Statistical thermodynamics 1: the concepts - W.H. Freeman

More documents

Recommendations

Info

$Microscopy Math - W.H. Freeman$

PC8eC16 1/26/06 14:34 Page 569 16.2 THE MOLECULAR PARTITION FUNCTION 569 Justification 16.2 The partition function for a particle in a one-dimensional box The energy levels of a molecule of mass m in a container of length X are given by eqn 9.4a with L = X: n 2 h 2 E n = n = 1, 2,... 8mX 2 The lowest level (n = 1) has energy h 2 /8mX 2 , so the energies relative to that level are ε n = (n 2 − 1)ε ε= h 2 /8mX 2 The sum to evaluate is therefore q X = ∞ ∑ e −(n2 −1)βε n=1 The translational energy levels are very close together in a container the size of a typical laboratory vessel; therefore, the sum can be approximated by an integral: ∞ ∞ q X = e −(n2−1)βε dn ≈ e −n2βε dn 1 0 The extension of the lower limit to n = 0 and the replacement of n 2 − 1 by n 2 introduces negligible error but turns the integral into standard form. We make the substitution x 2 = n 2 βε, implying dn = dx/(βε) 1/2 , and therefore that A 1 D q X = B E C βε F 1/2 ∞ 0 π 1/2 /2 5 6 7 A 1 D e −x2 dx = B E C βε F 1/2 A D A 2πmD B E = B E C 2 F C h 2 β F π 1/2 1/2 X Another useful feature of partition functions is used to derive expressions when the energy of a molecule arises from several different, independent sources: if the energy is a sum of contributions from independent modes of motion, then the partition function is a product of partition functions for each mode of motion. For instance, suppose the molecule we are considering is free to move in three dimensions. We take the length of the container in the y-direction to be Y and that in the z-direction to be Z. The total energy of a molecule ε is the sum of its translational energies in all three directions: ε n1 n 2 n 3 = ε (X) n1 + ε (Y) (Z) n2 + ε n3 (16.16) where n 1 , n 2 , and n 3 are the quantum numbers for motion in the x-, y-, and z-directions, respectively. Therefore, because e a+b+c = e a e b e c , the partition function factorizes as follows: q = ∑ e −βε (X) n 1 −βε (Y) (Z ) n2 −βε n3 all n = ∑ all n e −βε (X) n 1 (Y) (Z) e −βε n 2 e −βε n 3 A = C∑ (X) n 1 e −βε n 1 D F A C ∑ (Y) n 2 e −βε n 2 D F A C ∑ (Z) n 3 e −βε n 3 D F (16.17) = q X q Y q Z It is generally true that, if the energy of a molecule can be written as the sum of independent terms, then the partition function is the corresponding product of individual contributions.
PC8eC16 1/26/06 14:34 Page 570 570 16 STATISTICAL THERMODYNAMICS 1: THE CONCEPTS Equation 16.15 gives the partition function for translational motion in the x- direction. The only change for the other two directions is to replace the length X by the lengths Y or Z. Hence the partition function for motion in three dimensions is 3/2 A 2πm D q = XYZ (16.18) C h 2 β F The product of lengths XYZ is the volume, V, of the container, so we can write 1/2 V A β D h q = Λ = h = (16.19) Λ 3 C 2πm F (2πmkT) 1/2 The quantity Λ has the dimensions of length and is called the thermal wavelength (sometimes the thermal de Broglie wavelength) of the molecule. The thermal wavelength decreases with increasing mass and temperature. As in the one-dimensional case, the partition function increases with the mass of the particle (as m 3/2 ) and the volume of the container (as V); for a given mass and volume, the partition function increases with temperature (as T 3/2 ). Illustration 16.3 Calculating the translational partition function To calculate the translational partition function of an H 2 molecule confined to a 100 cm 3 vessel at 25°C we use m = 2.016 u; then 6.626 × 10 −34 J s Λ = {2π×(2.016 × 1.6605 × 10 −27 kg) × (1.38 × 10 −23 J K −1 ) × (298 K)} 1/2 = 7.12 × 10 −11 m where we have used 1 J = 1 kg m 2 s −2 . Therefore, 1.00 × 10 −4 m 3 q = =2.77 × 10 26 (7.12 × 10 −11 m) 3 About 10 26 quantum states are thermally accessible, even at room temperature and for this light molecule. Many states are occupied if the thermal wavelength (which in this case is 71.2 pm) is small compared with the linear dimensions of the container. Self-test 16.5 Calculate the translational partition function for a D 2 molecule under the same conditions. [q = 7.8 × 10 26 , 2 3/2 times larger] The validity of the approximations that led to eqn 16.19 can be expressed in terms of the average separation of the particles in the container, d. We do not have to worry about the role of the Pauli principle on the occupation of states if there are many states available for each molecule. Because q is the total number of accessible states, the average number of states per molecule is q/N. For this quantity to be large, we require V/NΛ 3 >> 1. However, V/N is the volume occupied by a single particle, and therefore the average separation of the particles is d = (V/N) 1/3 . The condition for there being many states available per molecule is therefore d 3 /Λ 3 >> 1, and therefore d >> Λ. That is, for eqn 16.19 to be valid, the average separation of the particles must be much greater than their thermal wavelength. For H 2 molecules at 1 bar and 298 K, the average separation is 3 nm, which is significantly larger than their thermal wavelength (71.2 pm, Illustration 16.3).
Page 1 and 2: PC8eC16 1/26/06 14:34 Page 560 16 S
Page 3 and 4: PC8eC16 1/26/06 14:34 Page 562 562
Page 9: PC8eC16 1/26/06 14:34 Page 568 568
Page 29: PC8eC16 1/26/06 14:34 Page 588 588

Statistical thermodynamics 1: the concepts - W.H. Freeman

Create successful ePaper yourself

Delete template?

Save as template?