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

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