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Answer :
Final answer:
The decay constant of a radioactive material that decays from 500 to 200 decays per second over 50 minutes is approximately 0.0002 sec^-1, calculated using the formula for first-order decay.
Explanation:
The subject pertains to the concept of radioactive decay, specifically, determining the decay constant of a sample. We start with the formula for the number of radioactive nuclei remaining at time t: N = N0e^-λt, where N0 is the initial count rate, N is the count rate at time t, λ is the decay constant, and t is the decay time.
Given that N0 is 500 decays/second, N is 200 decays/second, and t is 50 minutes (which we convert to seconds by multiplying by 60 to get 3000 seconds), we rearrange the formula to isolate λ: λ = -ln(N/N0) / t. Substituting the given numbers, we find that λ is approximately 0.0002 sec^-1. This value is the decay constant of the radioactive sample, meaning approximately 0.02% of the sample decays each second.
This application of first-order kinetics and knowledge of radioactive decay gives us insight into decay constants and half-lives, crucial in various fields, such as nuclear power management and medical imaging techniques.
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