The excluded volume b, introduced by vander Wall's as an empirical correction term, can be related to the size gas molecules. To do so, we assume the excluded volume is the result of the pairwise coming together of molecules. This assumption is justified when b values are obtained from second viral coefficient data. Fitting values for the empirical constants are derived from van der Waal's equation. Some b values obtained in this way are given in table.
So that we need to deal with a single molecular size parameter, we treat molecules as spherical particles. The diameter of a molecule is d. the volume of a molecule is v
The volume in which a pair of molecules cannot move because of each other's presence is indicated by the lightly shaded region. The radius of this excluded volume sphere is equal to the molecule diameter d. the volume excluded to the pair of molecules is 4/3πd^{3}. We thus obtain,
= 4[4/3π (d/2)]_{3}
The expression in brackets is the volume of a molecule.vander Waal's b term is the excluded volume per mole of the molecules. Thus we have, with N representing Avogadro's number,
B= 4n [4/3π (d/2)^{3}] = 4N (vol. of molecule)
Molecular size and Lennard Jones intermolecular Attraction term based on second virial coefficient data:
Gas |
Excluded volume B, L mol-1 |
Molecular diam. D, pm |
ELJ, J × 10-21 |
He |
0.021 |
255 |
0.14 |
Ne |
0.026 |
274 |
0.49 |
Ar |
0.050 |
341 |
1.68 |
Kr |
0.058 |
358 |
2.49 |
Xe |
0.084 |
405 |
3.11 |
H_{2} |
0.031 |
291 |
0.52 |
N_{2} |
0.061 |
364 |
1.28 |
O_{2} |
0.058 |
358 |
1.59 |
CH_{4} |
0.069 |
380 |
1.96 |
C(CH_{3})_{4} |
0.510 |
739 |
3.22 |
Van der Waal's equation and the Boyle temperature:
Gas |
Tboyle, K |
Tboyle/TC |
H_{2} |
110 |
3.5 |
He |
23 |
4.5 |
CH_{4} |
510 |
2.7 |
NH_{3} |
860 |
2.1 |
N_{2} |
330 |
2.6 |
O_{2} |
410 |
2.7 |
Example: calculate the radius of the molecule from the value of 0.069 L mol^{-1} for the excluded volume b that is obtained from the second virial coefficient data.
Solution: the volume of 1 mol of methane molecules is obtained by dividing the b value of 0.069 L mol^{-1} = 69 × 10^{-6} m^{3} mol^{-1} value by 4. Then division by Avogadro's number gives the volume per molecule. We have:
Volume of methane molecule = 69 × 10^{-6 }m^{3}/4 × 6.022 × 10^{23 }
= 2.86 × 10^{-29} m^{3}
The volume is equal to 4/3∏r^{3 }and on this basis we calculate:
r = 1.90 × 10^{-10} m and d = 3.80 × 10^{-10} m = 380 pm