Question

1. (A) Find the de Broglie wavelength (in nm) associated with an electron that is moving with a velocity of 2310 km/s. The electron rest mass is 9.11 x 10^-31 kg.

Note, electrons having this speed would need to be treated as waves in atoms because the wavelength is on the order of the size of atoms.

(B) A baseball weighs 220 g. Top speed for a professional pitcher is about 100 mph when he throws a fast ball. Find the de Broglie wavelength (in nm) associated with a baseball that is moving with a velocity of 87 mph.

(C) What is the wavelength (in nm) of the line in the spectrum of the hydrogen atom that arises from the transition of the electron from the orbital with n =3 to the orbital with n = 2?

Answer 1

1.(A) We shall use the formula λ= h/ρ, where λ is the de Broglie wavelength, h is Planck's constant(6.6626x10^-34) and ρ is the momentum of the electron.

ρ= mv, where m is the mass of the electron and v is its velocity.

λ= h/mv
= (6.626x10^-34)/(9.11x10^-31)(2310 x 10³)
= 3.14 A⁰
= 3.14 x10^-10m
= 0.314x10^-9m
= 0.314 nm

(B) De Broglie wavelength = Plank's constant/momentum
Momentum = speed * mass
Speed = 87 miles per hour = 140012.9 meters per hour = 38.8 meters per second
Mass = 0.220 kilograms
Momentum = 8.536 kg meters/sec

Planks constant = 6.626 x 10^-34 J
Wavelength = 6.626 x 10^-34 /8.536meters

=7.76 x 10^-35 metres
= 7.76 x 10^-26 nm

(C) 1 / λ = Rh x (1 / n1² - 1/n2²)

Rydberg's constant = 10 973 731,6 m^-1

. . . then

λ = 1 / ( Rh x (1 / n1² - 1/n2²) )
λ = 1 / ( 10 973 731,6 x (1 / 3² - 1/2²) )
λ = 6.56 x 10^-7 m

=> λ = 656 nm