ACER 9/16 Lecture #4: Atomic and Molecular Structure

Photoelectric Effect (what is an equation that can describe photoelectric effect?)

Light energy absorbed by a metal creates a photoelectron which, if enough energy is absorbed, is ejected from the metal.

E_photon = E’ + φ + KE_PE

E’ is the energy needed for the photoelectron to reach the surface of the metal.

φ is the work function, the minimum energy needed to remove an electron from the surface of the metal.

KE_PE is the remaining, kinetic energy of the photoelectron after it escapes the metal.

If the electron is already at the surface of the metal, then E’ = 0 and KE_max = hv_photon - φ

How does frequency affect the photoelectric effect? What is the threshold frequency?

Electrons are only ejected if the frequency of the photon is equal to or exceeds a certain threshold frequency unique to each metal, v₀. After this threshold, photocurrent and light intensity have a proportional relationship.

How does intensity relate to the photoelectric effect?

Intensity refers to the number of photons. It only affects the number of electrons ejected.

If light was a wave, then intensity should determine the KE of the electrons and whether they escape. Instead, their kinetic energies are determined by the frequency of the photons.

This would only make sense if light could behave like a particle.

What is another form of Planck’s equation? What’s it used for?

Another form of the Planck-Einstein relation is E = nhν = nhc/λ where n is the number of photons.

This calculates the total energy of multiple photons or a quantized EM wave.

What was the significance of the photoelectron effect?

It proved that light could act as a particle. It also produced experimental evidence for Planck’s theory of quantization.

• Einstein mathematically predicted that maximum KE (of escaped electrons) plotted against frequency (of photons) should result in a slope that matches Planck’s constant h. Robert Millikan ran experiments and proved Einstein’s prediction.

Express the maximum kinetic energy of an emitted electron in photoelectric effect using classical mechanics.

What does this mean?

Based on the assumption that light is a particle, the kinetic energy of the photoelectrons KE_PE is equal to:

KE_max = ½ m_ev²

• m_e, the mass of the electron, is a constant value of 9.11 x 10^-31 kg

This means that ½ m_ev² ∝ hv.

Wave-Particle Duality

the idea that light can act as a wave and/or a particle

How do we know that light is a wave?

Through the double slit experiment and constructive and destructive interference

Constructive Interference

in-phase oscillation

Two waves that are in the same phase (peaks align with peaks, troughs align with troughs) can combine into one wave with combined amplitudes at that point where they align.

After passing each other, the waves continue with their original amplitudes.

Destructive Interference

out-of-phase oscillation

Two waves that are in opposite phases (180 degrees out of phase, peaks align with troughs) cancel each other out, resulting in a single wave with reduced or no amplitude at that point where they align.

Having no amplitude does not mean that the wave no longer exists. After passing each other, the waves continue with their original amplitudes.

Double-Slit Experiment

When electrons were fired at a wall through two slits, a striped wave pattern was created on the wall, indicating wave behavior (interference between the electrons would create a striped pattern).

When unobserved electrons were fired one by one to prevent interference between them, they still created that striped wave pattern. This meant that each electron behaved like a wave and interfered with itself.

Boundary Condition

a rule that limits a math or physics problem by stating what is actually happening at a specific boundary

For example, a string fixed at both ends leads to a condition on the allowed wavelengths.

Standing Waves (What is an equation that can describe standing waves? How do they relate to quantization?)

waves that are trapped in one spot by boundary conditions, vibrating in place

For a standing wave on a string fixed at both ends, the relationship between the length of the string L and the wavelength of the vibration λ is: L = nλ/2

n must be an integer (1, 2, 3 …)

They are examples of energy quantization because they can only exist at specific whole-number intervals.

What is the de Broglie wavelength? (describe the formula)

λ = h/p = h/mv

• λ is the wavelength of the moving object

• h is Planck’s constant

• p is momentum, which is how hard it is to stop a moving object (kg*m/s)

• m is mass (kg)

• v is velocity (m/s)

This equation is derived from E = mc².

Compare electrons to standing waves. Include De Broglie’s wave loop.

Electrons are quantized for the same reason as standing Waves. They can only exist in specific orbits if they fit perfectly in a closed loop without overlapping or destroying themself.

De Broglie’s wave loop can be described: 2πr = nλ

Heisenberg Uncertainty Principle

As a consequence of wave-particle duality, the position and momentum of a particle can’t be known simultaneously.

To calculate momentum, you need to calculate wavelength. In order to measure wavelength, the wave needs to be a long, repeating continuous ripple. If it is a repeating continuous ripple, you cannot measure the exact position of the wave.

To calculate position, the wave must be one wave packet, a single sudden ripple. It no longer has a measurable wavelength.

Represent Heisenberg uncertainty principle mathematically. What is a limitation of this relation?

When measuring two properties, A and B, whose product has dimensions of action (energy x time, position x momentum), their uncertainties must satisfy: ∆A x ∆B ≥ h/4π

∆x x ∆p ≥ h/4π

• ∆ represents the uncertainty or error in the measurement. ∆x is the uncertainty in position. ∆p is the uncertainty in momentum.

This equation is only relevant for microscopic things like electrons, whose wavelengths are small enough to be disrupted when observed.

Ψ(x) ≈ sin(x)

Since an electron can behave like a wave, it can be represented by a wave function similar to the functions describing EM or sound waves. A quantum wavefunction modeled as a sine wave can be represented: Ψ(x) ≈ sin(x)

This equation describes a quantum state where the electron is equally likely to be found almost anywhere in space (explains why orbitals are possible).

It is an exact spatial solution to the time-independent Schrödinger equation for specific systems, such as a particle in a 1D infinite potential well or a free particle.

What is an orbital, then?

the mathematical representation of the likelihood that the electron will be found in a specific region of space around the nucleus