Casting Equations of State into Virial Form Virial expansion

the 2nd , 3rd virial coefficients 12 fluids

the exponential terms in last 2 equations seem intimidating, , out of line virial expansion sequence. purpose correct third virial coefficient, isotherms in liquid phase represented correctly. actually, exponential term converges rapidly ρ increases, , if took first 2 terms in taylor expansion series,



1
−
γ

ρ

2




{\displaystyle 1-\gamma \rho ^{2}}

, , multiply



1
+
γ

ρ

2




{\displaystyle 1+\gamma \rho ^{2}}

, result



1
−

γ

2



ρ

4




{\displaystyle 1-\gamma ^{2}\rho ^{4}}

. contributed



c

/

r

t

3




{\displaystyle c/rt^{3}}

term third virial coefficient, , 1 term eighth virial coefficient, can ignored.

after expansion of exponential terms, benedict-webb-rubin , starling equations of state have interesting form:

z
=
1
+
b

ρ

r


+
c

ρ

r


2


+
f

ρ

r


5




{\displaystyle z=1+b\rho _{r}+c\rho _{r}^{2}+f\rho _{r}^{5}}

the fourth , fifth virial coefficients zero. after third virial term, next significant term sixth virial coefficient. seems first 3 virial terms dominant compressibility factor fluids, down



0.5

t

c




{\displaystyle 0.5t_{c}}

, ,



2.0

ρ

c




{\displaystyle 2.0\rho _{c}}

.

it interesting note in original study in 1901 kamerlingh onnes[1], omitted fourth virial coefficient d, , designated higher terms residue in virial equation. unfortunately, significance of first 3 third virial terms has never been appreciated, , effects on gaseous-liquid equilibrium masked other higher virial coefficients in blind search of precision, multi-variable optimization algorithms or likes.

it clear why benedict-webb-rubin improved on beattie-bridgeman equation of state adding complicated exponential term. ought have recognized third virial coefficient in gaseous phase small, had large in liquid phase. instead of enlarging third virial coefficient, chose add strange-looking exponential term, sole purpose make third virial coefficient larger @ lower temperatures. taylor expansion of exponential term reveals true intentions.

re-analyzing data reported starling[13], virial coefficients best represented:

b
=

b

0


+



b

1



t

r




+



b

2



t

r


2




+



b

3



t

r


3






{\displaystyle b=b_{0}+{\frac {b_{1}}{t_{r}}}+{\frac {b_{2}}{t_{r}^{2}}}+{\frac {b_{3}}{t_{r}^{3}}}}






c
=

c

0


+



c

1



t

r




+



c

2



t

r


2




+



c

3



t

r


3




:


{\displaystyle c=c_{0}+{\frac {c_{1}}{t_{r}}}+{\frac {c_{2}}{t_{r}^{2}}}+{\frac {c_{3}}{t_{r}^{3}}}:}






f
=

f

0


+



f

1



t

r






{\displaystyle f=f_{0}+{\frac {f_{1}}{t_{r}}}}

b , c determined using simple second order regression analysis experimental pρt isotherms.




b

0


−

b

3




{\displaystyle b_{0}-b_{3}}

,




c

0


−

c

3




{\displaystyle c_{0}-c_{3}}

determined using third order regression analysis on b , c.




f

0


−

f

1




{\displaystyle f_{0}-f_{1}}

determined analyzing residues in compressibility factor after first 3 virial terms removed virial equation. data reported starling[13] re-analyzed , results shown in following table. these coefficients dimensionless since scaled critical molar volumes , critical temperate.