Question

Be sure to answer all parts. Consider the following energy levels of a hypothetical atom: E4 −2.01 × 10−19 J E3 −5.71 × 10−19 J E2 −1.25 × 10−18 J E1 −1.45 × 10−18 J (a) What is the wavelength of the photon needed to excite an electron from E1 to E4? × 10 m (b) What is the energy (in joules) a photon must have in order to excite an electron from E2 to E3? × 10 J (c) When an electron drops from the E3 level to the E1 level, the atom is said to undergo emission. Calculate the wavelength of the photon emitted in this process. × 10 m

Answer 1

(a) 159 nm

First of all, let's calculate the energy difference between the level E1 and E4:

$\Delta E=E_4 -E_1 = -2.01\cdot 10^{-19}J-(-1.45\cdot 10^{-18} J)=1.25\cdot 10^{-18} J$

Now we know that this energy difference is related to the wavelength of the absorbed photon by

$\Delta E=\frac{hc}{\lambda}$

where

$h=6.63\cdot 10^{-34} Js$ is the Planck constant

$c=3\cdot 10^8 m/s$ is the speed of light

$\lambda$ is the wavelength of the photon

Solving for $\lambda$, we find

$\lambda=\frac{hc}{\Delta E}=\frac{(6.63\cdot 10^{-34} Js)(3\cdot 10^8 m/s)}{1.25\cdot 10^{-18} J}=1.59\cdot 10^{-7} m = 159 nm$

b) 293 nm

As done in part a), let's calculate the energy difference between the level E2 and E3:

$\Delta E=E_3 -E_2 = -5.71\cdot 10^{-19}J-(-1.25\cdot 10^{-18} J)=6.79\cdot 10^{-19} J$

this energy difference is related to the wavelength of the absorbed photon by

$\Delta E=\frac{hc}{\lambda}$

Solving for $\lambda$ again, we find

$\lambda=\frac{hc}{\Delta E}=\frac{(6.63\cdot 10^{-34} Js)(3\cdot 10^8 m/s)}{6.79\cdot 10^{-18} J}=2.93\cdot 10^{-7} m = 293 nm$

c) 226 nm

As done in part a) and b), let's calculate the energy difference between the level E1 and E3:

$\Delta E=E_3 -E_1 = -5.71\cdot 10^{-19}J-(-1.45\cdot 10^{-18} J)=8.79\cdot 10^{-19} J$

this energy difference is related to the wavelength of the emitted photon by

$\Delta E=\frac{hc}{\lambda}$

Solving for $\lambda$ again, we find

$\lambda=\frac{hc}{\Delta E}=\frac{(6.63\cdot 10^{-34} Js)(3\cdot 10^8 m/s)}{8.79\cdot 10^{-18} J}=2.26\cdot 10^{-7} m = 226 nm$