VIEWS: 10 PAGES: 30 POSTED ON: 1/24/2012
Two-phase behavior Behavior in the two phase region log Pvp A B T C Ln P Antoine’s Equation Enthalpy Antoine’s Equation • Antoines equation is used to predict the relationship between the T and P in the two phase region. P is usually called saturation pressure and T is usually called the vaporization temperature or vice-versa • For propane, the Antoines constants are • A = 6.80398, B =803.81, C= 246.99 (Psat in mmHg) • Calculate the vapor pressure for propane at 100, 200, 300 K and the critical point 369.8 K. Compare to the values from the chart. Antoine’s Equation Problem 6 The Antoine constants for n-butane are listed in a data base as A = 6.80896, B = 935.86, and C = 238.73 where Psat is in mm Hg: where T is in oC, and P is in mmHg. Predict the normal boiling point (temperature at which butane boils at 1 atm pressure.) Compare the predicted value to the value listed in the critical properties table given earlier. Gibbs Free energy The Gibbs free energy is especially useful for calculations in the two phase region. Why? ~ ~ ~ dG S dT VdP Since dT = 0 and dP = 0, Gliq = Gvap The Gibbs free energy of the vapor and liquid in a two phase system are equal. The criteria for equilibrium between two phases is that G1 = G2. Relationship of V and S Along each edge of the envelope: ~ ~ ~ ~ ~ ~ dGliq Sliq dT Vliq dP dGvap S vap dT Vvap dP Psat (liquid ) Psat ( vapor) Tvap( liquid ) Tvap( vapor) G( liquid ) G( vapor) ~ ~ ~ ~ 0 (Sliq S vap )dTvap (Vliq Vvap )dPsat ~ ~ ~ ~ 0 (Sliq S vap )dTvap (Vliq Vvap )dPsat ~ ~ ~ ~ (Sliq S vap )dTvap (Vliq Vvap )dPsat ~ ~ Psat ( S vap S liq ) ~ ~ Tvap (Vvap Vliq ) The way Psat changes with Tvap is directly related to DSvap and DVvap Finding the slope of Psat vs Tvap Psat Psat @T1 Tvap T1 Tvap Change in Psat with T B B log Psat A A Tvap C Tvap C Psat 10 A B Psat Tvap C B ln(10)10 Tvap C Tvap 2 ~ ~ Psat ( S vap S liq ) ~ ~ Tvap (Vvap Vliq ) Numerical (Easier) • Estimating the slope of Psat vs Tvap at a particular T. Psat Psat(T ) Psat(T ) Tvap 2 The smaller the value of , the more accurate the estimation. DSvap • Given the Antoine’s constants, one can calculate the way the saturation pressure changes with Tvap at any Tvap. • One can use an EOS to estimate Vvap and Vliq at that temperature Tvap.(eg 10- 24 PR) • From the difference in Vvap and Vliq one can obtain Svap – Sliq (DSvap) Shortcut Approximations • Since Vliq << Vvap, a shortcut is to find only Vvap ~ ~ Psat ( S vap S liq ) ~ Tvap (Vvap ) • Since Vvap ~ RT/P ~ ~ Tvap Psat ( S vap S liq ) Psat Tvap R Calculation of DSvap from dPsat/dTvap Problem 7 a. From Antoine’s equation and the P-R equation of state: Compute the difference in the vapor and liquid specific volumes at butane’s normal boiling point and at 25 C. b. Compute the difference in the entropies of liquid butane and butane vapor at each temperature using each of the 3 methods described in the notes. (Mathematically or numerically take the derivative of Antoine’s equation – pressure wrt temperature. Using this slope, calculate the ratio of the entropy to volume change at each temperature.) Calculation of DHvap • Since along the two phase region: ~ ~ Gliq Gvap ~ ~ ~ ~ H liq Tvap Sliq H vap Tvap S vap ~ ~ H vap H liq ~ ~ S vap S liq Tvap ~ ~ Psat ( H vap H liq ) ~ ~ Tvap Tvap (Vvap Vliq ) This can be used to find DHvap at a particular temperature. One calculates the change in Psat with Tvap at a particular Tvap, uses an EOS to get the volumes, then calculates DHvap. This is the best way to get DHvap. Clausius-Clapeyron Eq This is an approximate equation based on the fact that Vliq<< Vvap and Vvap is ~ RT/P ~ ~ ~ ~ ( H vap H liq ) Psat ( H vap H liq ) Psat ~ Tvap RTvap 2 Tvap Tvap (Vvap ) Psat ~ ~ ( H vap H liq ) ~ ~ ( H vap H liq ) 1 dPsat Psat 1 2 dTvap ln o o Psat RTvap Psat R Tvap T vap Clausius-Clapeyron ~ ~ Psat ( H vap H liq ) 1 1 ln o o Psat R Tvap T vap Given Psat at two different Tvaps, DHvap can be computed. c. Compute the enthalpy difference between the saturated vapor and saturated liquid butane at its normal boiling point and at 25 C. (How does DHvap relate to DSvap?) d. Using only these two temperatures and the vapor pressures corresponding to them, use the Clausius- Clapeyron equation to estimate the enthalpy of vaporization for butane. Summary • In the two phase region, there is an explicit relationship between P and T. As T increases so does the vapor pressure, according to Antoine’s equation. • The relationship between Pvap and Tsat can be used to compute the difference in the vapor and liquid volumes, entropy, and enthalpy. Gibbs Phase Rule The Gibbs phase rule is an equation that tells you now many properties you are free to designate in a chemical equilibrium situation. Df N 2 How many Number of “intensive phases in Number of contact properties” different you are free equilibrium chemicals or to designate “components” Single Phase Df N 2 A one component system, with a single phase (vapor, liquid, or supercritical) Df = 1 – 1 + 2 = 2 (specify 2 intensive variables and all others are fixed. Giving 1 does not fix the system and you cannot give 3 (the first 2 fix the third automatically) 2-phase Region Df N 2 In the two phase region, one is only free to designate 1 property: Df = 1 – 2 + 2 = 1. Designating the T, fixes the pressure, liquid and vapor specific volume, enthalpy, entropy, etc. Two phase region In the two phase region, there is only 1 temperature for each pressure. 2 Components – Two phases You can (must) designate 2 properties (T and P) this will fix the compositions, enthalpies, internal energies etc of both phases. Phase Rule in Multicomponent Systems Example MEEK and Water MEEK rich phase Df N 2 Water rich phase Df 2 2 2 2 Intensive variables – T, P, xm, ym, Vu,Vl,Su,Sl … 3 phases Question: How many things are you free to specify for 3 phases in equilibrium (V, L, G) if there is only one component? Df = 1 – 3 + 2 = 0 There is one and only one temperature, pressure, volume(s) entropies etc where solid, liquid, and gaseous water are in equilibrium. This is called the “triple point” (0.01C) You have nothing you can pick. Triple Point “Stability Analysis” In 304, you will be presented with calculational procedures that will tell you whether or not two phases will form at the conditions of a solution. Summary • There is an explicit “rule” that relates the number of “degrees of freedom” of multiphase and multicomponent systems at equilibrium. • This rule tells one how many of the “intensive” variables in a system can be independently specified, which keeps one’s thoughts organized about systems