Question

Part A

Compute the energy separation between the ground and second excited states for an electron in a one-dimensional box that is 7.70 angstroms in length. Express the energy difference in kJ⋅mol−1.

Express your answer to three significant figures and include the appropriate units.

Part B

Compute the wavelength of light (in nm) corresponding to this energy.

Express your answer to three significant figures and include the appropriate units.

Answer 1

The energy of a particle in the n^th level, according to the particle in a box model is given by:

$E_{n}=\frac{n^{2}\cdot h^{2}}{8\cdot m\cdot L^{2}}$

Where n is the level number, h is Planck's constant, m is the mass of the electron and L is the length of the box. The energy difference we are looking for is:

$\Delta E=E_{3}-E_{1}=(9-1)\cdot \frac{h^{2}}{8\cdot m\cdot L^{2}}=\frac{h^{2}}{m\cdot L^{2}}=\frac{(6.63x10^{-34}J\cdot s)^{2}}{9.11x10^{-31}kg\cdot (7.70x10^{-10}m)^{2}}=8.14x10^{-19}J$

In order to convert this to a wavelength, we need to use the relation:

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

Where c is the speed of light and lambda is the wavelength. We rearrange and solve:

$\lambda =\frac{h\cdot c}{E}=\frac{6.63x10^{-34}J\cdot s\cdot 3x10^{8}m/s}{8.14x10^{-19}J}=2.44x10^{-7}M=244nm$