2F2 :  An Application of Equations of State  6 pts 

Steam is
contained in a 203 L
tank at 500^{o}C.
The mass of steam in the tank is 12.4 kg. Determine the pressure in the tank using: a.) Ideal Gas EOS b.) Virial EOS c.) van der Waal EOS d.) RedlichKwong EOS 

e.) Compressibility
Factor EOS f.) Steam Tables. 

Read :  Not much to say here.  
Given:  m  12.4  kg  V  203  L  
T  500  ^{o}C  0.203  m^{3}  
Find:  P  ???  kPa  
Assumptions:  None.  
Equations / Data / Solve:  
Begin by collecting all of the constants needed for all the Equations of State in this problem.  
R  8.314  J/molK  T_{c}  647.4  K  
MW  18.016  g NH_{3} / mol NH_{3}  P_{c}  2.21E+07  Pa  
Part a.)  Ideal Gas EOS : 

Eqn 1  Solve for pressure : 

Eqn 2  
We must determine the molar volume before we can use Eqn 2 to answer the question.  
Use the definition of molar volume: 

Eqn 3  
Where : 

Eqn 4  
MW  18.016  g H_{2}O / mol H_{2}O  n  688.28  mol H_{2}O  
V  2.95E04  m^{3}/mol  
Now, plug values back into Eqn 2.  R  8.314  J/molK  
Be careful with the units.  T  773.15  K  
P  2.18E+07  Pa  
P  21.8  MPa  
Part b.)  van der Waal EOS : 

Eqn 5  
We can determine the values of a and b, which are constants that depend only on the chemical species in the system, from the following equations.  

Eqn 6  T_{c}  647.4  K  
P_{c}  2.21E+07  Pa  
a  0.5530  Pamol^{2}/m^{6}  

Eqn 7  b  3.04E05  m^{3}/mol  
Now, we can plug the constants a and b into Eqn 5 to determine the pressure.  
P  17.9  MPa  
Part c.)  RedlichKwong EOS : 

Eqn 8  
We can determine the values of a, b and a, which are constants that depend only on the chemical species in the system, from the following equations.  

Eqn 9 

Eqn 10  
Now, plug values into Eqns 8 10 :  
a  14.25855  Pam^{6}K^{1/2}/mol^{2}  
b  2.110E05  m^{3}/mol  P  18.0  MPa  
Part d.)  Compressibility EOS :  Given T_{R} and the ideal reduced molar volume, use the compressibility charts to evaluate either P_{R} or the compressibility, Z  

Eqn 11 

Eqn 12  
T_{R}  1.1942  
Defiition of the ideal reduced molar volume : 

Eqn 13  
V_{R}^{ideal}  1.2110  
Read the Generalized Compressibility Chart for P_{R} = 0 to 1 :  P_{R}  0.78  
Z  0.83  
We can use the definition of P_{R} to calculate P : 

Eqn 14  

Eqn 15  
P  17.2  MPa  
Or, we can use Z and its definition to determine P : 

Eqn 16  
P  18.1  MPa  
Part e.)  The Steam Tables provide the best available estimate of the pressure in the tank.  
Because T > T_{c}, the properties of the water in the tank must be obtained from the superheated vapor table, even though the water is actually a supercritical fluid in this system.  
At this point we can make use of the fact that we have a pretty good idea of what the actual pressure is in the tank (from parts ad) or we can scan the spuerheated vapor tables to determine which two pressures bracket our known value of the specific volume.  
In either case, we begin by converting the molar volume into a specific volume : 

Eqn 17  
Using the MW of water from part (a) yields :  v  1.637E05  m^{3}/g  
v  0.016371  m^{3}/kg  
The Superheated Steam Table gives us :  
At P =  20  MPa  and  At P =  40  MPa  
v =  0.014793  m^{3}/kg  v =  0.005623  m^{3}/kg  
We can determine the pressure in our tank by interpolation :  P  16.56  MPa  
P  16.6  MPa  
Verify:  No assumptions to verify.  
Answers :  a.)  P  21.8  kPa  d.)  P  18.1  kPa  
b.)  P  17.9  kPa  e.)  P  16.6  kPa  
c.)  P  18.0  kPa  
None of these Equations of State did very well because steam at high pressure behaves in a very nonideal manner due to the high polarity of the molecules and the resulting stron electrostatic interactions. 