Question

What wavelength photon is required to excite a hydrogen atom from the n = 1 state to the n = 3 state?

Answer 1

Answer:

The wavelength required is 102.9 nm.

Explanation:

The energy levels for the hydrogen atom are

$E(n) = \frac{-13.6 \ eV}{n^2}$

So, for a transition from the first level to the third level we got

$\Delta E = E(3) - E (1)$

$\Delta E = \frac{-13.6 \ eV}{ 3 ^2} - \frac{-13.6 \ eV}{1^2}$

$\Delta E = \frac{-13.6 \ eV}{ 9} - \frac{-13.6 \ eV}{1}$

$\Delta E = \frac{8}{9} 13.6 \ eV$

$\Delta E = 12.09 \ eV$

$\Delta E = 12.09 \ eV * \frac{1.6 \ 10^-19 \ Joules}{1 \ eV}$

$\Delta E = 1.93 \ 10^-18 \ Joules$

So we need a photon with this energy.

The energy of a photon its given by

$E = h \nu = h \frac{c}{\lambda}$

So, the wavelength will be

$\lambda = \frac{h c}{E}$

$\lambda = \frac{6.62 \ 10^{-34} \ \frac{m^2 kg}{s} \ * 3.00 \ 10^8 \ \frac{m}{s}}{1.93 \ 10^-18 \ Joules}$

$\lambda = 10.29 \ 10^{-8} m$

$\lambda = 1.029 \ 10^{-7} m$

$\lambda = 102.9 \ nm$