Question

Possible units for the disintegration constant λ are
1. kg/s
2. s/kg
3. hour
4. 1/day

Answer 1

Correct Answer - Option 4 : 1/day

CONCEPT:

Radioactivity:

• Radioactivity is a process by which the nucleus of an unstable atom loses energy by emitting radiation.
  • ​Two forces, namely the force of repulsion that is electrostatic and the powerful forces of attraction of the nucleus acts in the nucleus.
  • These two forces are considered extremely strong in nature.
  • The instability of the nucleus increases as the size of the nucleus increases because the mass of the nucleus becomes a lot to concentrate. That’s the reason why atoms of Plutonium, Uranium are extremely unstable and show radioactivity.
• The spontaneous breakdown of an atomic nucleus of a radioactive substance causing the emission of radiation from the nucleus is known as Radioactive decay.

​EXPLANATION:

• In radioactive decay, the number of remaining nuclei after decay time t is given as,

⇒ N = N_oe^-λt     -----(1)

Where N_o = initial amount of nuclei, and $\lambda =$ disintegration or decay constant

• We know that the exponential term is always unitless quantity. So the power of the exponential term is also a unitless quantity.

​The dimension of t is given as,

⇒ [t] = [M⁰ L⁰ T¹]     -----(2)

Let the dimension of disintegration constant is,

⇒ [λ] = [Mx Ly Tz]     -----(3)

The λt term is a unitless quantity. So,

⇒ [λt] = [M0 L0 T0]     -----(4)

by equation 2, equation 3, and equation 4,

⇒ [Mx Ly Tz][M0 L0 T1] = [M0 L0 T0]

⇒ [Mx Ly Tz+1] = [M0 L0 T0]     -----(5)

By equation 5,

⇒ x = 0,

⇒ y = 0,

⇒ z + 1 = 0

⇒ z = -1

Therefore the dimension of disintegration constant is,

⇒ [λ] = [M0 L0 T-1]     -----(6)

• By equation 6 it is clear that the possible units for the disintegration constant λ is 1/day.
• Hence, option 4 is correct.