Properties of Macroscopic Atomic Systems
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5.1 Equation of State for Gases and Vapors The equation of state for an ideal gas has the form p = NT ,
(5.1)
where p is the gas pressure, T is the temperature, and N is the number density of atoms or molecules. Conversion factors for this equation are given in Table 5.1. This equation may be represented in the form [210] pV = nRT ,
(5.2)
where n is a gas amount expressed in moles, V is the gas volume per mole, and the gaseous constant R is equal to R = 82.06
cm3 MPa cm3 atm = 8.314 . mol K mol K
Equation (5.2) describes a rare gas when an average distance between neighboring atoms or molecules N −1/3 significantly exceeds a distance R0 of a strong interaction between gas atoms or molecules U (R0 ) ∼ T , where U (R) is the interaction potential of two atoms at a distance R, and T is a thermal energy. A subsequent expansion over a small parameter N 1/3 R0 may lead to the van der Waals equation a (5.3) p + 2 (V − b) = T V using two van der Waals constants, a and b. This equation may be spread to the critical point and the liquid state of the system of atoms. The lower the number density Table 5.1. Conversion factors for the equation of gas state Number 1.
Formula N = Cp/T
2.
p = CN T
The proportionality factor C 7.339 × 1021 cm−3 9.657 × 1018 cm−3 1.036 × 10−19 Torr
Units p in atm, T in K p in Torr, T in K N in cm−3 , T in K
116
5 Properties of Macroscopic Atomic Systems
of atoms is, the better the van der Waals equation. If we use this equation at the critical point, the critical parameters [211] of this system of atoms are expressed through the van der Waals parameters. The critical point of an atom system is determined by the equations 2 ∂ p ∂p = 0, = 0, (5.4) ∂V T ∂V 2 T which allows us to express the critical parameters Vcr , pcr and Tcr through the parameters of the van der Waals equation (5.3) in the following manner Vcr = 3b,
pcr =
a , 27b2
Tcr =
8a . 27b
(5.5)
In particular, from this we obtain the relation between critical parameters within the framework of the van der Waals equation ζ =
Tcr 8 = . Vcr pcr 3
(5.6)
The accuracy of the van der Waals equation decreases with an increase of the gas density. Its accuracy is about 30% at the critical point, as follows from comparison of the data of Table 5.2 with formulas (5.5) and (5.6), where parameters of the van der Waals equation are used. In particular, the combination of critical parameters ζ defined according to formula (5.6) is equal for inert gases 3.38 ± 0.13 and for molecular gases 3.49 ± 0.15 according to the data of Table 5.2. These values coincide within the limits of their accuracy, but differ from that of formula (5.6) based on the van der Waals equation. Table 5.3 contains critical parameters for molecular systems consisting of molecules AF6 . These molecules are almost round, and therefore interaction between Table 5.2. Parameters of van der Waals equation and critical parameters of inert and molecular gases [2, 9, 10]
H2 He N2 O2 F2 Ne Cl2 Ar Br2 Kr J2 Xe Rn
a (105 MPa cm6 /mol2 ) 0.245 0.0346 1.370 1.382 1.
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