Theoretical calculation of the heat capacity Principle of equipartition of energy Heat capacity of ideal and real gases Heat capacity of solids Dulong Petit Einstein Debye models Heat capacity of metals electronic contribution Reading Chapter 6 2 of Gaskell MSE 3050 Phase Diagrams and Kinetics Leonid Zhigilei Degrees of freedom and equipartition of energy For each atom in a solid or gas phase three coordinates have to be specified to describe the atom s position a single atom has 3 degrees of freedom for its motion A solid or a molecule composed of N atoms has 3N degrees of freedom We can also think about the number of degrees of freedom as the number of ways to absorb energy The theorem of equipartition of energy classical mechanics states that in thermal equilibrium the same average energy is associated with each independent degree of freedom and that the energy is kBT For the interacting atoms e g liquid or solid for each atom we have kBT for kinetic energy and kBT for potential energy equality of kinetic and potential energy in harmonic approximation is addressed by the virial theorem of classical mechanics Based on equipartition principle we can calculate heat capacity of the ideal gas of atoms each atom has 3 degrees of freedom and internal energy of 3 2kBT The molar internal energy U 3 2NAkBT 3 2RT and the molar heat capacity under conditions of constant volume is cv dU dT V 3 2R In an ideal gas of molecules only internal vibrational degrees of freedom have potential energy associated with them For example a diatomic molecule has 3 translational 2 rotational 1 vibrational 6 total degrees of freedom Potential energy contributes kBT only to the energy of the vibrational degree of freedom and Umolecule 7 2kBT if all degrees of freedom are fully excited MSE 3050 Phase Diagrams and Kinetics Leonid Zhigilei Temperature and velocities of atoms At equilibrium velocity distribution is Maxwell Boltzmann 3 m v 2x v 2y v z 2 m 2 r exp dN v T 2k BT 2 k BT v2 i dv dv dv x y z 3k BT m If system is not in equilibrium it is often difficult to separate different contributions to the kinetic energy and to define temperature Acoustic emissions in the fracture simulation in 2D model Figure by B L Holian and R Ravelo Phys Rev B51 11275 1995 Atoms are colored by velocities relative to the left to right local expansion velocity which causes the crack to advance from the bottom up MSE 3050 Phase Diagrams and Kinetics Leonid Zhigilei Heat capacity of molecules straightforward application of equipartition principle does not work Classical mechanics should be used with caution when dealing with phenomena that are inherently quantized For example let s try to use equipartition theory to calculate the heat capacity of water vapor Motion Degrees of freedom U cv Translational 3 3 RT 1 5R Rotational 3 3 RT 1 5R Vibrational 3 6 RT 3R Total cv 6R But experimental cv is much smaller At T 298 K H2O gas has cv 3 038R What is the reason for the large discrepancy Rotation Vibration Translation MSE 3050 Phase Diagrams and Kinetics Leonid Zhigilei Heat capacity of molecules straightforward application of equipartition principle does not work continued What is the reason for the large difference between the prediction of classical calculations cv 6R and much smaller experimental cv 3 038R at 25 C Rotation Vibration The table shows the vibrational frequencies of water along with the population of the first excited state at 600 K Translation cm 1 Exp h kBT 3825 1 0 x 10 4 1654 1 9 x 10 2 3936 8 0 x 10 5 For the high frequency OH stretching motions there should be essentially no molecules in the first vibrational state even at 600 K For the lower frequency bending motion there will be about 2 of the molecules excited Contributions to the heat capacity can be considered classically only if En h kBT Energy levels with En kT contribute little if at all to the heat capacity Only translational and rotational modes are excited the contribution from vibrations is only 0 038R MSE 3050 Phase Diagrams and Kinetics Leonid Zhigilei Heat capacity of solids Dulong Petit law In 1819 Dulong and Petit found experimentally that for many solids at room temperature cv 3R 25 JK 1mol 1 This is consistent with equipartition theory energy added to solids takes the form of atomic vibrations and both kinetic and potential energy is associated with the three degrees of freedom of each atom P t K t 3 k BT 2 The molar internal energy is then U 3NAkBT 3RT and the molar constant volume heat capacity is cv U T v 3R Although cv for many elements at room T are indeed close to 3R low T measurements found a strong temperature dependence of cv Actually cv 0 as T 0 K MSE 3050 Phase Diagrams and Kinetics Leonid Zhigilei Heat capacity of solids Einstein model The low T behavior can be explained by quantum theory The first explanation was proposed by Einstein in 1906 He considered a solid as an ensemble of independent quantum harmonic oscillators vibrating at a frequency Quantum theory gives the energy of ith level of a harmonic quantum oscillator as i i h where i 0 1 2 and h is Planck s constant For a quantum harmonic oscillator the Einstein Bose statistics must be applied rather than Maxwell Boltzmann statistics and equipartition of energy for classical oscillators and the statistical distribution of energy in the vibrational states gives average energy U t h e h k B T 1 There are three degrees of freedom per oscillator so the total internal energy per mol is 3 N A h U h k BT e 1 2 U cV T V h h e 3 N A k B k BT 2 e h k B T 1 k BT Note you do not need to remember all these scary quantum mechanics equations for tests exams but you do need to understand the basic concepts behind them The Einstein formula gives a temperature dependent cv that approaches 3R as T and approaching 0 as T 0 MSE 3050 Phase Diagrams and Kinetics Leonid Zhigilei The High Temperature Limit of the Einstein Specific Heat Let s show that Einstein s formula approaches Dulong Petit law at high T For high temperatures a series expansion of the exponential gives e h k BT 1 h k BT The Einstein specific heat expression then becomes 2 cV h h e 3 N A k B k BT 2 e h k B T 1 2 k BT h h 1 3 N A k B k BT k BT 2 h k BT h 3 N A k B 3 R 3 N A k B 1 k BT In the Einstein treatment the appropriate frequency in the expression had to be determined empirically by comparison with experiment for each element Although the general match with experiment was reasonable it was not exact Einstein formula predicts