Answer
${\bf 5.0 \times 10^{16} }\, \text{photons/s}$
Work Step by Step
To calculate how many photons are emitted per second from atoms in the excited state, we need to use the relationship between the number of excited atoms, their lifetime in the excited state, and the rate of photon emission.
The decay rate $ r $, or the number of photons emitted per second by one excited atom, is given by the inverse of the lifetime $ \tau $:
$$
r = \frac{1}{\tau}
$$
We are given that $ \tau = 20\;\rm nm=20 \times 10^{-9} \, \text{s} $, so
$$
r = \frac{1}{20 \times 10^{-9}} = 5.0 \times 10^7 \, \text{s}^{-1}
$$
This means each excited atom emits $ 5.0 \times 10^7 $ photons per second.
So, the total number of photons emitted per second is the decay rate multiplied by the total number of excited atoms:
$$
N_\text{total} = r N_{\text{atoms}} $$
$$N_\text{total} = (5.0 \times 10^7 ) (1.0 \times 10^9) = \color{red}{\bf 5.0 \times 10^{16} }\, \text{photons/s}
$$
