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1 Average Particle Energy in an Ideal Gas - the total energy of the system is found by summing up (integrating) over all particles n(ε) at different energies ε - with the integral - we find - note: - the total energy is proportional to temperature T and the number of particles N - the energy is independent of the specifics of the particle, e.g. its mass - average energy per particle phys4.14 Page 1 - characteristic energy per particle at room temperature T = 300 K The Equipartition Theorem in thermal equilibrium at temperature T the average energy ε stored in a particle with f degrees of freedom is - there are f = 3 independent degrees of freedom for linear motion along the x, y, z coordinates - in a diatomic molecule there are additionally f = 2 rotational degrees of freedom along the two axis perpendicular to the bond axis - any harmonic oscillator has f = 2 vibrational degrees of freedom any independent degree of freedom per particle (e.g. position or momentum) that appears quadratically in the total energy (Hamiltonian) of a system contributes 1/2 k T to its average energy phys4.14 Page 2

2 Maxwell-Boltzmann Velocity Distribution - the velocity distribution of particles in an ideal gas can be found from the energy distribution - thus the number of particles with velocity in an interval dv around v is - plot of Maxwell's velocity distribution - rms (root-mean-squared) velocity of a molecule with average energy of 3/2 kt (from equipartition theorem) phys4.14 Page 3 Differences between most probable, average and rms particle velocity - rms velocity - most probable velocity - average velocity note: - both the average velocity and the rms velocity are larger than the most probable velocity phys4.14 Page 4

3 Variation of Velocity Distribution - velocity distribution of oxygen (O 2 ) at two different temperatures - velocity distribution of hydrogen (H) at 273 K - increase of velocity with decrease in mass ~ m -1/2 and increase of temperature ~ T 1/2 - average velocity: - example He: de Broglie wave length phys4.14 Page 5 Statistics of Quantum Particles - bosons are indistinguishable particles with symmetric wave functions - bosons can be in the same quantum state - for bosons the presence of a particle in a certain quantum state increases the probability of other particles to be found in that state - their statistics is described by the Bose-Einstein distribution function - for photons α = 0, generally α depends on the system parameters and is determined from the normalization condition - the distribution function is named after the Indian physicist Bose and after Einstein who extended Bose's original calculation for photons to massive particles phys4.14 Page 6

4 - fermions are indistinguishable particles with anti symmetric wave functions - particles with anti symmetric wave function cannot be in the same state - for fermions the presence of a particle in a certain state prevents any other particles from being in that state - their statistics is described by the Fermi-Dirac distribution function - the Fermi energy is defined by - the distribution function is named after Enrico Fermi and Paul Dirac who realized that the exclusion principle would lead to statistics different from that for bosons or classical particles phys4.14 Page 7 Comparison of the different distribution functions - f BE > f MB > f FD at any temperature - in the large kt limit all distribution functions approach the Maxwell- Boltzmann distribution Fermi-Dirac distribution function in terms of the Fermi energy ε F - consider the Fermi-Dirac distribution function in the zero temperature limit - this allows one to draw some important conclusions about the relevance of ε F for electrons in metals - later we will calculate the energy of free electrons in a metal and also their specific heat phys4.14 Page 8

5 FD - Distribution function at T = 0 - all states at ε < ε F are occupied f = 1 - all states at ε > ε F are empty f = 0 - for a systems with N electrons the Fermi energy is determined by filling up all states string from ε = 0 obeying the exclusion principle until all states are filled at ε = ε F - as T is increased from zero electrons below the Fermi energy will fill up states above the Fermi energy - for temperatures on the order of the Fermi energy the occupation even at low energies will be reduced and states at higher energies will be filled phys4.14 Page 9 Blackbody Radiation - consider a thermal emitter of radiation, called a blackbody, as a cavity at temperature T filled with photons - considering the cavity as a box with perfectly reflecting walls, then the radiation inside the box must be standing electromagnetic waves - the standing waves must have wavelengths λ the half integer multiples of which correspond to the cavity length L = j λ/2 - for a cubic cavity of dimension L 3 the number j of half integer wavelengths along each dimension x, y, z is - for a standing wave in an arbitrary direction phys4.14 Page 10

6 Counting the Number of Photon Modes in a Cavity - number of modes g(λ) dλ in the cavity with wavelengths λ in the interval dλ - consider a vector j with length - in the interval dj around j there is a number of combinations of j x, j y, j z that result in the same absolute value of j and thus the same wave length - the total number of j states with positive j i is - as there are two different polarizations for the standing waves - number of standing waves phys4.14 Page 11 Density of Standing Waves per Volume - the number of modes per unit volume is found by dividing by the cavity volume L 3 - the number of possible modes increases quadratically with frequency and is independent of the shape of the cavity - from the classical equipartition theorem we expect every mode in thermal equilibrium at temperature T to contribute an energy of kt (there are two degrees of freedom per mode that can be described as a harmonic oscillator) to the total energy - thus the energy density of the radiation in the cavity is - this is the Rayleigh-Jeans formula that is accurate only for ν small in comparison to kt - this energy density diverges (ultraviolet catastrophe) for large ν and thus must be wrong, this problem could not be resolved with just classical physics phys4.14 Page 12

7 Planck Radiation Law: - as photons are bosons the Bose-Einstein distribution function should hold for describing their statistical properties - the energy per mode is - the total energy density of the cavity is then - this formula is known as the Planck law of blackbody radiation (we will also discuss Einstein's derivation of this equation) - it describes the radiation spectrum emitted by any body at a thermal equilibrium temperature T (e.g. the sun, light bulbs) phys4.14 Page 13 Black Body Spectrum fusion (H 2 -> He) power ~ GW temperature T ~ 6000 Kelvin continuous spectrum power on earth 1 kw/m 2 largest intensity in visible part of the spectrum phys4.14 Page 14

8 Spectrum of the Sun phys4.14 Page 15 Wien's Displacement Law: - find the wavelength at which the spectral density of the emitted radiation is maximum - express the Planck law in terms of wave lengths λ - calculate the derivative - the result is Wien's displacement law - shows that the wavelength of maximum radiation emission shifts to lower values as T increases, e.g. bodies at a few thousand degrees emit visible radiation whereas bodies at room temperature emit infrared radiation - example: the sun phys4.14 Page 16

9 Stefan-Boltzmann Law - find the total energy density of a black body at temperature T - with a universal constant a - note that the total energy depends on the fourth power of temperature - the energy R radiated by an object per unit time and unit area is also proportional to T 4 as stated in the Stefan-Boltzmann law - with the Stefan constant - and the emissivity e of the blackbody which can range form 0.07 for polished steel to 0.97 for matte black paint phys4.14 Page 17

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