What is the probability that a radioactive atom having a mean life of 10 days decay during the fifth day ?

To determine the probability that a radioactive atom with a mean life of 10 days decays during the fifth day, we can utilize the concept of the exponential decay model, which is commonly applied in radioactive decay scenarios. The mean life, or average lifetime, of a radioactive atom is the expected time until it decays, and it is related to the decay constant.

Understanding the Exponential Decay Model

The decay of radioactive atoms can be described by the exponential decay formula:

N(t) = N0 * e^(-λt)

Where:

• N(t) is the number of atoms remaining at time t.
• N0 is the initial number of atoms.
• λ is the decay constant.
• e is the base of the natural logarithm.

The mean life (τ) is related to the decay constant (λ) by the formula:

τ = 1/λ

Given that the mean life is 10 days, we can calculate the decay constant:

λ = 1/τ = 1/10 days = 0.1 days^-1

Calculating the Probability of Decay

To find the probability that the atom decays during the fifth day, we need to calculate the probability that it has not decayed by the end of the fourth day and then decays before the end of the fifth day. This can be expressed mathematically as:

P(4 < t < 5) = P(t < 5) - P(t < 4)

Using the exponential decay formula, we can find these probabilities:

P(t < t') = 1 - e^(-λt')

Now, we can calculate:

• P(t < 4) = 1 - e^(-0.1 * 4) = 1 - e^(-0.4)
• P(t < 5) = 1 - e^(-0.1 * 5) = 1 - e^(-0.5)

Now, substituting these values into our probability equation:

P(4 < t < 5) = (1 - e^(-0.5)) - (1 - e^(-0.4))

This simplifies to:

P(4 < t < 5) = e^(-0.4) - e^(-0.5)

Final Calculation

Now, we can compute the values:

• e^(-0.4) ≈ 0.6703
• e^(-0.5) ≈ 0.6065

Thus:

P(4 < t < 5) ≈ 0.6703 - 0.6065 = 0.0638

Conclusion

The probability that a radioactive atom with a mean life of 10 days decays during the fifth day is approximately 0.0638, or about 6.38%. This calculation illustrates how the exponential decay model effectively describes the behavior of radioactive materials over time.