Definitions. Pure Component Phase Diagram. Definitions (cont.) Class 16 Non-Ideal Gases

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1 Sore 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Average = 85% Exam Rank Class 16 Non-Ideal Gases Definitions Critial emperature, ressure Vapor Gas Van der Waals EOS Other Equations of State Compressibility Fator riniple of Corresponding States Kay s Rule Water hase Change (set, inrease until ondensation ours) his point defines the ritial temperature ( ) and pressure ( ) of H 2 O! No ondensation ours as pressure inreases H 2 O above and is alled a superritial fluid, not a gas or a liquid. Definitions Critial emperature ( ): Highest temperature at whih a speies an oexist in two phases (liquid and vapor) Critial ressure ( ): ressure that orresponds to ritial temperature Vapor: Gas phase Usually used in the 2-phase region where liquid is present Gas: Gas phase Usually used when no liquid is present ure Component hase Diagram wo-phase region Definitions (ont.) Redued emperature ( r ): r = / Redued ressure ( r ): r = / Superritial fluid: > and > ressure-temperature phase diagram for CO 2 1

2 hase Diagram for H 2 O (hought Quiz) hase Diagram for H 2 O (Quiz Answers) 1. Label the following: a. 2-phase envelope b. Critial point. 2. Draw 3 isotherms a. hrough 2-phase b. hrough. Superritial 3. Label Superritial Fluid 1. lease draw 2. Label the following: a. 2-phase envelope b. Critial point. 3. Draw 3 isotherms a. hrough 2-phase b. hrough. Superritial 4. Label Non-Ideal Gases High Density Moleular interations Volume of eule beomes important Methods Equations of State More terms than ideal gas law Still relate,, and Generalized ompressibility hart z From p. 192, ideal when: ideal 5L / ( diatomi gases) 20L / ( other gases) ideal Ideal vs. Real Gases Ideal vs. Real Gases J. D. van der Waals Idea 1. Real pressure gas = Ideal gas pressure minus the ontrating fores per unit area due to the intereular attrations = ideal a/v 2, or ideal = + a/v 2 2. Atual ar volume = volume oupied by the ideal gas (with the eules oupying no spae) + volume of the eules themselves V = V ideal + b, or V ideal = V b From p. 221, less than 1% error with Ideal Gas Assumption when: V n 5 L for diatomi gases (H 2, N 2, O 2, et.) 20 L for other gases lug ideal and V ideal into the ideal gas equation: ideal V ideal = = ( + a/v 2 )(V b) Now solve for : Cubi form: a ˆ 2 V b ˆ 3 V ( b ) 2 a ab 0 his is one of the virial equations of state (meaning that it relates to eular interations) 2

3 Non-Ideal Eqns. of State (that are in the text) Virial 1 onstant = f(,,) able for Van der Waals 2 onstants a & b = f(, ) 1 a ˆ 2 V b Soave-Redlih-Kwong (SRK) a 3 onstants b a & b = f(, V ˆ b ) = f(,, ) (5.3.2) (5.3.6) (5.3.7) R a 64 b r r a = ( ) 2 / b = ( )/ m = = [1 + m(1 r 0.5 )] 2 1 See Exel Example See Mathad Example (A opy of this Mathad file is on the ChEn 273 web page under Class Notes) Van der Waals Results: CO V vdw ( ) Latm 33 V ideal ( ) 32 Latm 31 =400K V Van der Waals below (3 roots) ˆ 3 2 ( b ) a ab 0 Cubi EOS Guess Ideal Gas to start 0.01 atm 10 Does this inrease in non-ideal behavior with inreasing pressure make sense? ottom line: EOS an be used to try to alulate liquid densities How aurate are these nonideal equations of state? Need to ompare against reliable data Example in book for propane at 423 K and 70 atm Ideal gas Volume error = 92% SRK Volume error = 12%! Law of Corresponding States hysial properties of a gas depend on the proximity to its ritial state asis for using r and r 3

4 Compressibility Fator Ideal vs. Real Gases V z n z V zn Simple equation, similar to ideal gas law z aounts for real gas behavior z = 1 for ideal gas z 1 for nonideal gas How do you get values for z? Look up for partiular ompounds (e.g. erry s Handbook) Law of Corresponding States z! From p. 221, less than 1% error with Ideal Gas Assumption when: V n 5 L for diatomi gases (H 2, N 2, O 2, et.) 20 L for other gases Law of Corresponding States hysial properties of a gas depend on the proximity to its ritial state asis for using r and r to get z Objetive: Know two of,, and V, and want to find the third for a real gas roedure Calulate the redued,, and/or V ideal V ˆ r, r, and / or r Use the redued values to find z from Figs to Use z to find the missing value from eq Adjustments for H 2 or He: 8 K, adjusted 8atm, adjusted Note: the author uses V and V interhangeably Note: his hart was taken from the blue & red version of the book. It has too many divisions between 0.9 and 1.0!! he version in the blak & red book is orret. Find lines of onstant r r z V r ideal Use adjusted values on ompressibility harts his one is orret! If r = 4 and r = 1.4, find z. 4

5 Only good for this value of z, (i.e., most simple fluids) From Mathad example, r =.686, r = z z = 0.91 (Van der Waals = 0.89) Nonideal Gas Mixtures Mixtures require mixing rules hese an get quite ompliated Mixing rules for Law of Corresponding States (z) Use Kay s rule to alulate a seudo-ritial emperature and ressure for the mixture Determine the seudo-redued emperature and ressure Find z m for the mixture from the ompressibility hart as before his mixing rule not good for other EOS approahes You will get mixing rules for EOS in hermo or on web Corresponding States: Mixtures (Kay s Rule) Note that these are weighted by e fration r r y A A A y A / / y y y C y C C C Hint: We will be using this a lot for mixtures! Common Student Mistake on Kay s Rule Some students ompute a z for eah speies and then average the z s Get z i from r,i and r,i hen let z tot = y i z i Corresponding States: Mixtures (Kay s Rule Example) Kays rule example = 21 atm = 650 K yi (K) (atm) MW n Deane n Otane Isopentane seudoritial ( and ) avg MW= Redued seudoritial z= 0.84 V=zn Density = n*mw/v = *MW/(z) R= L atm/(k g) Density= 50.3 g/l Ideal = 42.2 g/l Hint: We will be using this a lot for mixtures! 5

6 Corresponding States: Mixtures (Kay s Rule Example) Mixing Rules for SRK EOS Kays rule example = 21 atm = 650 K Wrong Way!!! yi (K) (atm) r r zi MW n Deane n Otane Isopentane seudoritial ( and ) avg MW= Redued seudoritial z= 0.84 V=zn Density = n*mw/v = *MW/(z) R= L atm/(k g) Density= 50.3 g/l 54.5 Ideal = 42.2 g/l Next year in thermodynamis. You an find the mixing rules on the web as well Hint: We will be using this a lot for mixtures! Summary Non-ideal equations of state are iterative if you are alulating Use ideal gas as first guess Use and to get parameters Alternate: Calulate at a lot of different s and interpolate Compressibility fators are alternate approah Use and to get r and r Use z in modified ideal gas equation Mixing rules required for mixtures Kay s rule for ompressibility fator alulations Use, mix and, mix to get r,mix and r, mix, then get z for mixture Other mixing rules available (not in our book) for equations of state Homework (all workbook problems) 1. SRK Eq. of State 2. Compressibility Chart 3. Kay s Rule 6