Untitled - Aerobib - Universidad Politécnica de Madrid

2 CHAPTER 1. THERMOCHEMISTRY
3.0
2.5
2.0
1.5
ϕ/ε
1.0
0.5
0.0
−0.5
−1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
r/σ
Figure 1.1: Dimensionless Interaction Potential of Lennard-Jones, ϕ/ε, versus dimensionless
distance between molecules, r/σ.
the only needed except for very high pressures, is expressed in the form 2
B (T ) = 2π N M
∫ ∞
0
(
1 − exp(− ϕ(r)
kT ) )
r 2 dr, (1.2)
where N = 6.0288 × 10 23 mol −1 is the Avogadro number, M is the molar mass
of the gas and k = 1.38047 × 10 −16
erg/grad is the Boltzmann constant. It has
experimentally been checked that the interaction potential of Lennard-Jones
( (σ ) 12 ( σ
) ) 6
ϕ (r) = 4ε − , (1.3)
r r
represents, suitably, the actual behavior of many gases. In Eq. (1.3), σ is the radius
of the molecules and ε is a constant which has energy dimensions. Fig. 1.1 represents
Eq. (1.3). There, it is seen that if the distance r between the molecules is larger than
several times the radius of the molecule, the molecules attract each other whilst if
r < 1.12σ they reject with a force that increases very rapidly as the molecules get
closer. By substituting (1.3) into (1.2) one verifies that B (T ) can be expressed in the
form
where
2 See Ref. [3].
B (T ) = b 0
M B∗ (T ∗ ) , (1.4)
b 0 = 2 3 πNσ3 (1.5)