Theoretical calculation of the heat capacity

Transcription

1 eoretical calculation of te eat 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: Capter 6. of Gasell

2 Degrees of freedom and equipartition of energy For eac atom in a solid or gas pase, tree coordinates ave to be specified to describe te atom s position a single atom as 3 degrees of freedom for its motion. A solid or a molecule composed of N atoms as 3N degrees of freedom. We can also tin about te number of degrees of freedom as te number of ways to absorb energy. e teorem of equipartition of energy (classical mecanics) states tat in termal equilibrium te same average energy is associated wit eac independent degree of freedom and tat te energy is ½. For te interacting atoms, e.g. liquid or solid, for eac atom we ave ½ for inetic energy and ½ for potential energy - equality of inetic and potential energy in armonic approximation is addressed by te virial teorem of classical mecanics. ased on equipartition principle, we can calculate eat capacity of te ideal gas of atoms - eac atom as 3 degrees of freedom and internal energy of 3/. e molar internal energy U3/N A 3/R and te molar eat capacity under conditions of constant volume is c v [du/d] V 3/R In an ideal gas of molecules only internal vibrational degrees of freedom ave potential energy associated wit tem. For example, a diatomic molecule as 3 translational + rotational + 1 vibrational 6 total degrees of freedom. Potential energy contributes ½ only to te energy of te vibrational degree of freedom, and U molecule 7/ if all degrees of freedom are fully excited.

3 emperature and velocities of atoms At equilibrium velocity distribution is Maxwell-oltzmann, r dn v, ( ) m π 3 [ ] + v v m vx y + exp z dv x dv y dv z v i 3 /m If system is not in equilibrium it is often difficult to separate different contributions to te inetic energy and to define temperature. Acoustic emissions in te fracture simulation in D model. Figure by.l. Holian and R. Ravelo, Pys. Rev. 51, 1175 (1995). Atoms are colored by velocities relative to te left-to-rigt local expansion velocity, wic causes te crac to advance from te bottom up.

4 Heat capacity of molecules straigtforward application of equipartition principle does not wor Classical mecanics sould be used wit caution wen dealing wit penomena tat are inerently quantized. For example, let s try to use equipartition teory to calculate te eat capacity of water vapor. Motion Degrees of freedom U c v ranslational 3 3 ½R 1.5R Rotational 3 3 ½R 1.5R Vibrational 3 6 ½R 3R otal c v 6R ut experimental c v is muc smaller. At 98 K H O gas as c v 3.038R. Wat is te reason for te large discrepancy? Rotation Vibration ranslation

5 Heat capacity of molecules straigtforward application of equipartition principle does not wor (continued) Wat is te reason for te large difference between te prediction of classical calculations, c v 6R, and muc smaller experimental c v 3.038R at 5 C? Rotation Vibration e table sows te vibrational frequencies of water along wit te population of te first excited state at 600 K. ranslation ν (cm -1 ) Exp(-ν/ ) 1.0 x x x 10-5 For te ig frequency OH stretcing motions, tere sould be essentially no molecules in te first vibrational state even at 600 K. For te lower frequency bending motion, tere will be about % of te molecules excited. Contributions to te eat capacity can be considered classically only if E n ~ ν <<. Energy levels wit E n contribute little, if at all, to te eat capacity. Only translational and rotational modes are excited, te contribution from vibrations is only 0.038R.

6 Heat capacity of solids Dulong Petit law In 1819 Dulong and Petit found experimentally tat for many solids at room temperature, c v 3R 5 JK -1 mol -1 is is consistent wit equipartition teory: energy added to solids taes te form of atomic vibrations and bot inetic and potential energy is associated wit te tree degrees of freedom of eac atom. P(t) K(t) 3 e molar internal energy is ten U 3N A 3R and te molar constant volume eat capacity is c v [ U/ ] v 3R Altoug c v for many elements at room are indeed close to 3R, low- measurements found a strong temperature dependence of c v. Actually, c v 0 as 0 K.

7 Heat capacity of solids Einstein model e low- beavior can be explained by quantum teory. e first explanation was proposed by Einstein in He considered a solid as an ensemble of independent quantum armonic oscillators vibrating at a frequency. Quantum teory gives te energy of i t level of a armonic quantum oscillator as ε i (i + ½) were i 0,1,, and is Planc s constant. For a quantum armonic oscillator te Einstein-ose statistics must be applied (rater tan Maxwell-oltzmann statistics and equipartition of energy for classical oscillators) and te statistical distribution of energy in te vibrational states gives average energy: U(t) e ere are tree degrees of freedom per oscillator, so te total internal energy per mol is c V U V U 3N 3N e A A 1 1 e ( e 1) Note: you do not need to remember all tese scary quantum mecanics equations for tests/exams but you do need to understand te basic concepts beind tem. e Einstein formula gives a temperature dependent c v tat approaces 3R as, and approacing 0 as 0.

8 e Hig emperature Limit of te Einstein Specific Heat Let s sow tat Einstein s formula approaces Dulong Petit law at ig. For ig temperatures, a series expansion of te exponential gives e Einstein specific eat expression ten becomes ( ) + A A V 1 3N 1 e e 3N c 1 e + 3R 3N 1 N 3 A A + In te Einstein treatment, te appropriate frequency in te expression ad to be determined empirically by comparison wit experiment for eac element. Altoug te general matc wit experiment was reasonable, it was not exact. Einstein formula predicts faster decrease of c v as compared wit experimental data. Debye advanced te treatment by treating te quantum oscillators as collective modes in te solid - ponons.

9 Heat capacity of solids Debye model Debye assumed a continuum of frequencies wit a distribution of g(ν) aν, up to a maximum frequency, ν D, called te Debye frequency. is leads to te following expression for te Debye specific eat capacity: c V 9 N A θ D 3 θ D 0 / x 4 e x ( e 1) dx x were x ν/ and θ D ν D / Debye caracteristic temperature For low temperatures, Debye's model predicts - good agreement wit experimental results. c V 1π 4 NA 5 θd 3 We can see tat c v depends on /θ D wit θ D as te scaling factor for different materials. C v, J K -1 mol -1 θ D ~ ν D ~ strengt of te interatomic interaction, ~ 1/(atomic weigt). /θ D For a armonic oscillator 1 π force constant reduced mass

10 Heat capacity of metals electronic contribution c v [ U/ ] v terefore as soon as energy of electrons are canging wit, tey will mae contribution to c v. o contribute to bul specific eat, te valence electrons would ave to receive energy from te termal energy, ~. ut te Fermi energy is muc greater tan and te overwelming majority of te electrons cannot receive suc energy since tere are no available energy levels witin of teir energy. e small fraction of electrons wic are witin of te Fermi level (defined by Fermi-Dirac statistics) does mae a small contribution to te specific eat. is contribution is proportional to temperature, c v el γ and becomes significant at very low temperatures, wen c v γ + A 3 (for metals only).

11 Summary (1) Mae sure you understand language and concepts: Degrees of freedom Equipartition of energy Heat capacity of ideal gas Heat capacity of solids: Dulong-Petit law Quantum mecanical corrections: Einstein and Debye models Heat capacity of gas, solid or liquid tends to increase wit temperature, due to te increasing number of excited degrees of freedom, requiring more energy to cause te same temperature rise. e teoretical approaces to eat capacities, discussed in tis lecture, are based on rater roug approximations (anarmonicity is neglected, ponon spectrum is approximated by ν in Debye model, etc.). In practice c p () in normally measured experimentally and te results are described analytically, e.g. C p A + + C - for a certain range of temperatures.

12 Summary () Equipartition teory is only valid if all degrees of freedom are fully excited. Quantized energy levels 4.9 C v, J K -1 mol -1 Hig, C v 3R Low, C v 0 Energy P n ~ e ΔE/ ΔE << - classical beavior ΔE - quantum beavior