Chemistry 431 Chemistry 431 Lecture 2 Breakdown of classical - PowerPoint PPT Presentation

Chemistry 431 Chemistry 431 Lecture 2 Breakdown of classical physics Heat capacity Heat capacity Photoelectric effect Wave-particle duality Wave-particle duality Atomic spectra Semi-classical hydrogen atom S i l i l h d t NC State University NC State University

Breakdown of classical physics Aside from the ultraviolet catastrophe there were a number of experiment observations that did not p agree with classical physics: 1. The heat capacity approaches zero as the p y pp temperature approaches zero 2. The “photoelectric effect”. Ionization of a metal p depends on the frequency, rather the intensity of radiation. 3. Atomic and molecular spectra had discrete lines. 4. The wave-like properties of electrons and other particles.

Heat capacity • The heat capacity is the energy required to raise the temperature of substance. The definition is: C v , m = ∂ U m ∂ T = ∂ < E > ∂ T ∂ T ∂ T , • Solids, liquids and gases all have heat q g capacities. • U m is the molar energy of the substance. This is also known as the molar internal energy and is the same as the average energy < E >. • The heat capacity is given at constant volume.

Internal energy of a solid • Einstein first calculated the internal energy of a metal by treating it as a collection of oscillators, metal by treating it as a collection of oscillators, which represent the bonds between the atoms 3 N h ν 3 N A h ν U m = e h ν / kT – 1 • This expression assumes that the frequency of the oscillators is h ν . The expression has more p than superficial similarity to the Planck Law. The different is that the Planck Law refers to radiation modes and the Einstein formula refers to vibrational frequencies.

Internal energy of a solid • We can define the Einstein temperature as: θ E = h ν h ν k • Using the definition of the Einstein temperature • Using the definition of the Einstein temperature we can rewrite the internal energy as: θ E U m = 13 R e θ E / T – 1 e θ E / T

Limits of the function of internal energy f • As the temperature approaches 0, the value of As the temperature approaches 0, the value of e h ν /kT >> 1 so the expression becomes. 3 N A h ν U m ≈ e h ν / kT = 3 N A h ν e – h ν / kT – h ν / kT A U 3 N h • As the temperature becomes large (or As the temperature becomes large (or approaches infinity) we can use the expansion e h ν / kT = 1 – h ν h ν h ν / kT 1 kT + ... to show that to show that U m = 3 N A kT = 3 RT

Comparison of heat capacities • The classical heat capacity is C C v , m = 3 R 3 R • The classical heat capacity agrees with • The classical heat capacity agrees with experiment at room temperature. However, the classical heat capacity fails at low temperature. classical heat capacity fails at low temperature. • The Einstein heat capacity is 2 2 C v . m = θ E 2 2 e θ E /2 T 3 R e θ E / T – 1 T T e – 1 E

Comparison of heat capacities One can also write this as follows C C v , m = 3 Rf = 3 Rf where the function. 2 f = θ E θ E 2 e θ E /2 T e f e θ E / T – 1 T At high temperature f=1 (see page 248) At high temperature f=1 (see page 248). However, However at low temperature f ≈ θ E θ E 2 e – θ E / T θ / T f T This agrees with experiment This agrees with experiment. As the temperature As the temperature goes to zero the heat capacity goes to zero.

Photoelectric Effect • Electrons are ejected e - from a metal surface by from a metal surface by h ν e absorption of a photon. • Depends on frequency, p q y Metal Surface not on intensity. • Threshold frequency Kinetic Energy gy corresponds to h ν corresponds to h ν 0 = Φ = Φ h ν Φ is the work function. • Φ Φ h ν It is essentially equal to y q the ionization potential Insufficient energy Photoejection occurs for photoejection of the metal.

Photoelectric Effect Photoelectric Effect • The kinetic energy of the ejected particle is given The kinetic energy of the ejected particle is given by: 1/2 mv 2 = h ν - Φ • The threshold energy is Φ , the work function. • This demonstrates the particle-like behavior of This demonstrates the particle like behavior of photons. • A wave-like behavior would be indicated if the a e e be a o ou d be d cated t e intensity produced the effect.

The Wa e Particle D alit The Wave-Particle Duality • The fact that the DeBroglie wavelength Th f t th t th D B li l th explains the quantization of the hydrogen atom is a phenomenal h d t i h l success. • Other wave-like behavior of particles includes electron diffraction. • Particle-like behavior of waves is shown in the photoelectric effect p

De Broglie Relation • The wave-like properties of particles can be described very simply in the y p y relationship of wavelength and momentum: λ = h h p • The practical importance of this • The practical importance of this expression is realized in electron microscopy microscopy. By tuning the accelerating By tuning the accelerating voltage in an electron microscope we can alter the momentum and therefore can alter the momentum and therefore the wavelength of the electron.

The definition of a photon • The wave-particle duality goes both ways. • If a particle can act like a wave, then a wave can act like a particle. • Light particles are called photons. The g p p absorption of photons can explain how atoms and molecules can absorb discrete amounts of energy. • The energy of a photon is: The energy of a photon is: E = h ν

Experimental observation of h d hydrogen atom t • Hydrogen atom emission is “quantized”. It y g q occurs at discrete wavelengths (and therefore at discrete energies). • The Balmer series results from four visible lines at 410 nm, 434 nm, 496 nm and 656 nm. • The relationship between these lines was shown to follow the Rydberg relation.

Atomic spectra Atomic spectra • Atomic spectra consist of series of narrow lines. • Empirically it has been shown that the wavenumber of the spectral lines can be fit by ⎛ ⎞ ⎜ ⎟ 1 ~ ν = = − > ⎜ ⎟ 1 1 ( ) R n n ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ λ λ 2 2 1 1 2 2 2 2 n n ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ 1 2 ⎝ ⎠ where R is the Rydberg constant and n and n where R is the Rydberg constant, and n 1 and n 2 are integers.

The hydrogen atom: semi-classical approach • Why should the hydrogen atom care Why should the hydrogen atom care about integers? • What determines the value of the Rydberg constant R=109 677 cm -1 ? Rydberg constant R=109,677 cm 1 ? • Bohr model for the hydrogen atom. + e 2 m v 2 e m v f f = 4 π ε 0 r 2 = r r e - Coulomb Centrifugal • Balance of forces. Balance of forces • Assume electron travels in a radius r. • There must be an integral number of g wavelengths in the circumference. 2 π r = n λ n = 1,2,3….

The electron must not interfere with itself • The condition for a stable orbit is: 2 π r = n λ , n=1,2,3.. • The Bohr orbital shown has n = 16. • The DeBroglie wavelength λ λ = h/p or λ = h/mv λ h/ h/ gives: mvr = nh/2 π n=1,2,3… • This is a condition for quantization This is a condition for quantization of angular momentum

Example of self-interference p • According to the Bohr picture the condition shown will lead to diti h ill l d t cancellation of the wave and is not a stable orbit not a stable orbit. • The quantization of angular momentum implies quantization momentum implies quantization of the radius: r = 4 πε 0 n 2 h 2 2 me 2

The significance of quantized orbits g q • The Bohr model is consistent with quantized orbits of the electron around the nucleus. bit f th l t d th l • This implies a relationship between quantized angular momentum and the wavelength angular momentum and the wavelength. • Einstein argued (based on relativity) that λ = h/p, where the wavelength of light is λ , and the g g , momentum of a photon is p. • DeBroglie argued that the same should hold for all particles. ll ti l

The Bohr Model Predicts The Bohr Model Predicts Quantized Energies • The radii of the orbits are quantized and therefore the energies are quantized. • According to classical electrostatics: e 2 e 2 2 m v 2 – 1 E = T + V = = 4 π ε 0 r 8 π ε 0 r 2 4 8 Substituting in for r gives m e 4 2 1 E n = – 8 ε 0 8 ε 0 h n 2 2 h