Answer
(a) $9.0\times 10^5~~$ photons have been emitted.
(b) The lifetime of the excited state is $~~8.7~ns$
Work Step by Step
(a) Each time an atom undergoes a quantum jump to the ground state, one photon is emitted.
We can find the number of photons that have been emitted:
$(0.90)(1.0\times 10^6) = 9.0\times 10^5$
$9.0\times 10^5~~$ photons have been emitted.
(b) We can find the lifetime of the excited state:
$N = N_0~e^{-t/\tau}$
$0.10~N_0 = N_0~e^{-t/\tau}$
$0.10 = e^{-t/\tau}$
$ln(0.10) = -\frac{t}{\tau}$
$\tau = -\frac{t}{ln(0.10)}$
$\tau = -\frac{20~ns}{ln(0.10)}$
$\tau = 8.7~ns$
The lifetime of the excited state is $~~8.7~ns$
