Einstein Model for the Vibrational Heat Capacity of
Einstein Model for the Vibrational Heat Capacity of Solids
Einstein Model: Vibrational Heat Capacity of Solids • Starting as early as the late 1600’s, people started measuring the heat capacities Cp (or Cv) of various solids, liquids & gases at various temperatures. • 1819: Dulong & Petit observed that the molar heat capacity for many solids at temperatures near room temperature (300 K) is approximately independent of the material & of the temperature & is approximately R is the Ideal Gas Constant! • This result is called the “Dulong-Petit Law”
• Late 1800’s & early 1900’s: experimental capabilities became more sophisticated, & it became possible to achieve temperatures well below room temperature. • Then, measurements clearly showed that the heat capacities of solids depend on what the material is & are also strongly dependent on the temperature. • In fact, at low enough temperatures, the heat capacities of many solids were observed to depend on temperature as Cv = AT 3 (A is a constant) • Early 1900’s: A theory explanation for this temperature dependence of Cv was one of the major unsolved problems in thermodynamics.
Molar Heat Capacity of a Solid • Assume that the heat supplied to a solid is transformed into kinetic & potential energies of each vibrating atom. • To explain the classical Dulong Petit Law, use a simple classical model of the vibrating solid as in the figure. Assume that each vibrating atom is an independent simple harmonic oscillator & use the Equipartition Theorem to calculate the Thermal Energy E & thus the Heat Capacity Cv 4
Molar Thermal Energy of a Solid • Dulong-Petit Law: Can be explained using the Equipartition Theorem of Classical Stat Mech: Thermal Average Energy for each degree of freedom is (½) k. T • Assume that each atom has 6 degrees of freedom: 3 translational & 3 vibrational, the Thermal Average Energy (per mole) of the vibrating solid is: R NA k 5
The Molar Vibrational Heat Capacity • Using this approximation, the Thermal Average Energy (per mole) of the vibrating solid is: (1) • By definition, the Heat Capacity of a substance at Constant Volume is (2) • Using (1) in (2) gives: In agreement with Dulong & Petit! 6
Measured Molar Heat Capacities • Measurements on many solids have shown that their Heat Capacities are strongly temperature dependent & that the Dulong-Petit result Cv = 3 R is only valid at high temperatures. C = 3 R v 7
Einstein Model of a Vibrating Solid • 1907: Einstein extended Planck’s quantum ideas to solids. • He proposed that the energies of the lattice vibrations in a solid are quantized simple harmonic oscillators. 8
Einstein Model of a Vibrating Solid 1907 Einstein Assumptions: • Energies of lattice vibrations in a solid are quantized simple harmonic oscillators. : • Each vibrational mode is an independent oscillator • Each mode vibrates in 3 -dimensions • Each mode is a quantized oscillator with energy: or = ħ 9
Einstein Model of a Vibrating Solid 1907 Einstein Assumptions: • Energies of lattice vibrations in a solid are quantized simple harmonic oscillators. : • Each vibrational mode is an independent oscillator • Each mode vibrates in 3 -dimensions • Each mode is a quantized oscillator with energy: or = ħ actually Note! Einstein left the Zero Point energy (½ )ħ out En = (n + ½ )ħ of his model. It adds a constant to energy & so it doesn’t affect Cv. 10
• In effect, Einstein modeled one mole of solid as an assembly of 3 NA distinguishable oscillators. • He used the Canonical Ensemble to calculate the average energy of one oscillator in this model: 11
Einstein Model of a Vibrating Solid • To compute the average energy for one oscillator, note that it can be written as: where and b = 1/k. T Z is the partition function 12
Einstein Model • For b = 1/k. T & En = nħω, the partition function for one oscillator in the Einstein model is: which, follows from the result 13
Einstein Model • Differentiating with respect to b: 14
Einstein Model …and multiplying by – 1/Z: • This is the Einstein result for the average energy of one oscillator of energy = ħ . 15
Einstein Model • Einstein’s result for the average energy of one oscillator of energy = ħ. • Einstein then made the simple (& absurd & unphysical!) assumption that each of the 3 NA oscillators vibrates at the same frequency so each has the same energy = ħω! • With this assumption, the mean vibrational energy of the solid is just 3 NA times the above result! 16
• So, what is the frequency at which each of the 3 NA oscillators is vibrating? 17
• So, what is the frequency at which each of the 3 NA oscillators is vibrating? • In the Einstein Model this is an adjustable parameter which is fit to data & which depends on what solid is being considered. • It is usually denoted as E = ħ E where E is called “The Einstein Frequency” 18
• So, the molar heat capacity in Einstein’s model is: • Here TE (ħ E)/k. B is called the “Einstein temperature”. • Its not a true temperature, of course, but it is an energy in temperature units. 19
Einstein Model: CV for Diamond Einstein, Annalen der Physik 22 (4), 180 (1907) For diamond the Einstein Temperature TE = 1320 K 20
The Debye Model Vibrating Solids 21
Summary • The Dulong-Petit law is expected from the equipartition theorem of classical physics. However, it fails at temperatures low compared with the Einstein temperature TE • If the energy in solids is assumed to be quantized, however, models (such as Debye’s) can be developed that agree with the observed behavior of the heat capacity with temperature. • Einstein was the first to suggest such a model. • Next we will briefly discuss • The Debye Model. 22
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