Today: Bohr Model Details
1. Model for discrete electron orbits in atoms.
2. Prediction of allowed radii from new
assumptions.
3. Discrete electronic energies calculated.
• HWK 8 due Wed. 10AM.
• Week 8 online participation available until Tuesday
• Reading for Monday.: TZ&D Chap. 6.1-6.4.
Bohr model background
1. 1/λnm = R (1/m2 - 1/n2) – Big experimental target
2. Gravity -1/r2 force gives orbits. Planetary resonances
Coulomb -ke2/r2 force between electron and proton,
So might expect orbits.
3. Classical EM says electron going in circle should
radiate energy, and spiral in. (accelerating charge
radiates)
proton
+
-
Bohr’s additional hypothesisa. Fixed orbits are stable (quant.)
and at fixed energies
b. But WHY??
Important ideas about orbits arise from
classical physics (review of phys I- planets etc)
-
+
v
Basic connections between
r, v, and energy!
r
Important ideas about orbits arise from
classical physics (review of phys I- planets etc)
-
v
Basic connections between
r, v, and energy!
r
F = ma = Fcent = ?
(quick memory check)
a. -mv
b. -mv2/r
c. -v2/r2
d. -mvr
2
Ans b) Fcent = -mv /r
e. don’t remember learning
2
SO: Coulomb force, = kq+ q-/r , anything related to this
mv2/r =ke2/r2
mv2 = ke2/r
+
What does this say about total mechanical energy?
r
Fcent
v
Basic connections between
r, v, and energy!
mv2 =ke2/r
+
E = KE + PE = 1/2mv2 +PE
0 distance from proton
potential
energy
PE =?
PE = -ke2/r
so E = 1/2ke2/r -ke2/r = -1/2ke2/r
if you know E, you know r!
if you know r, you know E!
if you know r or E, you know v!
Nucleus
Electron
++
++
-
Higher
Energy
-
Energy
levels
When electron moves to an orbit further from the nucleus,
a. energy of electron decreases because energy is released as positive
and negative charges are separated, and there is a decrease in
electrostatic potential energy of electron since it is now further away
b. energy of electron increases because it takes energy input to separate
positive and negative charges, and there is an increase in the
electrostatic potential energy of the electron.
c. energy of electron increases because it takes energy input to separate
positive and negative charges, and there is a decrease in the
electrostatic potential energy of the electron.
Nucleus
Electron
++
++
F-
Higher
Energy
-
Energy
levels
Electron feels force toward nucleus. External agent must
work against that force to move electron farther away, so
there is an increase in the PE.
Also need a tangential force to change electron velocity to
sit in the new orbit. SLOWER at larger radius, so a decrease
in KE. PE increase is larger
So electrons at higher mechanical energy levels are further
from the nucleus!
Energy (total) levels
for electrons
3rd ex. lev.
2nd ex. lev.
1st excited
level
ground level
Bohr- “Electron in orbit with only certain particular energies”.
This implies that an electron in Bohr model of hydrogen atom:
a. is always at one particular distance from nucleus
b. can be at any distance from nucleus.
c. can be at certain distances from nucleus corresponding to
energy levels it can be in.
d. must always go into center where potential energy lowest
0 distance from proton
potential
energy
Warning:
Bad mix of representations
potential energy (curve)
total energy (lines)
Energy levels
for electrons
3rd ex. lev.
2nd ex. lev.
1st excited
level
ground level
“Electron in orbit with only certain particular energies”.
This implies that an electron in Bohr model of hydrogen atom:
a. is always at one particular distance from nucleus
b. can be at any distance from nucleus.
c. is at certain distances from nucleus corresponding to energy
levels it can be in.
d. must always go into center where potential energy lowest
v
r
-
+
so E = -1/2 ke2/r
Fcent
Only certain E levels should exist.
e can hop down to lowest level, giving off
photons when making a jump, stable in the
lowest level.
0 distance from proton
potential
energy
But what determines these “special” energies?
Complex argument based on idea that
at large sizes, electron should radiate
classically, differences only at small size.
(correspondence principle).
Quantized angular momentum L = mvr=nh
Predicted special E’s.
Bohr calculated special energies.
label energy level with n (n = 1, 2, 3, …)
involved bunch of constants, h, m, e, c
that when combined (see book) give
En = -13.6 eV/n2
This then predicts size of jumps between
levels.
Agreed with observed spectra/Balmer
series to four decimal places!!
(since E and r, connected, also predicts
radius of each orbit. Lowest orbit is “Bohr
radius”, ab=0.053 nm, rn =abn2)
Review Bohr Model – see book 5.6
Bohr started with 3 basic ideas:
Ordinary
Classical
Mechanics
1. Energy Cons.: E = KE + PE = ½mv2 - ke2/r
2. Centripetal Force: Fcent = mv2/r = ke2/r2
3. Angular Momentum Quantization L = n= Totally new idea:
Derived from
Correspondence
Principle
Solve 3 for v ⇒ mvr = n= ⇒ v = n=/mr
Sub 3 into 2, solve for r to get rn = n2=2/mke2 = n2aB
Hydrogen orbital radii
Sub 2 into 1 to get E = -ke2/2r
Hydrogen
Sub rn into E to get En = -mk2e4/2=2n2 = E1/n2 energies
where E1 = -13.6eV = ground state energy of H
& aB= =2/mke2 = Bohr radius = size of H in gnd state.
Note: k =1/4πε0 (textbook)
Successes of Bohr Model
• Explains source of Balmer formula and predicts
empirical constant from fundamental constants:
1/λ12 = R(1/n22 - 1/n12) ⇔ Ephoton = E1(1/n22 - 1/n12)
R = 1/(91.2nm) = mk2e4/4πc=3
• Explains variations in R for different single
electron atoms.
• Predicts approximate size of hydrogen atom
• Explains (sort of) why atoms emit discrete
spectral lines
• Explains (sort of) why electron doesn’t spiral into
nucleus
Which of the following principles of classical
physics is violated in the Bohr model?
A.
Opposite charges attract with a force inversely
proportional to the square of the distance between
them.
B. The force on an object is equal to its mass times its
acceleration.
C. Accelerating charges radiate energy.
D. Particles always have a well-defined position and
momentum.
E. All of the above.
Note that both A & B are used in derivation of Bohr model.
Bohr model is a weird mix of classical
physics and arbitrary rules…
• Why is angular momentum quantized yet
Newton’s laws still work?
• Why don’t electrons radiate when they are
in fixed orbitals yet Coulomb’s law still
works?
• No way to know a priori which rules to
keep and which to throw out…
What things CAN’T the Bohr model explain?
• WHY is angular momentum quantized?
• WHY don’t electrons radiate when they are in
fixed orbitals?
• How does electron know which level to jump to?
(i.e. how to predict intensities of spectral lines)
• Can’t be generalized to more complex (multielectron) atoms
• Shapes of molecular orbits and how bonds work
• Can’t explain doublet spectral lines
Ideas for how to resolve these
problems?