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Applied thermodynamics TMT4140 Lecture 4
Equations of state Tore Haug-Warberg Department
of Chemical Engineering Spring semester 2007
Haug-Warberg TMT4140 (EOS)
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Basic concept
Total differential in canonical variables
Euler integration requires N2 equations of state
in S,V,n
Very few EOS have S,V,n as free variables. In
fact T,V,n is more common which makes Helmholtz
energy a favorable starting point
Starting point
Haug-Warberg TMT4140 (EOS)
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Model contributions
Normally, several EOS are combined into a
thermodynamic model framework. The basic
contributions are built up from quantum
mechanical models of non-interacting gas
molecules or crystal lattices. To this
theoretical basis is added an ideal mixture law
(entropy of mixing) from statistical mechanics
combined with an empirical EOS for the real
mixture behaviour.

• Free translation
• Harmonic vibration
• Rigid rotation
• Electronic exitation
• Fonons (crystals)
• Free electrons (metals)

• Ideal gas mixture
• Van der Waals
• Virial equation
• Debye-Huckel

Beyond this course
Haug-Warberg TMT4140 (EOS)
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Ideal gas
Translational energy (Sackur-Tetrode equation) of
monatomic gases
Harmonic oscillator
Quantum mechanics
Rigid rotator
Statistical mechanics
Ideal gas
Haug-Warberg TMT4140 (EOS)
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Virial equation
The virial equation is an infinite power
expansion in density, but for most practical
purposes the expansion is truncated after the
second term
From statistical mechanics
E.g. hard spheres
The Lennard-Jones potential
Haug-Warberg TMT4140 (EOS)
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Van der Waals equation
The VDW equation (1873)
The so-called mixing rules for multicomponent
mixtures are much younger
Normally, the coefficients a and b are tied up to
the critical point (one set of parameters for
each component)
Critical constants
NB! important
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Corresponding state
Using critical constants in a and b makes VDW a
universal equation in reduced co-ordinates (no
arbitrary parameters)
Ideally, all components should conform with this
equation, but in practice not (of course). The
figure shows VLE-data for nitrogen.
Haug-Warberg TMT4140 (EOS)