(\frac{d N_{z}}{d t}=\lambda_{y} N_{y}\)

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Atomic bombs work on the principle, and the energy generated in the nuclear reaction is responsible for explosive power.

Nuclear power plant works on controlled nuclear reaction.

Sample Problems

Q.1. In a nuclear reaction, \(_{{\text{92}}}{{\text{U}}^{{\text{238}}}}{ \to _{\text{Z}}}{\text{T}}{{\text{h}}^{\text{A}}}{{\text{ + }}{\text{2}}}{\text{H}}{{\text{e}}^{\text{4}}}\) The value of \(A\) and \(Z\) are,
Ans: In the nuclear reaction
\(_{{\text{92}}}{{\text{U}}^{{\text{238}}}}{ \to _{\text{Z}}}{\text{T}}{{\text{h}}^{\text{A}}}{{\text{ + }}{\text{2}}}{\text{H}}{{\text{e}}^{\text{4}}}\)
Uranium decays into thorium, producing an alpha particle.
By the conservation of the proton and the neutron, comparing the mass number and the atomic number of the reactant and the product we get,
\(238=A+4 \Rightarrow A=234\)
and
\(92=Z+2 \Rightarrow Z=90\)

Q.2. In the reaction represented by equation \(_{\text{Z}}{{\text{X}}^{\text{A}}}{ \to _{{\text{Z – 2}}}}{{\text{Y}}^{{\text{A – 4}}}}{ \to _{{\text{Z – 2}}}}{{\text{Y}}^{{\text{A – 4}}}}{ \to _{{\text{Z – 1}}}}{{\text{K}}^{{\text{A – 4}}}}\) Which of the following will give the correct sequence radiations?
Ans: In alpha decay, mass number decreases by 4, and atomic number decreases by \(2\).
In beta decay, mass number remains the same, but the atomic number increases by \(1\).
In gamma decay, both mass number and the atomic number remains the same.
In the process,
\(_{\text{Z}}{{\text{X}}^{\text{A}}}{ \to _{{\text{Z – 2}}}}{{\text{Y}}^{{\text{A – 4}}}}\)
Mass number decreases by \(4\), and atomic number decreases by \(2\). Therefore, it is an alpha decay process.
In the process,
\(_{\text{Z}}{{\text{X}}^{\text{A}}}{ \to _{{\text{Z – 2}}}}{{\text{Y}}^{{\text{A – 4}}}}\)
Both mass number and atomic number remain the same. Therefore, it is a gamma decay.
In the process,
\(_{{\text{Z – 2}}}{{\text{Y}}^{{\text{A – 4}}}}{ \to _{{\text{Z – 1}}}}{{\text{K}}^{{\text{A – 4}}}}\)
Only atomic number increases by one. Therefore, it is beta decay.

Summary

In the given article, we learnt about the meaning of radioactivity and the reason for nuclear reaction or disintegration that is the nucleus is unstable and to attain stability, it disintegrates. We discussed different types of decays and the reaction. We also discussed how does the atomic and mass number changes in a different reaction. We then learnt about the mass defect and various other conservation laws that are followed during the reaction, such as law of conservation of charge, conservation of momentum. We also discussed the law of radioactive disintegration and also learnt about successive disintegration, and applied the radioactive decay law in the given case.
Nuclear reactions can be of two types:
Nuclear fusion: In this type of reaction, two lighter nuclei combine to produce a heavier nucleus. It occurs inside the sun.
Nuclear fission: In this type of reaction, a heavier nucleus breaks into two or more lighter nuclei. It is used in nuclear power plants.

PRACTICE QUESTIONS RELATED TO RADIOACTIVITY

FAQs on Radioactivity

Q.1. What is radioactivity?
Ans: Radioactivity is the property of the substance due to which it decays or disintegrates into daughter nuclei as its nuclei is not stable. Due to this nuclear reaction, the substance radiates alpha rays, electrons, or a positron and sometimes gamma rays as well.

Q.2. What is positron?
Ans: Positron is a particle like an electron, but instead of the negative charge, it possesses a positive charge equal to that of a proton.

Q.3. Does the law of radioactive decay or disintegration depend on the temperature or pressure?
Ans: The law of radioactive decay or disintegration does not depend on physical factors such as temperature or pressure.

Q.4. When is gamma rays produced?
Ans: When the daughter nuclei formed after the nuclear reaction is in the excited state, the gamma rays are produced, and the nucleus attains a stable state.

Q.5. Why does the successive disintegration occur?
Ans: Successive disintegration occurs because the daughter nuclei obtained from the first nuclear reaction is itself not stable and requires undergoing another disintegration in order to attain stability.

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