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n-fall08-week6-7 - J.I. Siepmann Chem 8561 46 IDEAL...

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Unformatted text preview: J.I. Siepmann Chem 8561 46 IDEAL POLYATOMIC GAS • a polyatomic molecule consists of three or more nuclei and many electrons • to derive the canonical partition function for polyatomic molecules, we will follow a similar approach as used for the diatomic molecule • use Born-Oppenheimer approximation to separate the motions of nuclei and elec- trons ◦ electrons in the field of the n nuclei at fixed conformation of molecule ◦ nuclei in the electronic potential u e 1 ( r ) • thereafter, we can separate the motion of the nuclei into translations of the center- of-mass, rotations, and intramolecular motions (vibrations) • again, the large (continuous) number of translational states allows us to use Boltz- mann statistics, thus Q ( N, V, T ) = ( q trans q rot , vib , elec , nucl ) N N ! (164) • each of the n nuclei of the polyatomic molecule requires 3 cartesian coordinates to describe its position, thus the configuration of the polyatomic molecule is specified by 3 n coordinates ◦ the position of the COM takes three coordinates ◦ the orientation of a given vector (bond of the molecule) in space is specified by 2 angles, thus 2 angles are needed for a linear polyatomic molecule, and 1 additional angle is needed for nonlinear polyatomic molecules to specify the rotation of the molecule around the axis (bond) given by the first 2 angles ◦ thus, the remaining 3 n − 5 or 3 n − 6 coordinates are needed to describe the conformation of the polyatomic molecule (the separations of the n nuclei with respect to each other); all these coordinates are conventionally referred to as vibrational degrees of freedom J.I. Siepmann Chem 8561 47 • the equivalent of the rigid rotor/harmonic oscillator approximation for diatomic molecules allows us to separate rotational and vibrational degrees of freedom • with these approximations all five types of degrees of freedom can be treated as independent, and the canonical partition function can be written as product of individual contributions Q = [ q trans ( V, T ) q rot ( T ) q vib ( T ) q elec ( T ) q nuclear ( T )] N N ! (165) Translational, electronic, and nuclear partition functions • the COM is a point particle with mass equal to the sum over all the individual masses of the molecule q trans = bracketleftbigg 2 π ( ∑ n i =1 m i ) k B T h 2 bracketrightbigg 3 / 2 V (166) • with the conventional choice that the zero of energy is defined by completely sep- arated atoms at rest, the global minimum of the (total) electronic potential has an energy of − D e (and the vibrational groundstate, sum of 3 n − 6 (nonlinear) or 3 n − 5 (linear) modes, has an energy of ∑ modes hν m / 2), then the electronic partition function is q elec = ω e 1 e D e /k B T + ··· (167) • and the nuclear partion functions can again be neglected q nucl = 1 (168) J.I. Siepmann Chem 8561 48 Rotational partition function • in case, the polyatmic molecule is linear, the rotational partition function can be calculated using the equations derived for the diatomic molecule,...
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n-fall08-week6-7 - J.I. Siepmann Chem 8561 46 IDEAL...

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