Bohr Model of Atom

Bohr proposed a model of the atom in which the electrons orbited the nucleus like planets around the sun

•Classical physics did not agree with his model. Why?

To overcome this objection, Bohr proposed that certain specific orbits corresponded to specific energy levels of the electron that would prevent them from falling into the protons

•As long as an electron had an ENERGY LEVEL that put it in one of these orbits, the atom was stable

Bohr introduced Quantization into the model of the atom

Bohr Model of Atom

By blending classical physics (laws of motion) with quantization, Bohr derived an equation for the energy possessed by the hydrogen electron in the nth orbit.

Bohr Model of the Atom

• The symbol n in Bohr’s equation is the principle quantum number– It has values of 1, 2, 3, 4, …– It defines the energies of the allowed orbits

of the Hydrogen atom– As n increases, the distance of the electron

from the nucleus increases

• Only orbits where n = some positive Only orbits where n = some positive integer are permitted.integer are permitted.

• The energy of an electron in an orbit has The energy of an electron in an orbit has a negative valuea negative value

• An atom with its electrons in the lowest An atom with its electrons in the lowest possible energy level is at possible energy level is at GROUND GROUND STATESTATE– Atoms with higher energies (n>1) are in Atoms with higher energies (n>1) are in

EXCITED STATESEXCITED STATES

Energy of quantized state = - Rhc/nEnergy of quantized state = - Rhc/n22

Atomic Spectra and Bohr

If e-’s are in quantized energy states, If e-’s are in quantized energy states, then ∆E of states can have only then ∆E of states can have only certain values. This explains sharp certain values. This explains sharp line spectra.line spectra.

Energy absorption and electron excitation

Spectra of Excited Atoms•To move and electron from the n=1 to an excited state, the atom must absorb energy

•Depending on the amount of energy the atom absorbs, an electron may go from n=1 to n=2, 3, 4 or higher

•When the electron goes back to the ground state, it releases energy corresponding to the difference in energy levels from final to initial

E = Efinal - Einitital

E = -Rhc/n2

E = -Rhc/nfinal2 - (-Rhc/ninitial

2) = -Rhc (1/ nfinal2 - 1/ninitial

2)(does the last equation look familiar?)

Origin of Line SpectraOrigin of Line Spectra

Balmer seriesBalmer series

Atomic Line Spectra and Atomic Line Spectra and Niels BohrNiels Bohr

Atomic Line Spectra and Atomic Line Spectra and Niels BohrNiels Bohr

Bohr’s theory was a great Bohr’s theory was a great accomplishment.accomplishment.

Rec’d Nobel Prize, 1922Rec’d Nobel Prize, 1922Problems with theory —Problems with theory —• theory only successful for H.theory only successful for H.• introduced quantum idea introduced quantum idea

artificially.artificially.• So, we go on to So, we go on to QUANTUMQUANTUM or or

WAVE MECHANICSWAVE MECHANICSNiels BohrNiels Bohr

(1885-1962)(1885-1962)

Wave-Particle DualityDeBroglie thought about how light, which is an electromagnetic wave, could have the property of a particle, but without mass.

He postulated that all particles should have wavelike properties

This was confirmed by x-ray diffraction studies

Wave-Particle DualityWave-Particle DualityWave-Particle DualityWave-Particle Duality

L. de BroglieL. de Broglie(1892-1987)(1892-1987)

de Broglie (1924) proposed de Broglie (1924) proposed that all moving objects that all moving objects have wave properties. have wave properties. For light: E = mcFor light: E = mc22

E = hE = h = hc / = hc / Therefore, mc = h / Therefore, mc = h / and for particles:and for particles: (mass)(velocity) = h / (mass)(velocity) = h /

Baseball (115 g) at 100 mphBaseball (115 g) at 100 mph = 1.3 x 10= 1.3 x 10-32-32 cm cm

e- with velocity = e- with velocity = 1.9 x 101.9 x 1088 cm/sec cm/sec = 0.388 nm= 0.388 nm

Wave-Particle DualityWave-Particle DualityWave-Particle DualityWave-Particle Duality

•The mass times the velocity of the ball is very large, so the wavelength The mass times the velocity of the ball is very large, so the wavelength is very small for the baseballis very small for the baseball•The deBroglie equation is only useful for particles of very small massThe deBroglie equation is only useful for particles of very small mass

1.6 The Uncertainty Principle

• Wave-Particle Duality– Represented a Paradigm shift for our understanding of

reality!

• In the Particle Model of electromagnetic radiation, the intensity of the radiation is proportional to the # of photons present @ each instant

• In the Wave Model of electromagnetic radiation, the intensity is proportional to the square of the amplitude of the wave

• Louis deBroglie proposed that the wavelength associated with a “matter wave” is inversely proportional to the particle’s mass

deBroglie Relationship• In Classical Mechanics, we caqn easily

determine the trajectory of a particle– A trajectory is the path on which the location and

linear momentum of the particle can be known exactly at each instant

• With Wave-Particle Duality:– We cannot specify the precise location of a

particle acting as a wave– We may know its linear momentum and its

wavelength with a high degree of precision• But the location of a wave? Not so much.

The Uncertainty Principle• We may know the limits of where an electron will be

around the nucleus (defined by the energy level), but where is the electron exactly?– Even if we knew that, we could not say where it would be the

next moment

• The Complementarity of location and momentum:– If we know one, we cannot know the other exactly.

Heisenberg’s Uncertainty Principle• If the location of a particle is known to within an

uncertainty ∆x, then the linear momentum, p, parallel to the x-axis can be simultaneously known to within an uncertainty, ∆p, where:

= h/2 = “hbar”

=1.055x10-34 J·s

• The product of the uncertainties cannot be less than a certain constant value. If the ∆x (positional uncertainty) is very small, then the uncertainty in linear momentum, ∆p, must be very large (and vice versa)

Wavefunctions and Energy Levels

• Erwin SchrÖdinger introduced the central concept of quantum theory in 1927:– He replaced the particle’s trajectory with a

wavefunction• A wavefunction is a mathematical function whose values

vary with position

• Max Born interpreted the mathematics as follows:– The probability of finding the particle in a region is

proportional to the value of the probability density (2) in that region.

The Born Interpretation

2 is a probabilty density:– The probability that the particle will be

found in a small region multiplied by the volume of the region.

– In problems, you will be given the value of 2 and the value of the volume around the region.

The Born Interpretation• Whenever 2 is large, the particle has a high

probability density (and, therefore a HIGH probability of existing in the region chosen)

• Whenever 2 is small, the particle has a low probability density (and, therefore a LOW probability of existing in the region chosen)

• Whenever , and therefore, 2, is equal to zero, the particle has ZERO probability density.– This happens at nodes.

SchrÖdinger’s Equation

• Allows us to calculate the wavefunction for any particle

• The SchrÖdinger equation calculates both wavefunction AND energy

Curvature of the wavefunction

Potential Energy (for charged particles it is the electrical

potential Energy)

Particle in a Box

• Working with SchrÖdinger’s equation

• Assume we have a single particle of mass m stuck in a one-dimensional box with a distance L between the walls.

• Only certain wavelengths can exist within the box.– Same as a stretched string can

only support certain wavelengths

Standing Waves

Particle in a Box

• The wavefunctions for the particle are identical to the displacements of a stretched string as it vibrates.

€

n (x) =2

L

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

sinnπx

L

⎛

⎝ ⎜

⎞

⎠ ⎟ where n=1,2,3,…

n is the quantum number•It defines a state

Particle In a Box

• Now we know that the allowable energies are :

€

En =n2h2

8mL2 Where n=1,2,3,…

• This tells us that:1. The energy levels for heavier particles are less than

those of lighter particles.2. As the length b/w the walls decreases, the ‘distance’

b/w energy levels increases.3. The energy levels are Quantized.

Particle in a Box:Energy Levels and Mass

• As the mass of the particle increases, the separation between energy levels decreases– This is why no one

observed quantization until Bohr looked at the smallest possible atom, hydrogen

m1 < m2

Zero Point Energy

• A particle in a container CANNOT have zero energy– A container could be an atom, a box, etc.

• The lowest energy (when n=1) is:

€

En =h2

8mL2Zero Point Energy

•This is in agreement with the Uncertainty Principle:•∆p and ∆x are never zero, therefore the particle is always moving

Wavefunctions and Probability Densities

• Examine the 2 lowest energy functions n=1 and n=2

• We see from the shading that when n=1, 2 is at a maximum @ the center of the box.

• When n=2, we

see that 2 is at a maximum on either side of the center of the box

Wavefunction Summary

• The probability density for a particle at a location is proportional to the square of the wavefunction at the point

• The wavefunction is found by solving the SchrÖdinger equation for the particle.

• When the equation is solved to the appropriate boundary conditons, it is found that the particle can only posses certain discrete energies.