An electron in a one-dimensional box requires a wavelength of 8170 nm?

Question

An electron in a one-dimensional box requires a wavelength of 8170 nm to excite an electron from the n = 2 to the n = 5 energy level. Calculate the length of this box.

______nm

Answer 1

your answer is almost correct, (25-4) = 21.

then plug it in and you're good to go.

also, you have to convert 8170 nm to meters.

(8170x10^-9)

Answer 2

The energy levels of a quantum particle in a 1-dimensional "box" are given by:

E(n) = (h^2)*(n^2)/(8*m*L^2)

where h is Planck's constant, m is the rest mass of the particle, and L is the "length" of the "box".

The energy difference between the n=5 and the n=2 levels is then given by:

E(n=5  -> n=2) =  (25 - 4)*(h^2)/(8*m*L^2)

The wavelength and energy of a photon are related by:

E = h*c/wavelength, where c is the speed of light in a vacuum.  Equating these two enegies and solving for L gives:

h*c/wavelength = 19*(h^2)/(8*m*L^2)

L = sqrt[(19*h*wavelength)/(8*m*c)]

The mass of an electron is = 9.109*10^-31 kg, c = 2.998*10^8 m/s, and h = 6.626*10^-34 (kg*m^2)/s.  Plugging these values, along with the given value for the wavelength of the photon causing the transition, into the above equation gives:

L = 6.86*10^-9 m  = 6.86 nm

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Edit added in response to khuezi's answer:  

khuezi is absolutely correct.  I can't subtract 4 from 25 and get the correct answer!  Obviously, the correct equation and answer for the length of the box is:

L = sqrt[(21*h*wavelength)/(8*m*c)] = 7.2 nm