In[]:=
Clear[A1,B1,C1,D1,E1,F1]
AntoineEquation​​
log10

sat
P
i
=
A
i
-
B
i
C
i
+T
In[]:=
data={{60.3,70.3,80.3,90.3,100.3},{397,556,764,1031,1370}}
Out[]=
{{60.3,70.3,80.3,90.3,100.3},{397,556,764,1031,1370}}
In[]:=
DATA=Transpose[data]
Out[]=
{{60.3,397},{70.3,556},{80.3,764},{90.3,1031},{100.3,1370}}
In[]:=
Datafinal=Table[{DATA[[i]][[1]]+273.15,Log10[DATA[[i]][[2]]/760]},{i,1,5}]//N
Out[]=
{{333.45,-0.282023},{343.45,-0.135739},{353.45,0.00227977},{363.45,0.132445},{373.45,0.255907}}
In[]:=
nlm1=NonlinearModelFit[Datafinal,A1-B1/(T+C1),{A1,B1,C1},T]
Out[]=
FittedModel
4.72-
1660.
-1.54+T

In[]:=
nlm1["RSquared"]
Out[]=
1.
Our data agree well with coefficients obtained by NIST shown below:
​​
In[]:=
Normal[nlm1]
Out[]=
4.719-
1659.87
-1.54355+T
ModifiedAntoineEquation​​
ln

sat
P
i
=
A
i
+
B
i
C
i
+T
+
D
i
ln(T)+
E
i
 
F
i
T
with
C
i
=0and
F
i
=2
In[]:=
Clear[A1,B1,C1,D1,E1,F1]
In[]:=
Datafinal2=Table[{DATA[[i]][[1]]+273.15,Log[DATA[[i]][[2]]/760101.325]},{i,1,5}]
Out[]=
{{333.45,3.96895},{343.45,4.30578},{353.45,4.62358},{363.45,4.9233},{373.45,5.20758}}
In[]:=
C1=0;F1=2;
In[]:=
nlm2=NonlinearModelFit[Datafinal2,A1+B1/(T+C1)+D1Log[T]+E1T^F1,{A1,B1,D1,E1},T]
Out[]=
FittedModel
57.2-
5.40×
3
10
T
+8.62×
-6
10
2
T
-6.54Log[T]

In[]:=
Normal[nlm2]
Out[]=
57.2092-
5404.12
T
+8.61546×
-6
10
2
T
-6.53955Log[T]
In[]:=
nlm2["RSquared"]
Out[]=
1.
Clausius-ClapeyronEquation​​
ln

sat
P
i
=
A
i
-
B
i
RT
In[]:=
Clear[A1,B1,C1,D1,E1,F1]
In[]:=
data={{60.3,70.3,80.3,90.3,100.3},{397,556,764,1031,1370}}
Out[]=
{{60.3,70.3,80.3,90.3,100.3},{397,556,764,1031,1370}}
In[]:=
DATA=Transpose[data]
Out[]=
{{60.3,397},{70.3,556},{80.3,764},{90.3,1031},{100.3,1370}}
In[]:=
Datafinal=Table[{DATA[[i]][[1]]+273.15,Log[DATA[[i]][[2]]1.01325/760]},{i,1,5}]//N
Out[]=
{{333.45,-0.636219},{343.45,-0.299387},{353.45,0.0184123},{363.45,0.318129},{373.45,0.602411}}
In[]:=
R=8.314;
In[]:=
nlm3=NonlinearModelFit[Datafinal,A1-B1/(RT),{A1,B1},T]
Out[]=
FittedModel
10.9-
3860.
T

In[]:=
3855.69×8.314
Out[]=
32056.2
In[]:=
Normal[nlm3](*Heatofvaporizationofbenzene=3855.69x8.314=32056.2kJ/kmol*)
Out[]=
10.9269-
3855.7
T
In[]:=
nlm3["RSquared"]
Out[]=
1.
Rule: The model with the lowest AIC or BIC offers the best fit.​
​
Hence, model 1 (Antoine Equation) is better than model 2 (Modified Antoine Equation) and model 3 (Clausius-Clapeyron Equation)
In[]:=
nlm1["AIC"]
Out[]=
-73.9994
In[]:=
nlm1["BIC"]
Out[]=
-75.5617
In[]:=
nlm2["AIC"]
Out[]=
-63.6624
In[]:=
nlm2["BIC"]
Out[]=
-65.6152
In[]:=
nlm3["AIC"]
Out[]=
-67.5963
In[]:=
nlm3["BIC"]
Out[]=
-68.768

Plots of vapor pressure versus temperature show the typical exponential increasing trend

In[]:=
p1=Plot[10^(4.719-1659.87/(-1.54355+T+273.15)),{T,55,105},GridLinesAutomatic,FrameTrue,AspectRatio1,FrameLabel{Style["Temperature / °C ",14],Style["C6H6 Vapor Pressure / bar",14]}];
In[]:=
p2=Plot[Exp[57.2092-5404.12/T+8.61546*10^-6T^2-6.53955Log[T]]/101.325/.T(273.15+a),{a,55,105},GridLinesAutomatic,FrameTrue,AspectRatio1,PlotStyle{Dashed,Darker@Red}];
In[]:=
Show[p1,p2]
Out[]=

Reference

For experimental vapor pressure data versus temperature
◼
  • [1] C. Eon, C. Pommier, and G. Guiochon, Vapor pressures and second virial coefficients of some five-membered heterocyclic derivatives, Journal of Chemical & Engineering Data, 1971, 16 (4), pp. 408-410, DOI: 10.1021/je60051a008

    • CITE THIS NOTEBOOK

    AIC and BIC criteria for model selection applied to benzene vapor pressure versus temperature​
    by Housam Binous and Ahmed Bellagi
    Wolfram Community, STAFF PICKS, September 15, 2024
    ​https://community.wolfram.com/groups/-/m/t/3273909