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40-K decays by β− decay with a branching ratio of 89%. One gram of natural potassium emits 25 electrons per second from this decay. Calculate the lifetime of 40-K.

Answer :

Final answer:

The lifetime of 40-K is calculated using its half-life of 1.277 × 10⁹ years along with the decay rate, leading to a mean life determination.

Explanation:

The question asks us to calculate the lifetime of 40-K based on its decay properties. This involves using the concept of half-life, and the decay rate of the isotope. To perform the calculation, we can use the relationship between decay rate (activity), amount of substance, and its half-life. The half-life of 40-K is given as 1.277 × 10⁹ years.

The decay rate provided in the question is 25 electrons per second, which refers to 25 Becquerels since 1 Bq is equivalent to 1 decay per second. From this information, we can determine the lifetime or the mean life (τ) of 40-K using the formula τ = T1/2 / ln(2), where T1/2 is the half-life of the isotope. By substituting the value of T1/2 with 1.277 × 10⁹ years, we find that τ = (1.277 × 10⁹ years) / ln(2), which yields the lifetime of 40-K.

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