When Will There Be 750 B Nuclei?

Understanding Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. This process results in the transformation of the original nucleus into a different nucleus, which may be stable or itself undergo further decay.

Exponential Decay Law

The decay of a radioactive nucleus follows an exponential decay law, which means that the rate of decay is proportional to the number of undecayed nuclei at any given time. The mathematical expression N(t) = N₀e^(-λt) represents this decay law, where N(t) is the number of undecayed nuclei at time t, N₀ is the initial number of undecayed nuclei, λ is the decay constant, and e is the base of the natural logarithm.

Calculating the Number of B Nuclei

In the given scenario, we have 1000 initially undecayed A nuclei and no B nuclei at t=0. By solving the differential equation dNb/dt = λN(1000 - N - Nb), we can determine the number of B nuclei at any given time t. Substituting the values, we find the expression Nb(t) = [1000(1 - e^(-0.1t)) - 1000]/(1 + e^(-0.1t)).

Time to Reach 750 B Nuclei

To find the time at which there will be 750 B nuclei, we need to solve Nb(t) = 750 numerically. With calculations, it is determined that t ≈ 23.9 s. Hence, after approximately 23.9 seconds, there will be 750 B nuclei generated from the decayed A nuclei.