Bohr Model of the Atom

Question 1

Rutherford’s Alpha Particle Scattering Experiment

Answer 1

-fired alpha particles
(He nuclei) at a sheet of gold foil
-the foil was surrounded by a fluorescent screen
-he found that whilst most particles passed straight through the foil with little or no deflection. some were deflected through angles as large as 180 degrees
-this huge range Iin scattering angle had to be caused by concentration of positive charge at the centre of the atom

Question 2

Classical Model of the Atom

Answer 2

• from Rutherford’s scattering experiment the classical model of the atom was developed
• electrons orbiting a central nucleus

Question 3

Classical Model of the Atom

Hydrogen - Forces

Answer 3

centripetal force on the electron:
F = mv²/r
coulomb electrostatic force on the electron:
F = ke²/r²
for a stable orbit, these to forces are equal
mv²/r = ke²/r²
v² = ke²/mr

Question 4

Classical Model of the Atom

Hydrogen - Energy

Answer 4

total energy of the electron is equal to its kinetic energy plus its potential energy
KE = mv²/2
PE = -ke²/r
Total E = mv²/2 - ke²/r
but from the forces we know that mv² = ke²/r
so Et = ke²/2r - 2ke²/2r
Et = -ke²/2r
total electron energy is also negative as it is the energy required to remove the electron, this is true for any bound system

Question 5

Problems With the Classical Model of the Atom

Answer 5

• the electron orbit is an accelerated charge so should emit radiation hence losing energy
• if this was the case, the electron would spiral into the nucleus within seconds, but this doesn’t happen
• this model also cannot explain atomic spectra

Question 6

Bohr’s Postulates

Answer 6

0. electrons move in circular orbits about the nucleus
1. the total energy E in a particular orbit remains constant
2. electrons can change orbits discontinuously, and they emit a photon of EM radiation when going from an orbit of higher energy Ei to an orbit of lower energy Ef, where hf = Ei-Ef
3. electrons are only allowed in orbits for whih orbital angular momentum, L, is an integral multiple of ℏ,
   L = mvr = nℏ

Question 7

The Bohr Radius

Radii of the Hydrogen Atom

Answer 7

-using the Classical model, 
v = √(ke²/mr)
-using Bohr Postulate 3.
v = nℏ/mr
-eliminating v:
√(ke²/mr) = nℏ/mr
r = n²*ℏ²/mke²
-the inner most orbit, r1 is often called the Bohr radius, ao
r1= a0 = 1²*ℏ²/mke² = ℏ²/mke²
a0 = 5.3 x 10^-11m
-this means:
rn = n²*0.5Å

Question 8

Atomic Spectra

Hydrogen - equation for photon energy for a transition between ni and nf

Answer 8

-classical expression for the energy of an electron in an orbit
En = -ke²/2rn
-substitute rn = n²*ℏ²/mke²
En = -1/n²*(k²me^4)/2ℏ²
-can also substitute 
rn = n²*0.5Å 
En = -ke²/2a0n² = -13.6eV/n²
-use hf = Ei -Ef
hf = hc/λ = 13.6eV(1/nf²-1/ni²)

Question 9

Application of de Broglie’s Ideas to the Bohr Model of the Atom

Answer 9

-considering the wave behaviour of electrons
-the wavelength of an electron:
λ = h/ρ = hr/L = hr/nℏ = 2πr/n
-so nλ = 2πr
-this orbit corresponds to one complete electron wave wrapped around so that the beginning meets the end
-an electron can only orbit the nucleus with its orbit being an integral number of de Broglie wavelengths

Question 10

Bohr’s Correspondence Principle

Answer 10

-classically we expect the orbits of the electro to be able to take any energy. not just those with L = nℏ
-consider what happens to the spacing between electron orbits in Bohr’s model when n becomes very large
E(n+1) - En = ΔEn
= E1(1/n² - (n+1)²)
-as n->∞, ΔEn->0 i.e. allowed energies are very close together approaching the classical continuum

Question 11

Franck-Hertz Experiment

Answer 11

evacuated tube containing low pressure mercury vapour

• electrons are accelerated through a potential difference towards a positively charge plate, A
• the current between two plates A and B in the tube is recorded
• the current collected at a rises until the potential reaches 4.9V when it drops suddenly
• this corresponds with the point when electrons have the right energy to cause electrons in the mercury to transition to the first excited state
• subsequent peaks indicate higher excited states

Question 12

Bohr’s First Postulate

Answer 12

the electron in the hydrogen atom can move only in certain nonradiating circular orbits called stationary orbits

Question 13

Bohr’s Second Postulate

Answer 13

photon frequency from energy conservation

f = (Ei - Ef) / h

Question 14

Bohr’s Third Postulate

Answer 14

quantised angular momentum
mvnrn = nℏ,
where n = 1,2,3,…