A heavy nucleus $Q$ of half-life $20$ minutes undergoes alpha-decay with probability of $60 \%$ and beta-decay with probability of $40 \%$. Initially, the number of $Q$ nuclei is $1000 $. The number of alpha-decay of $Q$ in the first one hour is
- 50
- 75
- 350
- 525
The Correct Option is D
Solution and Explanation
Step 1: Understand the problem
We are given an initial number of radioactive nuclei and the time interval over which the decay occurs. The number of decays over time follows the exponential decay law based on the concept of half-life.
Step 2: Total number of nuclei at the start
Initial number of nuclei = 1000
Step 3: Calculate how many half-lives occur in 60 minutes
Let’s say the half-life of the substance is 20 minutes. Then in 60 minutes:
Number of half-lives = 60 / 20 = 3
Step 4: Find the number of undecayed nuclei after 3 half-lives
Using the decay formula:
Remaining nuclei = Initial nuclei × (1/2)n
Remaining = 1000 × (1/2)3 = 1000 × 1/8 = 125
Step 5: Calculate total number of decayed nuclei
Decayed nuclei = Initial − Remaining = 1000 − 125 = 875
Step 6: Determine the number of α-decays
It is given that 60% of the decays are α-decays:
α-decays = 875 × 0.6 = 525
Final Answer:
The number of α-decays in 60 minutes is 525
Correct option: Option (D)
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Concepts Used:
Nuclear Physics
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies the atom as a whole, including its electrons
Radius of Nucleus
‘R’ represents the radius of the nucleus. R = RoA1/3
Where,
- Ro is the proportionality constant
- A is the mass number of the element
Total Number of Protons and Neutrons in a Nucleus
The mass number (A), also known as the nucleon number, is the total number of neutrons and protons in a nucleus.
A = Z + N
Where, N is the neutron number, A is the mass number, Z is the proton number
Mass Defect
Mass defect is the difference between the sum of masses of the nucleons (neutrons + protons) constituting a nucleus and the rest mass of the nucleus and is given as:
Δm = Zmp + (A - Z) mn - M
Where Z = atomic number, A = mass number, mp = mass of 1 proton, mn = mass of 1 neutron and M = mass of nucleus.