Consider the following PT projection of a phase diagram typical of the CH./ heptene system
(a) Plot T-x|-y diagrams for the indicated pressures and P-x|-y diagrams for the indicated
temperatures Clearly indicate any three-phase lines azeotropes, critical end points, and 2-phase
(b) With help of the Gibbs phase rule. plain why the three-phase region shows up LiL2G
line instead fan area PT diagramofa binary mixture
III-a phase behavior
According simple principle o states, the equation of state for fluid written
reduced form. i.e. function of and P' will be valid for any other fluid in that
group of substances,i i.e.,
where fit universal function i.e. the same function for all substances in the corresponding
states group. In eqn. (1). p' Po³(e, T and v 'vv/N,o: where N. Avogadro's
constant; kgis Boltzmann's constant; and are the size parameter and energy parameter for
(a) eqn. (1) valid. show that an equally valid form of the simple corresponding states
where P. P/P and =V/V. Further show that the following equations are valid:
where are universal constants for the substances in the corresponding states group
(b) Eqn. (1) and(2). in reality turns out to be valid only for the classical inert gases (Ar. Kr. Xe).
Pitzer introduced "acemaric factor account for the departure from simple -fluid behavior For
simple fluids, has been observe that
Pitzer defines the acentric factor as
(i) What the physical meaning Pitzer' sacentric factor?
(ii) the definition the acentric factor' unique? Why did Pitzer define it at T/T =0.7?
(iii) Can H,andH,Obe described by egn (1) and (2)? Explain the reason
(c) Use the following parameters, where is molecular size (diameter) and is
intermolecular energy parameter related to the strength of intermolecular forces, The density of
xenon given in the table for the liquid 167 K and bar pressure Using the density xenon
as reference, calculate the density gem for Ar, H2O and H, the same reduced conditions
of temperatur and pressure from the simple corresponding states theory
Avogadro's number, mol
For many applications that we deal with in statistical mechanics it is possible to write the
canonical partition function i the form
provided that intramolecular rotations can be neglecred Here Zis the configuration integral,
and integrations over the dr are over the volume of the system and integrations over the do are
over all possible values the molecular orientation angles.
(a) What assumptions are implicit in writing O in this form?
(b) Which parts of the partition function above can usually be treated classically at room
temperature, and which cannot? What the criterion that determines whether we can
classical mechanics for given part the partition function not?
(c) Which parts of the partition function above depend on the volume of the system? Which
parts of the partition function above dependo the temperature?
(d) Which part(s) the partition function above must be known to determine the pressure.
(i.e the equation state)? this difficult to evaluate with accuracy for liquid?
(e) Which part(s) of the partition function above must be known to determine the internal
Starting from the canonical ensemble for binary mixture(T V. N1,N2). for which the
thermodynamic potential (Helmholtz free energy) use the Legendre method ts
derive expressionsi the grand canonical ensemble, with independent variables(T for
the following quantities:
(a) thermodvnamic potential (n) (Your result should not include any other thermodynamic
potential: U.A.H.G. etc.)
(b) partition function(E)
(c) probability distribution law
For function f(r,y) the exact differemial of f due to small changes in andy is
where m=(df/dx) and are the variables conjugate to and respectively We
now wish change variables from ie replace conjugate variable the
slope of with respect constant Further, we wish to retain all o the information
contained the original function f(x.y). The Legendre transform off. which we will denote by
the new function g(m.y ) isgiven by
Note that must have the same units as f.
The generalization the Legendre transform the case where there arean arbitrary
number of independent variables, No. is straightforward.
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