Question

Two adjacent allowed energies of an electron in a one-dimensional box are 5.4 eV and 9.6 eV .

What is the length of the box?

Answer 1

E_n = energy of n^th state = 5.4 eV

E_n+1 = energy of (n + 1)^th state = 9.6 eV (given)

L = length of the box (need to determine)

m = mass of electron = 9.1 x 10^-31 kg

h = plank's constant = 6.626 x 10^-34

Energy in nth state is given as

E_n = n²h²/(8 m L²) equation-1

Energy in (n+1) th state is given as

E_n+1 = (n + 1)²h²/(8 m L²)                                    equation-2

dividing equation-1 by equation-2

E_n/E_n+1= n²/(n + 1)²

putting the given values

5.4 /9.6 = n²/(n + 1)²

n = 3

using equation-1

E = n²h²/(8 m L²)

putting value in the equation.

5.4 x 1.6 x 10⁻¹⁹ = (3)² (6.63 x 10⁻³⁴)²/(8 (9.1 x 10⁻³¹) L²)

L = 7.9 x 10⁻¹⁰ m Answer