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thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
, a departure function is defined for any thermodynamic property as the difference between the property as computed for an
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is ...
and the property of the species as it exists in the real world, for a specified temperature ''T'' and pressure ''P''. Common departure functions include those for
enthalpy Enthalpy () is the sum of a thermodynamic system's internal energy and the product of its pressure and volume. It is a state function in thermodynamics used in many measurements in chemical, biological, and physical systems at a constant extern ...
,
entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
, and
internal energy The internal energy of a thermodynamic system is the energy of the system as a state function, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accoun ...
. Departure functions are used to calculate real fluid extensive properties (i.e. properties which are computed as a difference between two states). A departure function gives the difference between the real state, at a finite volume or non-zero pressure and temperature, and the ideal state, usually at zero pressure or infinite volume and temperature. For example, to evaluate enthalpy change between two points ''h''(''v''1,''T''1) and ''h''(''v''2,''T''2) we first compute the enthalpy departure function between volume ''v''1 and infinite volume at ''T'' = ''T''1, then add to that the ideal gas enthalpy change due to the temperature change from ''T''1 to ''T''2, then subtract the departure function value between ''v''2 and infinite volume. Departure functions are computed by integrating a function which depends on an
equation of state In physics and chemistry, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal energy. Most mo ...
and its derivative.


General expressions

General expressions for the
enthalpy Enthalpy () is the sum of a thermodynamic system's internal energy and the product of its pressure and volume. It is a state function in thermodynamics used in many measurements in chemical, biological, and physical systems at a constant extern ...
''H'',
entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
''S'' and
Gibbs free energy In thermodynamics, the Gibbs free energy (or Gibbs energy as the recommended name; symbol is a thermodynamic potential that can be used to calculate the maximum amount of Work (thermodynamics), work, other than Work (thermodynamics)#Pressure–v ...
''G'' are given byPoling, Prausnitz, O'Connell: ''The Properties of Gases and Liquids'', 5th Ed., McGraw-Hill, 2001. p. 6.5. \begin \frac &= \int_V^\infty \left T \left(\frac\right)_V \right\frac + 1 - Z \\ ex\frac &= \int_V^\infty \left T \left(\frac\right)_V - 1 + Z\right\frac - \ln Z \\ ex\frac &= \int_V^\infty (1-Z) \frac + \ln Z +1 - Z \end


Departure functions for Peng–Robinson equation of state

The Peng–Robinson equation of state relates the three interdependent state properties pressure ''P'', temperature ''T'', and molar volume ''V''''m''. From the state properties (''P'', ''Vm'', ''T''), one may compute the departure function for enthalpy per mole (denoted ''h'') and entropy per mole (''s''):Kyle, B.G.: ''Chemical and Process Thermodynamics'', 3rd Ed., Prentice Hall PTR, 1999. p. 118-123. :\begin h_-h_^ &= RT_C\left _r(Z-1)-2.078(1+\kappa)\sqrt\ln\left(\frac\right)\right\\ .5exs_-s_^ &= R\left ln(Z-B)-2.078\kappa\left(\frac-\kappa\right)\ln\left(\frac\right)\right\end where \alpha is defined in the Peng-Robinson equation of state, ''Tr'' is the reduced temperature, ''Pr'' is the reduced pressure, ''Z'' is the
compressibility factor In thermodynamics, the compressibility factor (Z), also known as the compression factor or the gas deviation factor, describes the deviation of a real gas from ideal gas behaviour. It is simply defined as the ratio of the molar volume of a gas ...
, and :\kappa = 0.37464 + 1.54226\;\omega - 0.26992\;\omega^2 :B = 0.07780\frac{T_r} Typically, one knows two of the three state properties (''P'', ''Vm'', ''T''), and must compute the third directly from the equation of state under consideration. To calculate the third state property, it is necessary to know three constants for the species at hand: the
critical temperature Critical or Critically may refer to: *Critical, or critical but stable, medical states **Critical, or intensive care medicine *Critical juncture, a discontinuous change studied in the social sciences. *Critical Software, a company specializing in ...
''Tc'', critical pressure ''Pc'', and the acentric factor ''ω''. But once these constants are known, it is possible to evaluate all of the above expressions and hence determine the enthalpy and entropy departures.


References


Correlated terms

* Residual property (physics) Thermodynamics Fluid mechanics Equations