Homework Assignment Number Two. where, for the oxygen molecule κ = ,

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1 D. eer, MSE 506, Dept. o Materials Siene & Engineering, University o Tennessee, noville Homework Assignment Numer Two Prolem. Consider the Peng-Roinson Equation o state as given elow. The ritial temperature and pressure o oygen are also given elow. Eah root o this equation is a molar volume. Find all o the roots o the Peng-Roinson equation or oygen at the temperatures given elow and or a pressure o.0 atmosphere. (a) T 98.0 () T 98.0 (For those o you who have not had thermodynamis, there an only e one phase aove the ritial temperature.) ( T ) P V V ( V ) ( V ) R T where, R 8. J/mol/, and a( T ) 0.57 a( T ), P RT, P ( ) T α T κ T where, or the oygen moleule κ , P Pa. Prolem. Consider the van der Waals equation o state. p V V T, and Vapor-liquid equilirium (VLE) at a given temperature, T, ours at a speii pressure, the vapor pressure, p vap. Thereore, we an onsider the vapor pressure our unknown. However, you an t alulate the vapor pressure without knowing the molar volumes o the vapor, V vap, and liquid, V liq, phases. Thereore, in a standard VLE prolem, we have three unknowns: p vap, V vap and V liq. These three variales require three equations. The irst equation is the relationship etween p vap, V vap or V liq., given y the equation o state, ( p V, V ) p 0 vap, vap liq vap Vvap Vvap The seond equation is the thermodynami onstraint estalishing mehanial equilirium, (, ) p V V p p 0 vap, vap liq vap liq V vap Vvap Vliq Vliq The third equation is the thermodynami onstraint estalishing hemial equilirium, ( p, V, V ) µ 0 vap vap liq vap liq µ where the hemial potential or a van der Waals gas is given y

2 D. eer, MSE 506, Dept. o Materials Siene & Engineering, University o Tennessee, noville V a µ RT ln Λ V VRT This epression or the hemial potential o a van der Waals gas introdues a new onstant, the thermal de Broglie wavelength, Λ, ut we don t need it eause it drops out when we equate hemial potentials. µ liq µ vap V liq a Vvap a RT ln RT ln Λ Vliq VliqRT Λ Vvap VvapRT a a RT ln( Vliq ) RT ln( Vvap ) Vliq VliqRT Vvap VvapRT I we so hoose, we ould solve the VLE prolem y inding the roots to these three equations. This requires good guesses or all three unknown variales. Here is a etter alternative solution approah. Solving a system o three equations and three unknowns is muh harder than solving a system o equations with one equation and one unknown. Thereore, a useul trik in this prolem is to reognize that the van der Waal s equation o state is a ui equation o state and the roots o a ui equation o state are easily otained using the Matla roots untion. pv ( p RT ) V av a 0 Thereore, we an pose the solution o the VLE prolem o the van der Waals luid as a single equation with a single equation, in whih we guess the vapor pressure, use the roots ommand to solve or the V vap and V liq. and sustitute them into the equation stating that the hemial potentials are equal. So, in order to solve this prolem as a single nonlinear algerai equation with a single variale, p vap, we have an equation, ( ) µ 0 µ p vap liq vap where we solve or the molar volumes eah iteration and then sustitute their values into the untion aove. Find the vapor pressure o Argon at T77. The van der Waals onstants or argon are a0.8 m 6 /mol and.80-5 m /mol. The gas onstant is R8. J/mol/. Also report the liquid and vapor molar volumes. Prolem. Find the solution to the ollowing system o nonlinear algerai equations near (,,). sin

3 D. eer, MSE 506, Dept. o Materials Siene & Engineering, University o Tennessee, noville Prolem. This prolem requires steady state mass alanes on a single-stage liquid-liquid etrator. Consider an etrator: Etrat {,, etrator Solvent S, { S,, S,, S, Feed F, { F,, F,, F, Rainate {,, In this system, a solvent stream, S, removes an impurity, enzene, rom a eed stream, F. Eiting the etrator are two streams, the rainate stream, whih is the leaned-up eed stream, and the etrat stream, whih now ontains the impurity. This unit uses a reyled urural stream as the solvent to etrat enzene rom a yloheane produt stream. F, S and R are eed rates with units o moles per hour. The variales are mole rations. The mole rations have two indies. The irst upperase letter indiates the stream and the seond lowerase letter indiates the omponent. The data you are given is as ollows. F 00 mol / hr F, F, F, S 50 mol / hr S, S, S, R? mol / hr??? E? mol / hr??? The equilirium onstants are: 5. 0, 0. 0 and You have eight unknowns, the lowrates o the rainate and etrat and the ompositions o the rainate stream and the omposition o the etrat stream. (a) Write a set o eight independent algerai equations desriing this system. () Is the set o equations linear or non-linear? () Solve or the 8 unknowns. Clearly identiy eah variale. For those unamiliar with liquid-liquid etrators, the relative steady state mass alane has the orm traditional mole alane: aumulation in out generation onsumption In this system, sine it is at steady state, there is no aumulation term. There is no hemial reation so the generation and onsumption terms are zero. Eah omponent has two inputs and two outputs. Furthermore, the sum o the mole rations in eah stream must e zero.

4 D. eer, MSE 506, Dept. o Materials Siene & Engineering, University o Tennessee, noville Prolem 5. Consider a system with two anions, A and B, and two ations, C and H. The ollowing reations are possile AH A H reation. AC A C reation. BH B H reation. BC B C reation. There are a total o eight speies: AH, AC, BH, BC, A -, B -, H and C. Potentially the omposition o eah o these eight speies is unknown. In general, i we have eight unknowns, then we need eight equations to solve or a unique solution. Four o the equations ome rom thermodynami equilirium oeiients. Equilirium oeiients govern the distriution o eah o these reations. Presumaly the equilirium oeiients are given. [ A ][ H ] [ A ][ H ] AH [ AH ] [ A ][ C ] [ A ][ C ] AC [ AC] [ B ][ H ] [ B ][ H ] BH [ BH ] [ B ][ C ] [ B ][ C ] BC [ BC] Four more reations ome rom molar alanes on A, B, C and H. [ ] [ AH ] [ AC] [ A ] [ ] [ BH ] [ BC] [ B ] [ ] [ AC] [ BC] [ C ] [ ] [ AH ] [ BH ] [ H ] equation. equation. equation. equation. A equation 5. B equation 6. C equation 7. H equation 8. Presumaly the total amounts o eah omponent added to the solution are given. Oten the ph o the solution is given, whih eetively provides the value o [ H ], in whih ase we have one less unknown and equation 8 is redundant. You ould hoose to solve this as 8 (or 7) equations with 8 (or 7) unknowns. Sine the relationships in the equilirium oeiients are so simple, one an alternatively eliminate some o the ions (AH, AC, BH and BC) rom the molar alanes. This leaves just our equations (the molar alanes) (three i the ph is given) and our unknowns (A -, B -, C and H.), (three i the ph is given).

5 D. eer, MSE 506, Dept. o Materials Siene & Engineering, University o Tennessee, noville Assume the ph is given. Solve or the onentration o the other seven ions as a untion o ph rom to 7 or the ollowing parameter values. Both the equilirium oeiients and the onentrations have units o mol. 0, 0, [ ] C. 0 and 5, [ A ].0 0, [ B ].0 0 0, and 5