Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (4th Edition)

Published by Pearson
ISBN 10: 0133942651
ISBN 13: 978-0-13394-265-1

Chapter 41 - Atomic Physics - Exercises and Problems - Page 1209: 57

Answer

It takes $~~t = 5.75~ns~~$ for 25% of the sample to decay.

Work Step by Step

Let $P$ be the probability that an excited atom will emit a photon. We can find the lifetime of the excited state: $P = \lambda~\Delta t$ $P = (\frac{1}{\tau})~(\Delta t)$ $\tau = \frac{\Delta t}{P}$ $\tau = \frac{0.20~ns}{0.010}$ $\tau = 20~ns$ We can find the time it takes for 25% of the sample to decay: $N = N_0~e^{-t/\tau}$ $0.75~N_0 = N_0~e^{-t/\tau}$ $0.75 = e^{-t/\tau}$ $ln(0.75) = -\frac{t}{\tau}$ $t = -ln(0.75)~(\tau)$ $t = -ln(0.75)~(20~ns)$ $t = 5.75~ns$ It takes $~~t = 5.75~ns~~$ for 25% of the sample to decay.
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