Question

Calculate the wavelength of the electromagnetic radiation required to excite an electron from the ground state to the level with n = 5 in a one-dimensional box 45.7 pm in length.

Answer 1

Answer: wavelength λ = 2.9Å

Explanation:

Using the particle in a box model. The energy level level increases with n^2

En = (n^2h^2)/ 8mL^2 .....1

For the ground state, n = 1 to level n= 5, the energy level changes from E1 to E5

∆E = (5^2 - 1^2)h^2/8mL^2

but 5^2 - 1^2 = 24.

so,

∆E = 24h^2/8mL^2 .....2

And the wavelength of the radiation can be derived from the equation below:

E = hc/λ

λ = hc/E .......3

Substituting equation 2 to 3

λ = hc/[(24h^2)/ 8mL^2]

λ = 8mcL^2/(24h)

λ = 8mcL^2/24h .....4

Where,

n = energy state

h = Planck's constant = 6.626 × 10^-34 Js

m= mass of electron = 9.1 × 10^-31 kg

L = length = 45.7pm = 45.7×10^-12 m

E = energy

c= speed of light = 3.0 ×10^8 m/s

λ= wavelength

Substituting the values into equation 4 above

λ = [(8×9.1×3×45.7^2)/(24×6.626)] × 10^(-31+8-24+34)

λ = 2868.285390884 × 10^-13 m

λ = 2.9 × 10^-10 m

λ = 2.9Å

Answer 2

Answer:

2.11 m

Explanation:

Given data

h=6.626×$10^{-34\\}$(plank constant)

L=45.7 pm

n=n2-n1

n=5-1

n=4

where,

n2=5

n1=1

To find:

E=?

λ=? (Wavelength)

solution

The energy stored in an electron at a specified level is given by;

E=$h^{2}$×$n^{2}$÷8m$l^{2}$..........(1)

m=mass of electron(9.1×$10^{-31}$)

l=length of box

To find E

putting the value of given data in eq(1)

E=9.41×$10^{-16}$

To find λ

λ=hc/E............................(2)

c=3×$10^{8}$(speed of light)

putting the value in eq 2 to find wavelength

λ=2.11 m

Note:

There is a chance in calculation error. but the method is correct to solve the problem.