At low temperatures and high pressures, the ideal gas equation ceases to apply. Under these conditions, you must deal with "real" gases.
A quick way of checking the validity of an ideal gas assumption is to look at the specific molar volume of the gas. The ideal gas equation is good to within about 1% if:
Real gases require more complex equations of state than do ideal gases. There are many options, but we will focus on:
Virial equations are a family of equations of state of the general form:
The parameters in the equation (B,C,D = ci) are called "virial coefficients". If ci=0 for i>0, the virial equation reduces to the ideal gas equation. Just as with the ideal gas equation, the temperatures and pressures used must be absolute.
The accuracy required determines the number of terms that are kept -- more terms makes the equation more accurate, but also more complicated to work with. Virial coefficients are different for each gas, but other than that are functions of temperature only.
Coefficients are normally obtained by making measurements of P, V, and T, and fitting the equation. These values are then published so that others may use them.
Many forms of the virial equation exist. Often, we will truncate the virial equation to
Virial equations cannot represent thermodynamic systems where both liquid and vapor are present. A "cubic" EoS is need to do this. One such is the Soave-Redlich-Kwong (SRK) equation.
In this equation, the b term is a volume correction, while the a is a molecular interaction parameter. The constants all depend on the critical temperature and pressure of the gas. These can be looked up easily in a data table.
The "acentric factor", omega, is also easily looked up. It is related to the geometry of the gas molecule.
To use the SRK equation:
Strategy: The pressure and temperature are known, so look up the critical properties and acentric factor, find the SRK equation constants, and solve the SRK equation for the specific molar volume.
The critical properties (from the back of the book) are
Solution of the Equation: This equation is cubic in V, so trying to solve it algebraically is a bit complicated. Consequently, we will use an iterative approach.
We can implement the iterative strategy in a number of ways -- and using a computer or a calculator solver is strongly recommended. All the approaches use the same basic approach. A value of V is guessed and the right hand side (RHS) of the equation calculated. The result will be compared to zero and a new value of V chosen and tried.
The ideal gas law is always a good place to get an initial value, because it will usually be "close" to the real solution:
You also need to determine how close to the "exact" answer you want to be. This error tolerance is usually built-in to the solver routine, although you usually can adjust it (in Mathcad, look at the value of TOL). If you've got a particularly difficult system, sometimes it is useful to relax the tolerance to get a rough solution, then update your estimate, tighten the tolerance, and repeat the solution for a final value.
This is the type of problem that Mathcad handles very well:
Download Example as Mathcad File
Engineering calculations often require a tradeoff between ease of use and accuracy. The ideal gas equation is very easy to use, but of questionable accuracy for many cases. Virial and cubic equations of state are accurate, but not particularly convenient. A good compromise is a generalized compressibility factor equation.
The "compressibility factor", z, is defined so that
Most gas properties depend on composition, but according to the "Law of Corresponding States" a few properties are the same for all gases when expressed in terms of deviation from the critical point. We do this by using the "reduced temperature" and "reduced pressure"
Warning: Hydrogen and Helium are special cases. They require a correction when calculating reduced properties.
Using the ideal gas law:
Using the compressibility factor equation
You can also enter the compressibility factor charts using the reduced volume. Critical volume isn't tabulated -- it must be calculated from the critical temperatures and pressures.
Mixtures of real gases can be treated using any equation of state; however, finding virial equation coefficients for gas mixtures is a subject for Thermo II. For this course, we will use an approximate approach using the compressibility factor equation and "Kay's Rule", which states that PV=zmRT. The mixture compressibility factor, zm, is found from the generalized compressibility factor chart using the system temperature and pressure reduced using "pseudocritical constants" given by:
Basis: 100 moles mixture
10 mol propane * 44 lb/mole = 440 lbs
20 mol butane * 58 = 1160
70 mol pentane * 72 = 5040
Total 6640 lbs, AMW 66.4
Mass fractions: 0.0663 propane, 0.175 butane, 0.759 pentane.
Basis: 300 lbs mixture
300 * 0.0663 / 44 = 0.452 lbmoles propane fed
300 * 0.175 / 56 = 0.905 lbmoles butane
300 * 0.759 / 72 = 3.16 lbmoles pentane
4.53 lbmoles total
Check the ideality of the gas
The physical properties are:
| Propane | Butane | Pentane | |
|---|---|---|---|
| Tc | 369.9 K | 425.17 K | 469.8 K |
| 665.8 R | 765.31 R | 845.64 R | |
| Pc | 42.0 atm | 37.47 atm | 33.3 atm |
References:
R.M. Price
Original: 6/20/94
Modified: 10/6/95, 10/17/96, 10/22/98; 12/25/2004, 3/2/2005
Copyright 1996, 2005 by R.M. Price -- All Rights Reserved