Stat Mech of Non-Ideal Gases: Equations of State

This state is called a supercritical fluid. Real Gases Many real gases do not come even close to obeying The Ideal Gas Law. At modest temperatures & high pressure, the molecules get close enough together that intermolecular attractive forces become significant. Two Things Can Happen: At low temperatures the gas can turn into a liquid. At higher temperatures the gas stays a gas but behaves a lot like a liquid. This state is called a supercritical fluid.

Critical Temperature (Tc) & Pressure (Pc) For all gases, there is a temperature at which it is impossible to ever form a liquid, regardless of the pressure. That temperature is called The Critical Temperature  Tc. For all gases, there is a pressure at which there is both vapor and liquid. That pressure is called The Critical Pressure  Pc. Tc & Pc. are key parameters for calculating the relationship between P, V, and T for non-ideal gases using empirical Equations of State. Tc & Pc are tabulated for many gases & fluids.

Empirical Equations of State Several empirical “cubic” equations have been invented to relate P to V and T for non-ideal gases: 1. Van der Waals 2. Redlich-Kwong 3. Peng-Robinson 4. Redlich-Kwong-Suave Now, brief discussions of #1 & #3.

Van der Waals Equation of State This one of the earliest & simplest empirical Equation of State. It was briefly discussed earlier in the course. It has the form: Comes from intermolecular attractions & is most important at small molar volumes. Molar Volume (Volume per Mole) Comes from the volume of the molecules themselves & is most important at small molar volumes.

The values of the empirical parameters a & b are different for different gas molecules, but they are related in the same way to the Tc & Pc for each gas. Note again that the critical properties Tc & Pc are tabulated for the common gases. It can be shown that the Van der Waals parameters a & b are related to Tc & Pc in the following way:

Peng-Robinson Equation of State The Van der Waals Equation of State is OK but other empirical Equations of State are more accurate. Of course, that also means that they are more complicated! An empirical Equation of State that tries to achieve a balance of accuracy vs complexity is the Peng-Robinson Equation of State. It has the form:

Peng-Robinson Equation of State The empirical parameters a & b are related to the critical temperature & pressure Tc & Pc in the following way: The empirical parameter a is temperature dependent & also depends on another tabulated, chemical specific, parameter, the so-called “acentric factor” :

Peng-Robinson It is usually a good idea to program the more complex equations into a spreadsheet or Maple. Because of the way the equation is written, finding the volume when T and P are given or finding the temperature when P and V are given requires trial and error calculations (root finding)

Beware Multiple Roots When the object is to find V with T and P known, then it is possible to get 3 answers (roots) that all satisfy the equation. This will only happen for T below the critical temperature. The smallest value is a volume that corresponds to the liquid at that T and P The largest value is a volume that corresponds to the vapor (most accurate). The middle value has no physical meaning (just a mathematical artifact). In trial and error programs like Solver, one must achieve the desired root by an initial guess that is close to desired root.

Cubic EOS roots Pressure P Isotherm above Tc Specific Volume (V)

Below Tc Three roots (3 V’s are predicted by equation) Pressure P Isotherm below Tc T1 Specific Volume (V)

Root Evaluation Below Tc – care must be taken to make sure that the right root is obtained There is one root near the ideal gas law (The large volume) There is one root near b This is the liquid root and is hard to get There is one root near 3b This is a physically meaningless root (the middle one)

Virial Equations For sophisticated calculations fitting equations with more adjustable parameters are used. These are called virial equations. Some equations (like those for water) might have 20 or more adjustable constants…

Summary EOS are more accurate representations of fluid PVT relationships than the simple ideal gas law. Cubic equations of state have a good balance between simplicity and accuracy. The other main type of empirical equation is a “virial” equation that attempts to fit the PVT behavior with a long series of “adjustment” terms: