Non ideal equations of state
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Non-ideal Equations of State. Review Real Fluids. Gases do not always obey the ideal gas law. At modest temperatures but high pressures, the molecules get close enough together that intermolecular attractive forces become significant. Two things can happen –

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Review real fluids
Review Real Fluids

  • Gases do not always obey the ideal gas law.

    • At modest temperatures but high pressures, 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


  • The temperature at which it becomes impossible to ever form a liquid regardless of the pressure is called the critical temperature. (Tc)

  • The pressure at which there is just last both vapor and liquid is called the critical pressure. (Pc)

Use of tc and pc
Use of Tc and Pc

  • The critical temperature and pressure are key parameters for calculating the relationship between P, V, and T for non-ideal fluids using empirical EOS’s.

Empirical equations
Empirical Equations

  • Several empirical “cubic” equations have been invented to relate P to V and T for non-ideal gases.

    • van der Waals

    • Redlich Kwong

    • Peng Robinson

    • Redlich Kwong Suave

Van der waals
van der Waals

At very small specific volumes, the molecules begin to touch which causes the pressure to rise sharply.

At small specific volumes, the attractive term is significant.

One of the earliest and simplest

Van der waals1
van der Waals

  • The values of a and b are different for different chemicals, but they are related in the same way to each chemical’s Tc and Pc. Critical properties are tabulated.

van der Waals EOS

Peng robinson

  • The vdW equation is Ok but other empirical EOS’s are more accurate (but more complicated) One that has a nice balance of accuracy vs complexity is the Peng-Robinson EOS.

Peng robinson1

  • The a and b parameters are related empirically to the critical properties:

Peng robinson2

  • The a parameter is temperature dependent and also depends on another tabulated,chemical specific, parameter called the “acentric factor”

acentric factor

Peng robinson3

  • 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
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
Cubic EOS roots



Isotherm above Tc

Specific Volume (V)

Below tc
Below Tc

Three roots (3 V’s are predicted by equation)



Isotherm below Tc


Specific Volume (V)

Root evaluation
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)

      • In Excel, make the first guess the ideal gas law – program will find the “gas root”

    • There is one root near b

      • This is the liquid root and is hard to get

    • There is one root near 3xb

      • This is a physically meaningless root (the middle one)

Example with p r eos
Example with P-R EOS

  • Connection

Problem 1.

Find the specific volume of propane gas at 1000 psia and 260 C using the PR equation of state.

w = 0.152; Tc = 369.8 K; Pc= 42.48 bar

Compare this value to the value obtained from the ideal gas law.

Peng robinson4

  • Connection

Problem 2

Using the PR equation of state compute how much methane one could put into a 100,000 m3 storage tank so that the pressure would not exceed 20 atm at 25 C?

Assuming this value is accurate, compute the % error one would get if she used the ideal gas law instead of the PR equation.

Virial equations
Virial Equations 10 bar, 60 C.

  • 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 adjustible constants…

Summary 10 bar, 60 C.

  • 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: