Anlony F. Saturno
Daniel Berthelot's Equation of State
University of Tennessee Knoxville
The Berthelot equation of state, for one mole of gas, is usually presented in one or both of the following forms:
accurately given by the factor 3219. Finally from arriving at 1,he expression
Equation (2) is often referred to as the "low pressure" form. In the above equations, the symbols P, V , and T have their usual meaning. The quantities, ir and T are the reduced pressure and temperature, respectively. The constant a is determined by the force law operating between the molecules and the constant b represents the volume unavailable for compression, i.e., the "excluded volume." The purpose of this paper is to point out the fact t,hat the two eauations above are really " auite . independent of one another. This is not anything new, but nevertheless, it is interesting to review the historical development of these equations and to present a more efficient "derivation" of equation (2) starting with equation ( 1 ) . Finally a few comments on the applicability of equation (2) will be given. We start by first determining the constants a, b, and R as given by equation ( I ) . At the critical point, i.e., P = P,,V = V,, and T = T., the following conditions hold, namely:
With these conditions, together with equation ( I ) , we obtain
Actually one might have arrived at the proper adjustments solely from the fact that 4b gave better results than 36 for the critical volumes. Thus in equation (4.1) we make the adjustment 3b 4b or equivalently b 4/8b to arrive a t 'V. = 3b 4b (7.1)
-
-
-
-
Furthermore, since Vc is seen to be directly proportioual to b, let us make the adjustment V c 4/3Vc, thus obtaining in equations (4.2) and (4.3) a = 3P.VC1T.
R = 8P,V, = 3T.
--
16P.Vc2T./3
(7.2)
32P.V0/9T.
(7.3)
Substitutiou of the equations (7) into the equation ( I ) gives the reduced equation ( 6 ) . I t is this equation (6) moreover, that reduces to equation (2) at low pressures. This can be seen as follows: multiply out and expand equation ( 6 ) .
+
At low pressures = V / V eis very large, thus we may neglect the term in +-2 in equation (8) to give
From equation (7.3)we have RT = 32PeV,/9r
Substitution of the above equation into equation (9) gives after some minor algebra These equations give a reduced equation of state when substituted back into equation (1): (r
+ 3/rd2)(d - 1/3)
=
8713
(5)
where .$ is the reduced volume, V / V V . This equation ( 5 ) , however, does not fit the observed data for low pressures (as noted by Berthelot). As a result Berthelot1 made some adjustment of the equations (4) to bring the equation (5) into agreement with the low pressure data. These adjustments were based upon Berthelot's observations that the critical volumes were more accurately given by 4b instead of 3b; that the critical vapor density predicted by equations (4) to be 8 / 3 times that given by the ideal gas laws was more 1
BERTHELOT, D., T ~ vmem. .
bur. intern. poids mesures, No.
13 (1907).
464 / Journal of Chemical Education
Rewriting equation (11.1) as
+
And recalling the series expansion ( 1 - x)-' .;: 1 x and x small, we obtain upon identifying x = (1144 16/3a#), the equation
Returning again to equation (6) we note that a t low pressures we may write
Subst,itution of this approximate equation into equat,ion (12) and performing some algebra gives the desired equation:
Returning to equation (6) we note that at the critical poiut (r = 6 = r = I), we obtain the absurd result that 4.75 = 3.56. Thus we are forced to cousider equation (6) as an empirical one which cannot hold near the critical point. This discrepancy in no way reduces the value of equation (6) which is very accurate at moderate to low pressures at room temperature. Note that so long as
this equation shows (in accordance with experience) that at low pressures gases are more compressible than corresponds to Boyle's Law; hydrogen shows the opposite behavior even a t ordinary temperatures since the above inequality is not fulfilled. Nitrogen also behaves similarly at higher temperatures. Moreover, it is evident that equation (2) is in the virial form with just a second virial coefficient whose value is
This equation has proved to be quite satisfactory for the estimation of small deviations from the ideal gas law when data are not available for the particular gas in question.
Volume
39, Number 9, September 7 962
/
465
