Comparison of equations of state for gaseous CO2
Comparison of equations of state for gaseous
CO
2
This exploration compares common equations of state with experimental data for gaseous . The ideal gas model is a rather harsh approximation and assumes that particles don’t interact with each other and that gas particles are essentially point particles with no volume. The van der Waals equation of state corrects for non-zero volume and also includes a pressure correction that allows for inter-particle attraction. The attraction would cause the measured pressure to be lower than that predicted by the ideal gas equation of state. Redlich Kwong equation of state is an empirical equation that fits experimental measurements above the critical temperature quite well. The model equations are provided below and all symbols have their standard meaning.
CO
2
Ideal Gas Equation
P=nRT
V
m
(
1
)van der Waals equation
P+(-b)=RT
a
2
V
m
V
m
(
2
)Redlich Kwong equation
P=-b-(+b)√T
RT
V
m
a
V
m
V
m
(
3
)Functions for Calculation
Functions for Calculation
We have analyzed experimental data for at three different temperatures: 293 K , 323 K, and 373 K. The critical temperature of is 304.25 K.
CO
2
CO
2
◼
Ideal gas equation of state: The model assumes that gas particles are point particles and do not interact with each other. The function calculates the pressure according to the ideal gas model. rgas is the Universal gas constant.
In[1]:=
rgas=Quantity0.08206,;idealgasP[t_,v_]:=
"Liters""Atmospheres"
"Moles""Kelvins"
rgast
v
◼
van der Waals equation of state: This model assigns a non-zero volume (parameter b) for particles and also includes attractive interaction (parameter a) between gas particles.
In[3]:=
vdwgasP[t_,v_,a_,b_]:=-aCO2=Quantity3.952,"Atmospheres";bCO2=Quantity0.042,;
rgast
v-b
a
2
v
2
"Liters"
2
"Moles"
"Liters"
"Moles"
◼
Redlich Kwong equation of state. This model also assigns non-zero volume and attractive interaction between particles. It is generally seen as a huge improvement over the VdW model.
In[6]:=
rkgasP[t_,v_,a_,b_]:=-v(v+b)ACO2=Quantity63.752,;BCO2=Quantity0.029677,;
rgast
v-b
a
1/2
t
"Atmospheres"
2
"Liters"
1/2
"Kelvins"
2
"Moles"
"Liters"
"Moles"
293 Kelvin
293 Kelvin
◼
Organizing the data in list format for computations and plotting.
In[9]:=
t293=Quantity[293,"Kelvins"];p293=QuantityArray[{35.,37,40,44,45,48,50,53,55,56,56.5,56.5,57,60,75,100},"Atmospheres"];v293=QuantityArray{0.533,0.493,0.441,0.380,0.366,0.320,0.304,0.269,0.245,0.231,0.224,0.0583,0.0580,0.0566,0.0536,0.0511},;
"Liters"
"Moles"
◼
Look at the P vs V plot for the gas and compare the experimental data with ideal gas model
In[52]:=
ListPlot[Transpose[{v293,#}]&/@{p293,idealgasP[t293,v293]},PlotLegends{"Experiment","Ideal Gas"},AxesLabel{"V(L/mol)","P(atm)"},PlotMarkersAutomatic,PlotRangeAll]
Out[52]=
Clearly the agreement between the ideal gas model and experimental data is not good. The model tracks the data qualitatively, however, it gets worse as molar volume is decreased.
◼
Let’s analyze the VdW model and compare it with the experimental values.
In[51]:=
ListPlot[Transpose[{v293,#}]&/@{p293,vdwgasP[t293,v293,aCO2,bCO2]},PlotLegends{"Experiment","VdW Gas"},AxesLabel{"V(L/mol)","P(atm)"},PlotMarkersAutomatic,PlotRangeAll]
Out[51]=
A cursory look tells us that the model is in quantitative agreement at higher molar volumes and in qualitative agreement at lower molar volumes.
◼
Let’s check the RK model
In[49]:=
ListPlot[Transpose[{v293,#}]&/@{p293,rkgasP[t293,v293,ACO2,BCO2]},PlotLegends{"Experiment","Redlich Kwong model"},AxesLabel{"V(L/mol)","P(atm)"},PlotMarkersAutomatic,PlotRangeAll]
Out[49]=
We can see that there is quantitative agreement at larger volumes and even at smaller volumes the difference between experimental and model calculated pressures is much lower than vdW model.
◼
Let’s put all the plots together so that we can compare all models with experimental data.
In[48]:=
ListPlot[Transpose[{v293,#}]&/@{p293,idealgasP[t293,v293],vdwgasP[t293,v293,aCO2,bCO2],rkgasP[t293,v293,ACO2,BCO2]},PlotLegends{"Experiment","Ideal Gas","vdW","Redlich Kwong"},AxesLabel{"V(L/mol)","P(atm)"},PlotMarkersAutomatic,PlotRangeAll]
Out[48]=
This composite plot makes it clear that the Redlich Kwong model has the best agreement with the experimental data.
323 K
323 K
◼
Organize the data so that we can plot it.
◼
Check the behavior of the ideal gas model.
Notice that the curves are slightly closer as the temperature has increased. There is still qualitative agreement, however, we can see a clear shift.
◼
Check the vdW model behavior
The VdW model still shows close agreement with experimental data at larger molar volumes.
◼
Check the RK model
The model seems to be tracking experimental data more closely as the temperature increased.
◼
Let’s look at all the models in one view.
Here we can see that the VdW model is completely out of sync with experimental data and the RK model.
373 K
373 K
◼
Organize the data for 373 K (100 C)
◼
Compare the ideal gas model behavior with experimental data
Note the interesting change at this temperature. The ideal gas model starts to track experimental data with reasonable agreement.
◼
Analyze the VdW model
The VdW model is still in very good agreement at larger molar volumes.
◼
Analyze the RK model
At higher temperatures the RK model has quantitative agreement with experimental values.
◼
Adding all the models and data on a single plot.
At the highest temperature (373K) the models seem to perform better. At high temperatures, the molecules are on average moving faster and the models in general perform better
Hands-on Exploration
Hands-on Exploration
◼
Notice that if you increase the attraction parameter, the calculated pressure decreases.
◼
Modify the size parameter to notice that it shows the opposite behavior to a.
Further Explorations
Authorship information
06/23/2017
aksharma@wagner.edu
Future improvements
Add quiz functionality to check for comprehension