A radioactive parent nucleus of Uranium-238 (^{238}_{92}\text{U}) decays through a sequenc… — JAMB UTME

A radioactive parent nucleus of Uranium-238 ( $^{238}_{92}\text{U}$ ) decays through a sequence of natural radioactive emissions to become a stable daughter nucleus of Lead-206 ( $^{206}_{82}\text{Pb}$ ). How many $\beta^-$ -particles are emitted in total during this decay process?

Answer: 6

Answer

The decrease in mass number from

238

206

32

, which corresponds to

8

alpha emissions (

32 / 4 = 8

). Emitting

8

alpha particles reduces the atomic number from

92

76

(

92 - 16 = 76

). To reach the atomic number of Lead (

82

), the atomic number must be increased by

6

(

82 - 76 = 6

), which requires emitting

6

beta-minus particles.

Step-by-Step Solution

Calculate the total change in mass number and deduce the number of alpha particles emitted.

The total mass number decreases by

238 - 206 = 32

. Each alpha particle (

^{4}_{2}\text{He}

) decreases the mass number by

4

, while beta particles have a mass number of

0

. Therefore, the number of alpha particles is

\frac{32}{4} = 8

Beta particles carry negligible mass compared to nucleons, making alpha emission the sole mechanism for mass number reduction.

Calculate the intermediate atomic number after all alpha decays.

Each alpha particle removes

2

units of atomic number. The total reduction in atomic number due to

8

alpha particles is

8 \times 2 = 16

. The intermediate atomic number is

92 - 16 = 76

Alpha particles consist of 2 protons and 2 neutrons.

Determine the number of beta-minus particles required to reach the final atomic number.

The final atomic number of Lead-206 is

82

. The atomic number must increase by

82 - 76 = 6

. Since each

\beta^-

-particle increases the atomic number by

1

, exactly

6

beta-minus particles are emitted.

A beta-minus decay converts a neutron into a proton, increasing the atomic number by 1 without altering the mass number.

Key Concept

Balancing mass numbers and atomic numbers in radioactive decay series

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