Waals Critical Point
The constants of van der Waal’s equation can be evaluated from critical point data.
Van der Waal’s equation or any of the other two parameter equations cannot describe the detailed PV behaviour of a gas in the region of liquid vapor equilibrium. In that region, straight line constant pressure sections interrupt the smooth curves produced by simple functions. There will, however, be one temperature for which van der Waals equation, with given values of a and b shows the horizontal point of inflection in PV displays that is to be identified with the critical isotherm. This identification provides a convenient way of obtaining values for a and b.
Equation can be written for 1 mol and rearranged to give
To investigate the horizontal point of inflection on a plot of P versus V, we obtain
At the critical point, the first and second derivatives are zero, and the pressure, volume per mole and temperature can be written as PC, VC and TC respectively. At the critical point,
These three equations can be solved for a, b, and R in terms of PC, VC, and TC. After some manipulation, the following relations are obtained:
b = 1/3 V_{C}, a = P_{C} V^{2}_{c}, R = (8P_{C} V_{C})/(3T_{C} )
the last of these equations can be rearranged to get
(P_{C} V_{C})/[RT]_{C} = 3/8
The critical point data give PCVC/RTC values such as those conform only approximately. Thus, to make the point of inflection of the van der Waal’s isotherm lie at the critical point, i.e. at the value of PC, VC and TC, R must be treated along with a and b as another adjustable constant. Adjusting the value of R to give a good fit at the critical point leads, however, to other problems. In particular, van der Waal’s equation does not then reduce to the ideal gas equation in the low pressure, ideal gas limit.
Usually the gas constant value of R is maintained, and adjusted values are assigned only to a and b. often the values of these empirical parameters are chosen so that the van der Waal’s PV isotherm with a horizontal point of inflection occurs at the critical temperature, and the pressure at which this point of inflection occurs is the critical pressure. This procedure can be followed by eliminating the VC term, by using VC = 3b, from the other expressions. The resulting expressions for a and b are
in much of our subsequent work we use the ideal gas expression PV = nRT instead of van der Waal’s equation or other semipermeable equations. At low pressures but not too low temperatures, deviations from this relation are often not appreciable. Furthermore, the simplicity of the ideal gas expression and the fact that it can be used for all gases without adjustment of any parameters make its use very convenient.
Substance |
A, bar L2 mol-2 |
B, L mol-1 |
H_{2} |
0.248 |
0.0266 |
He |
0.034 |
0.024 |
CH_{4} |
2.29 |
0.0428 |
NH_{3} |
4.25 |
0.0374 |
H_{2}O |
5.52 |
0.0304 |
CO |
1.47 |
0.0304 |
Ne |
0.197 |
0.0158 |
N_{2} |
1.37 |
0.0387 |
NO |
1.42 |
0.0283 |
O_{2} |
1.38 |
0.0317 |
HCl |
3.72 |
0.0408 |
Ar |
1.36 |
0.0322 |
CO_{2} |
3.66 |
0.0428 |
SO_{3} |
6.86 |
0.0568 |
n-C_{5}H_{12} |
19.2 |
0.145 |
Cl_{2} |
6.58 |
0.0562 |
C_{6}H_{6} |
18.9 |
0.120 |
Kr |
2.32 |
0.0396 |
Xe |
4.17 |
0.0513 |
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