Question

An electron confined to a one-dimensional box has energy levels given by the equation

En=n2h2/8mL2

where n is a quantum number with possible values of 1,2,3,…,m is the mass of the particle, and L is the length of the box.   

Calculate the energies of the n=1,n=2, and n=3 levels for an electron in a box with a length of 180 pm .

Enter your answers separated by a comma.

Calculate the wavelength of light required to make a transition from n=1→n=2 and from n=2→n=3.

Enter your answers separated by a comma.

In what region of the electromagnetic spectrum do these wavelengths lie?

Answer 1

Length of box = 180 pm = 1.80x10^-10 m

Mass of electron = 9.1x10^-31 kg

En = n²h² / 8mL²   

Fon n = 1

E1 = 1²*(6.626x10^-34)² / 8*9.1x10^-31 * (1.80x10^-10)²

E1 = 1.86x10^-18 J

Fon n = 2

E2 = 2²*(6.626x10^-34)² / 8*9.1x10^-31 * (1.80x10^-10)²

E2 = 7.44x10^-18 J

Fon n = 3

E3 = 3²*(6.626x10^-34)² / 8*9.1x10^-31 * (1.80x10^-10)²

E3 = 1.67x10^-17 J

n =1 n = 2

We know that,

1 / = R [1/(n1)² - 1/(n2)²]

where,

= wavelength

R = Rydberg's constant (1.097 x 10⁷ m⁻¹)

1 / = (1.097 x 10⁷ m⁻¹) [1/(1)² - 1/(2)²]

= (1.097 x 10⁷ m⁻¹) [1 - 1/4]

= (1.097 x 10⁷ m⁻¹) * 0.75

= 0.82275 x10⁷ m^-1

= 1.215 x 10^-7 m

= 121.5 nm (ultraviolet region)

For, n =2 n = 3

1 / = (1.097 x 10⁷ m⁻¹) [1/(2)² - 1/(3)²]

= (1.097 x 10⁷ m⁻¹) [1/4 - 1/9]

= (1.097 x 10⁷ m⁻¹) * 0.138

= 0.15138 x10⁷ m^-1

= 6.605 x 10^-7 m

= 660.5 nm (visible region)