Equations of State

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Equations of State

• In order to calculate properties such as internal
  energy and enthalpy, an accurate relationship
  between readily measurable properties such as P,
  V, T is necessary
• OBJECTIVES
• Introduce the virial equations
• Introduce the compressibility factor
• Discuss cubic equations of state
• Van der Waals, Redlich-Kwong
• Introduce the generalized compressibility factor
  and principle of corresponding states
• Discuss generalized equations of state

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Properties of Real Fluids - PV diagram
isotherms of CO2
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Virial Equations and Real Gases

• Consider the gas region of a PV diagram. Along
  an isotherm, the diagram indicates that a
  polynomial equation can be used to describe PV
• what happens as P ? 0?
• Real gases differ from ideal gases primarily
  because of the existence of attractive and
  repulsive forces.

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Compressibility Factor, Z

• used to describe degree of deviation of gas from
  ideal behavior

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Cubic Equations of State

• Van der Waals equation (1873)
• included in the ideal gas equation the effect of
  attractive-repulsive forces between molecules

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Van der Waals Equation

• compare to the ideal gas equation

Volume occupied by molecules
Forces of attraction
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Cubic Equations of State

• Van der Waals equation

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Cubic Equations of State

• Redlich-Kwong Equation (1949)
• There have been many equations of state
  published. In general

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Principle of Corresponding States

• cubic equations of state make it possible to
  solve problems for real fluids, if values for
  parameters are given.
• values for these parameters not available for all
  fluids
• to increase utility, need to be able to predict
  these values
• approach is to use generalized property
  correlations
• This approach is based on two findings
• Fluids can be grouped into classes based on their
  properties and within any class molecular
  interactions are similar.
• If a given equation of state fits the data for
  one member of a class, the same equation of state
  will fit the data for other members of that class

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Generalized Compressibility Factor
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Generalized Equations of State

• Recall from the P-V diagram of fluids, at the
  critical point, the isotherm undergoes an
  inflection. Mathematically this can be expressed
  as
• consider the van der Waals equation of state at
  the critical point

at Pc and Vc
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Generalized Redlich-Kwong
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Generalized Equations of State

• Van der Waals predicts that Zc 0.375, while
  Redlich-Kwong predicts 0.333.
• Zc for most fluids falls within the range 0.23 -
  0.31.

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Generalized Virial Equation

• Based on the use of the acentric factor, w, and
  is only applicable to gases.

Z0 and Z1 can be found in tables or
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EXAMPLE

• A 125 cm3 cylinder is rated at 200 atm. It
  presently contains 1 mole of methane at room
  temperature. If the temperature is raised to 50
  ºC, will the cylinder rupture? Use the ideal gas
  equation, and the Redlich-Kwong correlation.

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Example

• Find the molar volume of methane at 280 K and
  9.87 atm. Use the ideal gas equation, the
  Redlich-Kwong equation, and the Generalized
  Virial approach.