Question

A radioactive decay forms an isotope of the original nucleus with the emission of the following particles

• A. one $\alpha$- and four $\beta^-$-particles
• B. one $\alpha$- and one $\beta^-$-particles
• C. one $\alpha$- and two $\beta^-$-particles
• D. four $\alpha$- and one $\beta^-$-particles

Hint:

Decay Chains and Isotopes

• $\alpha$ decay: $A - 4$, $Z - 2$
• $\beta^-$ decay: $A$ unchanged, $Z + 1$
• Isotopes: Same $Z$, different $A$
• To restore $Z$ after one $\alpha$ decay, emit two $\beta^-$ particles.

Answer 1

The Correct Option is C

Let the initial nucleus be $^A_ZX$. Emission of:

• One $\alpha$-particle decreases $A$ by 4 and $Z$ by 2.
• Each $\beta^-$-particle increases $Z$ by 1.

For the final nucleus to be an \textit{isotope}, $Z$ must remain unchanged. Let one $\alpha$ and $x$ $\beta^-$ particles be emitted. Change in $Z$: \[ Z_\text{final} = Z - 2 + x = Z \Rightarrow x = 2 \] So, one $\alpha$ and two $\beta^-$ emissions restore $Z$, making the final nucleus an isotope (same $Z$, different $A$). Hence, option (3) is correct.

Answer 2

Step 1: Understand radioactive decay and particle emission
Radioactive decay is a process where an unstable nucleus loses energy by emitting radiation in the form of particles such as alpha (α) particles, beta (β) particles, or gamma rays. This transforms the original nucleus into a different isotope or element.

Step 2: Types of particles emitted in decay
- An alpha (α) particle consists of 2 protons and 2 neutrons and reduces the atomic number by 2 and mass number by 4.
- A beta (β⁻) particle is an electron emitted when a neutron converts into a proton, increasing the atomic number by 1 without changing the mass number.

Step 3: Formation of a new isotope
The emission of one alpha particle reduces the nucleus’ size and atomic number significantly, while the emission of two beta minus particles increases the atomic number by 2 (one for each beta particle).

Step 4: Net effect on nucleus
The combination of one α-particle and two β⁻ particles emission changes the nucleus to form a new isotope of the original element (same mass number adjusted by α decay, and atomic number adjusted by β decays).

Step 5: Conclusion
Hence, the radioactive decay that forms an isotope of the original nucleus involves emission of one α-particle and two β⁻ particles.