58. One gram of radioactive nuclei with a half life of 300 days is kept in an open container. The weight of nuclei remaining after 900 days (correct to 1 decimal place) is       mg.

Radioactive Decay Half-Life: Calculate the Remaining Mass After 900 Days

Correct Answer

125.0 mg

Concept of Half-Life

The half-life of a radioactive substance is the time required for half of its radioactive nuclei to decay. After every half-life, the remaining mass becomes half of its previous value.

The mathematical expression for radioactive decay is

N = N₀ × (1/2)^t/T

where:

N = Remaining mass after time t
N₀ = Initial mass
T = Half-life
t = Time elapsed

Step-by-Step Solution

Given:

Initial mass = 1 g = 1000 mg
Half-life = 300 days
Time elapsed = 900 days

First, calculate the number of half-lives.

Number of half-lives = 900 ÷ 300 = 3

This means the radioactive sample undergoes three successive halvings.

After the first half-life:

1000 mg → 500 mg

After the second half-life:

500 mg → 250 mg

After the third half-life:

250 mg → 125 mg

Using the formula directly:

N = 1000 × (1/2)³

N = 1000 × 1/8 = 125 mg

Therefore, the remaining mass of radioactive nuclei after 900 days is

125.0 mg

Why This Answer is Correct

Since 900 days correspond to exactly three half-lives, the original sample is reduced by half three times. Each half-life removes half of the remaining radioactive nuclei, not half of the original amount. This exponential decay is the defining characteristic of radioactive substances.

Alternative Shortcut Method

Instead of applying the formula, simply divide the initial mass by 2 for every half-life:

1000 mg → 500 mg → 250 mg → 125 mg

This method is extremely useful during competitive examinations because it saves valuable time.

Important Formula

Remaining Mass = Initial Mass × (1/2)^t/T

where t/T represents the number of half-lives completed.

Final Answer

Weight of radioactive nuclei remaining after 900 days = 125.0 mg

Answer: 125.0 mg