Physics 2000 JAMB Past Questions - Question 37: If the fraction of the atoms of a radioactive material left after 120 years is V , what is the half-life of the material?

Correct Answer

C

Explanation

1→ 1/2→1/4→1/32→1/16→1/8→1/64half life =total time /stages =120/6=20yearsThe half-life of a radioactive material is the time it takes for half of the radioactive atoms in a sample to decay. You can calculate the half-life using the following formula: If the fraction of the atoms of a radioactive material left after 120 years is V , what is the half-life of the material?

Half-life (tâ‚�/2) = (ln(2) / λ)

Where:
- t�/2 is the half-life.
- ln(2) is the natural logarithm of 2, which is approximately 0.693.
- λ is the decay constant.

The decay constant (λ) can be determined from the fraction of atoms remaining after a certain time. In your case, you mentioned that after 120 years, the fraction of atoms remaining is V. The fraction of atoms remaining after a certain time (N(t)) can be expressed as:

N(t) = Nâ‚€ * e^(-λt)

Where:
- N(t) is the number of radioactive atoms remaining after time t.
- Nâ‚€ is the initial number of radioactive atoms.
- e is the base of the natural logarithm.
- λ is the decay constant.

In your case, N(t) is V (the fraction remaining), Nâ‚€ is 1 (assuming you start with 100% of the atoms), and t is 120 years.

V = 1 * e^(-λ * 120)

Now, you can solve for λ:

e^(-λ * 120) = V

Take the natural logarithm of both sides to solve for λ:

-λ * 120 = ln(V)

Now, divide by -120:

λ = -ln(V) / 120

Now that you have the value of λ, you can calculate the half-life using the formula mentioned earlier:

Half-life (t‚�/2) = (ln(2) / λ)

Half-life (t/2) = (0.693 / (-ln(V) / 120))

Simplify further:

t/2 = (0.693 * 120) / -ln(V)

So, the half-life of the radioactive material can be calculated using the formula above with the given fraction of atoms remaining after 120 years (V).