Problem 69 - A Complete Resource Book in Chemistry for JEE Main

Question

In natural radioactive disintegration, U-238 emits one \(\alpha\) and two \(\beta\) and then five \(\alpha\) particle successively. The end product obtained is (a) \({ }_{82} \mathrm{~Pb}^{218}\) (b) \({ }_{82} \mathrm{~Pb}^{214}\) (c) \({ }_{84} \mathrm{~Pb}^{218}\) (d) \({ }_{82} \mathrm{~Pb}^{216}\)

Step-by-Step Solution

Answer

The end product is \( ^{214}_{82}\text{Pb} \), option (b).

Step 1: Understanding Alpha Decay

Alpha decay involves the emission of an \( \alpha \) particle which consists of 2 protons and 2 neutrons. Thus, the atomic number (Z) decreases by 2 and the mass number (A) decreases by 4.

Step 2: Understanding Beta Decay

Beta decay involves the emission of a \( \beta \) particle, resulting in the conversion of a neutron into a proton. As a result, the atomic number increases by 1 while the mass number remains unchanged.

Step 3: Applying Alpha and Beta Decays

Start with U-238: \( ^{238}_{92}\text{U} \). After emitting 1 \( \alpha \) and 2 \( \beta \), apply the changes: \( \alpha \) emission decreases the atomic number Z by 2 to 90 and mass number A by 4 to 234, making it \( ^{234}_{90}\text{Th} \). After 2 \( \beta \) decays, Z becomes 92 (since 90 + 1 + 1 = 92) and A remains 234, resulting in \( ^{234}_{92}\text{U} \).

Step 4: Successive Alpha Decays

The problem then states that five \( \alpha \) particles are emitted successively. Each \( \alpha \) decay decreases Z by 2 and A by 4. So, after 5 \( \alpha \) decays, Z will be 92 - (5*2) = 82 and A will be 234 - (5*4) = 214, making it \( ^{214}_{82}\text{Pb} \).

Step 5: Conclusion

The end product of this series of radioactive decays, starting from \( ^{238}_{92}\text{U} \), is \( ^{214}_{82}\text{Pb} \). The most likely corresponding answer from the given options is option (b) \( ^{214}_{82}\text{Pb} \).

Key Concepts

Alpha Decay

Alpha decay is a type of radioactive decay where an unstable atomic nucleus transforms into a different nucleus, with the emission of an alpha particle. An alpha particle's composition includes two protons and two neutrons, identical to the nucleus of a helium atom. When an atom undergoes alpha decay, the:

• atomic number (Z) decreases by 2 because it loses two protons,
• mass number (A) decreases by 4 due to the loss of two protons and two neutrons.

This form of decay is common in heavy elements like uranium-238. It's a crucial process in the transformation pathways of radioactive substances. As alpha decay occurs, the substance gradually becomes more stable, moving down the chain of decay products, each a step closer to a stable end product.

Beta Decay

In beta decay, a neutron in the nucleus is transformed into a proton, and a beta particle, which is an electron or positron, is emitted. This decay process is essential in neutron-rich nuclei striving to reach stability. During beta decay:

• the atomic number (Z) increases by 1, as a neutron converts to a proton,
• the mass number (A) remains unchanged, since the total nucleons are unaffected.

Beta decay plays a critical role in radioactive decay series, like the uranium-238 decay chain. It allows unstable isotopes to adjust the balance between protons and neutrons, leading toward stability. Despite no change in mass number, the alteration in atomic number signifies the formation of a new element.

Uranium-238 Decay Chain

The decay chain of uranium-238 is a sequence of radioactive decays, beginning with uranium-238. This series includes various emissions of alpha and beta particles until reaching a stable isotope. In this specific problem, U-238 undergoes one alpha and two beta decays initially, altering its atomic and mass numbers.
This then follows with five successive alpha decays. The step-by-step emission in the uranium-238 decay chain leads ultimately to a stable isotope of lead. Understanding this decay process helps illustrate the eventual transformation and disposal of nuclear matter via nature’s gradual stabilization mechanisms.

Atomic Number Changes

Atomic number changes are central to understanding radioactive decay. The atomic number (Z) indicates how many protons are present in the nucleus and defines the element. During decay:

• An alpha decay reduces the atomic number by 2,
• A beta decay increases the atomic number by 1.

These changes mean that the original element often transforms into a completely new one. As seen in the uranium-238 decay problem, the initial and successive decay processes gradually transform uranium into lead by altering the number of protons in the nucleus. Tracking these changes is vital for recognizing how elements evolve and stabilize.

Mass Number Changes

Mass number changes during radioactive decay reflect the total count of protons and neutrons in the nucleus. Different decay processes affect this number differently:

• In alpha decay, the mass number decreases by 4, as two protons and two neutrons are ejected from the nucleus.
• In beta decay, the mass number remains constant since a neutron changes into a proton (or vice versa) without altering the total nucleon count.

In the described decay sequence of uranium-238, the mass number changes after each alpha decay, reducing the total count of the particular isotopic state. Understanding these changes provides insights into the dynamics of radioactive decay and the eventual formation of stable elements from unstable isotopes.