1.3. Equipartition Of Energy Per Molecule And Its Constituent Parts  A Fundamental Problem

Gas molecules composed of several atoms Equipartition theorem (see Part III) Each quadratic term in H contributes kT /2 to <ε> For each atom <ε> = 3kT /2. For each molecule <K.E.>C.M. = 3kT /2. Ideal gas law: PV = NkT = 2E / f. E = total energy = N f kT /2 f = effective degrees of freedom of each molecule

Proof of (2) : <K.E.>C.M. = 3kT /2 From (1): Hence:

vcm and vrel are independent:  

Proof of (3): PV = NkT = 2E / f Elastic collisions: For r atoms: Adiabatic process: See Exercise 1.8

Rotations and vibrations of a triatomic molecule DoF = 33 = 9 Translation: 3 Rotation: 3 Vibration: 3

Heat Capacity Problem Experiment (Theory): For He,   1.660 For O2 ,   1.4  f  5 (6)

Heat Capacity Problem  Failure of the Equipartition Theorem Atom = ( Protons + Neutrons ) + electrons  f  Z Hadrons = Quarks f not known exactly Explanation: States are quantized  Degree of freedom is “frozen” if Excitation energy >> Thermal energy

Diatomic Gas Low T Experiment: High T Experiment:  f  5  f  7 Theory: K.E. + Vvib f = 5 +1 +1 = 7 Theory: K.E. + Vvib f = 5 + (1+1) = 5