Lecture 5: Diatomic gases (and others)

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1 Lecture 5: Diatomic gases (and others) General rule for calculating Z in complex systems Aims: Deal with a quantised diatomic molecule: Translational degrees of freedom (last lecture); Rotation and Vibration. Partition function as a product of independent factors. Law of equipartition of energy as the hightemperature limit of a quantum system. Black-body radiation Planck s formula for the spectrum of black-body radiation. Heat capacity of solids. February 05 Lecture 5 1

2 Diatomic gas in a box Quantum states in a cube of side a. Z Z n, J, r ( ) l m n J ( J + 1) + ( r + 1/ ) ma I + + tr translation rot rotation Partition function, Z, is n, J, r n, J, r n Z tr. exp ( β ) (J+1) states with with the the same energy i.e.different m j j A simple product of separate partition functions Average energy follows from Z. 1 dz dln Z tr + rot Z d β d β + ration ( J + 1) exp( β ) exp( β ) exp( β ) exp ( β ) ( J + 1) exp( β ) exp( β ) Z rot n, J, r tr. J Z tr Rotational degeneracy rot rot r February 05 Lecture 5

3 General result The result applies generally.. If the energy is a sum of components + + A B C and Z Z. Z. Z A B C + + A B C p For a contribution (Q), depending only on Q, the probability of finding a value, Q i, is i ( Q ) g exp( ( Q ) kt ) g exp( ( Q ) kt ) i i i independent of other parts of the system. i i February 05 Lecture 5

4 Diatomic molecule again Translational energy as before, tr kt Rotational energy Zrot J + 1 exp βj ( J + 1) I J T rot Ik High temperature limit is a characteristic temp. T >> T rot. Occupied levels closely spaced (compared with T rot ). Change Σ to an integral over variable to x J(J+1), dx (J+1)dJ. Z rot exp β x I d x I Low temperature limit ( ) ( ) ( ) β 0 1 d Zrot 1 rot high T Zrot d β β T << T rot. Only J0 and J1 states are occupied. ( ) Zrot 1+ exp β I 1 d Z rot exp β Z d β I February 05 Lecture 5 4 kt ( ) I Classical result ( ( orthogonal rotations) 0 T 0

5 Rotational contribution to specific heat Overall picture Quenched at at low temperatures Classical limit Typically T rot is a few K. H has the smallest moment of inertia and T rot 88K. General conclusion All quantum systems behave in a similar way. In the low temperature limit, kt<< E, the excitation is quenched. i.e. the excitation plays no part in the thermal properties of the system. In the high temperature limit kt>> E, the system behaves classically. Quantum of of energy February 05 Lecture 5 5

6 Vibrational energy Quantum rations Energy levels r r Z exp r r 0 ( β ) 1 d Z Z d β exp( kt ) 1 High temperature limit T >> k T exp( kt ) 1 kt + kt Low temp. limit exp( kt ) 0 ( +1 ) 1 1 exp ( β ) Typically T >1000K Hence for typical diatomic at room temp < tot > (5/)kT. neglect 1/ Geometric series Planck s formula for a single oscillator Characteristic Temp February 05 Lecture 5 6

7 Black body radiation Planck spectrum of black-body body radiation Radiation occupies standing wave states in a box of side, a. Energy of the modes: k g ( k)d k g( l, m, n n a ck ( ) l + m + n n a Planck formula gives us the average energy in mode, j. j exp ( kt ) 1 Total energy comes from a sum over all the modes First get density of states using the result calculated in the last lecture viz. a g( k)d k. k ) d g( )d a c ( ) l + m + n 1 February 05 Lecture 5 7 j j d d k polarisations

8 Energy density of black-body body radiation Energy density U ( T ) T ) T, )d V T, )d 0 1 c d exp( kt ) 1 g()d/a No. of of states per unit vol. in in d dat T, ) 1 exp( kt ) 1 Total power in the radiation U ( T ) T ) V c exp T ) T ) c c kt 4 0 ( 15 c )( kt ) 4 0 x exp Mean energy of of single oscillator at at.. ( kt ) ( x) d x 1 Planck s radiation formula d 1 4 /15 Stefan s Law February 05 Lecture 5 8

9 Black-body radiation Classical formula (Rayleigh-Jeans) kt per mode T,) kt g()d. A result know as the ultra-violet catastrophe. Cosmic microwave background. Cosmic Background Explorer, (COBE) T o (.74±0.06)K λ5mm ν60ghz λ0.5mm ν600ghz February 05 Lecture 5 9

10 Heat capacity of solids See later lectures for the full story. Vibrations in a solid Dulong and Petit: rational modes per atom <> kt Molar heat-capacity of all simple solids ~R. Works at high temperatures. Fails, for example, for diamond, which has a much lower value. Einstein: Realised that the new quantum theory could explain the phenomena. He quantised the interatomic rations. Assumed all oscillators have frequency E. Using our previous result (multiplied by, for the -modes per atom) Low temperature limit, exp(t), improved by Debye T T February 05 Lecture 5 10

Lecture 6: Diatomic gases (and others)

Lecture 6: Diatomic gases (and others) Lecture 6: Datomc gases and others) General rule for calculatng n complex systems Ams: Deal wth a quantsed datomc molecule: Translatonal degrees of freedom last lecture); Rotaton and Vbraton. Partton functon