# Problem Consider the following PT projection of a phase diagram ty...

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Problem 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 (e.g. regions your diagrams (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 P& P, T2T3 class III-a phase behavior Given: # degrees species; Problem 2 According simple principle o states, the equation of state for fluid written in reduced form. i.e. function of and P' will be valid for any other fluid in that group of substances,i i.e., P (1) 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 the intermolecular potential respectively. (a) eqn. (1) valid. show that an equally valid form of the simple corresponding states principleis (2) where P. P/P and =V/V. Further show that the following equations are valid: "=k,J/c=cT,, 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 px/p-1/10 T/T_0.7 (3) Pitzer defines the acentric factor as @= -1.0 (4) (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 an 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 elay sk(K) MW (g/mol) Xe p(s/em) 39.948 H-O 809. 2.827 59.7 2.016 Given: Avogadro's number, mol Problem For many applications that we deal with in statistical mechanics it is possible to write the canonical partition function i the form 2xmk7 o NA 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 use 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. P (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 energy. Given: Q onn alnQ Problem 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 Given: For function f(r,y) the exact differemial of f due to small changes in andy is dy=mdx+ndy 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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