Q. in a radioacative sample at a given instant 100 atoms decay in one half life.

then the nmber of atoms decaying in one mean life at that instant is

200 126 75 150

pls give the correct reason and will rate if satisfied

To determine how many atoms decay in one mean life, we first need to understand the relationship between half-life and mean life in radioactive decay. The half-life is the time required for half of the radioactive atoms in a sample to decay, while the mean life (or average life) is the average time an atom exists before it decays. The mean life is related to the half-life by a specific factor.

Understanding Half-Life and Mean Life

The half-life (T_1/2) is defined as the time it takes for half of the radioactive atoms to decay. In your example, if 100 atoms decay in one half-life, that means you started with 200 atoms. After one half-life, 100 atoms remain, and 100 have decayed.

The mean life (τ) can be calculated from the half-life using the formula:

• τ = T_1/2 / ln(2)

Here, ln(2) is approximately 0.693. This means that the mean life is longer than the half-life, specifically about 1.442 times the half-life.

Calculating Decay in Mean Life

Now, to find out how many atoms decay in one mean life, we can use the decay constant (λ), which is related to the half-life:

• λ = ln(2) / T_1/2

In one mean life, the number of decays can be calculated using the formula:

• Number of decays = N₀ * (1 - e^-λτ)

Since τ is the mean life, and we know that in one half-life, 100 atoms decay, we can deduce that in one mean life, the number of decays will be approximately double that of one half-life. This is because the mean life accounts for the entire decay process, not just the point at which half the atoms have decayed.

Final Calculation

Given that 100 atoms decay in one half-life, we can conclude that:

• In one mean life, approximately 200 atoms will decay.

Thus, the correct answer to your question is 200. This result aligns with the understanding that the mean life encompasses a broader time frame, allowing for more decays to occur compared to just a single half-life.

Summary

In summary, if 100 atoms decay in one half-life, then in one mean life, you can expect about 200 atoms to decay. This relationship highlights the difference between the two time measures in radioactive decay and illustrates how the decay process works over time.