Today: Bohr Model Details

• HWK 8 due Wed. 10AM.• Week 8 online participation available until Tuesday • Reading for Monday.: TZ&D Chap. 6.1-6.4.

1.Model for discrete electron orbits in atoms.

2.Prediction of allowed radii from new assumptions.

3.Discrete electronic energies calculated.

1. 1/λnm = R (1/m2 - 1/n2) – Big experimental target2. 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)

+ -

protonBohr’s additional hypothesis-

a. Fixed orbits are stable (quant.) and at fixed energies

b. But WHY??

Bohr model background

Important ideas about orbits arise fromclassical physics (review of phys I- planets etc)

+

-

r

vBasic connections betweenr, v, and energy!

Important ideas about orbits arise fromclassical physics (review of phys I- planets etc)

+

-

r

vBasic connections betweenr, v, and energy!

F = ma = Fcent = ?(quick memory check)a. -mvb. -mv2/rc. -v2/r2

d. -mvre. don’t remember learninganything related to this

Ans b) Fcent = -mv2/rSO: Coulomb force, = kq+ q-/r2,

mv2/r =ke2/r2

mv2 = ke2/r

What does this say about total mechanical energy?

vBasic connections betweenr, v, and energy!

mv2 =ke2/r+

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rFcent

E = KE + PE = 1/2mv2 +PEPE =?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!

distance from proton0

potentialenergy

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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.

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NucleusElectron

-EnergylevelsHigher

Energy

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NucleusElectron

-Energylevels

So electrons at higher mechanical energy levels are further from the nucleus!

F

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.

HigherEnergy

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

Energy (total) levelsfor electrons

ground level

1st excitedlevel

2nd ex. lev.3rd ex. lev.

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 nucleusb. 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

Energy levelsfor electrons

ground level

1st excitedlevel

2nd ex. lev.3rd ex. lev.

“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 nucleusb. 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

distance from proton0

potentialenergy

Warning:Bad mix of representations

potential energy (curve)total energy (lines)

v

+

-

rFcent so E = -1/2 ke2/r

distance from proton0

potentialenergy

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.

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But what determines these “special” energies?Complex argument based on idea thatat large sizes, electron should radiateclassically, differences only at small size.(correspondence principle). Quantized angular momentum L = mvr=nhPredicted 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/Balmerseries 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 5.6Bohr started with 3 basic ideas:

1. Energy Cons.: E = KE + PE = ½mv2 - ke2/r2. Centripetal Force: Fcent = mv2/r = ke2/r2

3. Angular Momentum Quantization L = n

Solve 3 for v ⇒ mvr = n ⇒ v = n /mrSub 3 into 2, solve for r to get rn = n2 2/mke2 = n2aB

Sub 2 into 1 to get E = -ke2/2rSub rn into E to get En = -mk2e4/2 2n2 = E1/n2

where E1 = -13.6eV = ground state energy of H& aB= 2/mke2 = Bohr radius = size of H in gnd state.

Ordinary Classical Mechanics

Totally new idea: Derived from Correspondence Principle

Hydrogen orbital radiiHydrogen energies

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/n2

2 - 1/n12) ⇔ Ephoton = E1(1/n2

2 - 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.

• 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…

Bohr model is a weird mix of classical physics and arbitrary rules…

• 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 (multi-

electron) atoms• Shapes of molecular orbits and how bonds work• Can’t explain doublet spectral lines

What things CAN’T the Bohr model explain?

Ideas for how to resolve these problems?