University of Washington Department of Chemistry Chemistry 452/456 Summer Quarter 2014

Transcription

1 Lecture 16 8/4/14 Unversty o Washngton Department o Chemstry Chemstry 452/456 Summer Quarter 214. Real Vapors and Fugacty Henry s Law accounts or the propertes o extremely dlute soluton. s shown n Fgure 1 Henry s Law only accounts or the propertes o solutons n the encrcled areas. For solutons that are not dlute, the physcal propertes o these solutons startng wth ther chemcal potentals, s not accurately descrbed by ether Raoult s Law or Henry s Law. To accurately descrbe the ree energy o a concentrated soluton o a non-deal vapor n contact wth a non-deal, concentrated soluton o a non-electrolyte, the concept o actvty s ntroduced. Fgure 1: The vapor pressure above mxtures o o CS2 and acetone devate strongly rom both Raoult s Law (dashed lnes) and Henry s Law (encrcled areas) or mole ractons between.2 and.8. In the two smple soluton models we have convered we assumed the vapor behaves deally so that the chemcal potental o each component n the vapor phase s obtaned by a smple ntegraton... dµ ( v) = Vd = d µ ( v) µ = ln (16.1) I V V or V = = + α n n (16.2)

2 where α s an emprcal correcton term ntended to account or all eects (.e. ntermolecular nteractons, nte molecular sze etc. ) that cause devatons rom deal gas behavor.. The expresson or the chemcal potental o the vapor s now d dµ ( v) = Vd = d + α d µ ( v) µ = αd + µ ( v) µ = ln αd + I we know the equaton o state a can be determned explctly. Otherwse a can be measured drectl as a uncton o and the ntegral can be evaluated numercally. See examples below. vap vap, In analog wth the deal vapor expresson µ µ = ln, or a real vapor we dene the ugacty. The ugacty s the vapor pressure...or escape tendency... corrected or the act that the gas s not deal. (16.3) dµ = dln = d+ αd= dln + αd (16.4) α d ln = d ln + αd d ln = d ln + d The reerence state can be dened as the lmt o very low pressure where the vapor behaves deally and so =. We obtan 1 dln dln = dln + αd 1 ln ln d α = + (16.5) We can rearrange equaton 16.5 and ater a lttle algebra we get 1 exp d α = γ (16.6) where γ s the ugacty coecent. The lower lmt o the ntegraton approaches zero n many practcal applcatons. The ugacty coecent can be determned by measurng the quantty V = α over a range o pressures and determnng the ntegral αd numercally. Note that γ measures the degree to whch the component o the vapor devates rom the deal gas law at and. Thereore

3 Lm = 1or Lm( γ ) = 1 (16.7). Examples o Fugacty Calculatons Example 1: I the equaton o state s known the ugacty can be determned by dervng the quantty α rom the equaton o state. Suppose the equaton o state s ( V nb) = n (16.8) where b s a constant that relects the volume excluded by nte molecules n the gas phase. Solve or V/n: V = V = + b. Then α=b and the ugacty coecent s easy to n calculate. ssumng or convenence that << 1 b b ( )/ b / γ = exp αd exp d e e = = (16.9) ll you need to calculate the ugacty s the pressure and the value or b. Note n some texts the ugacty s dened a lttle derently. Suppose we dene the ugacty as α V = + α or V = + α dµ = ln = dln + d 1 α (16.1) dln = dln + d 1 α = exp d Ths alternatve denton makes no derence to the result. gan or ( V nb) = n we now get wth the alternatve denton α = b. ut we get the same nal result as we dd usng the other denton: 1 α 1 b = exp d exp d = (16.11) b( ) b / = exp e

4 Example 2: The ugacty can also be calculated rom expermental data usng the equaton α = V o r α = V. elow s a study o the ugacty o ntrogen gas as a uncton o pressure. Noteworthy eatures: V 1 1 α The quantty = s tabulated n the second column and plotted as a uncton o pressure. Note the area under the curve s = 8atm 1 α ln γ = d The thrd column s the ugacty coecent explctly and the degree to whch F/ devates rom 1 measures the devaton rom deal gas behavor. The ourth column tests the degree to whch the data ts a model ( V nb) = n where b<<. The closer the value n column 4 s to 1 the closer s the agreement to the model. C. Real Soluton Equaton o State To account or non-dealty o the vapor above a real soluton, we substtute the ugacty or the pressure. We also ntroduce the term actvty, assumed to be the rato o the ugacty at a gven pressure to the ugacty at a reerence pressure; = a (16.12)

5 I the soluton were deal the actvty would just be the mole racton o component I n the soluton. ut n analogy to the ugacty coecent, devatons rom dealty are relected n a actvty coecent: = a = γ x (16.13) Equaton s the equaton o state or real solutons. Lke the ugacty coecent, the actvty coecent can be measured drectly or calculated a model or the real soluton s avalable. We wll consder a real soluton model n the next lecture, called the regular soluton model. In the next secton we show how actvty and ugacty are ntroduced nto equlbrum constant expressons. D. Summary o Equlbrum Relatonshps or Chemcal Reactons wth Real Vapors and Real Solutons (non-electrolyte) Consder a reacton ν + ν νcc+ νdd where a, b, c, and d are stochometrc coecents. The condton or equlbrum s: ν µ + ν µ = ν µ + ν µ C C D D ( ln a ) ( ln a ) ( ln a ) ( ln a ) ν µ + + ν µ + = ν µ + + ν µ + C C C D D D We now rearrange the equaton: G = ν µ + ν µ ν µ ν µ = ν ln a ν ln a ( ) ( ) ( ) ( ) ( ) ( ) C C D D ( ln a ) ν ( ln a ) + ν + C C D D a a νc ν D ln C ln ln ln D C D G a ν ν = + a ν ac a ν D = ln ν ν a a ν C ν D ac ad In general K = where a a ν a ν s the actvty o speces. ν ν ν ν C D Ideal Gases: a = K = C D D C D Real Gases: a = K = C D Non-electrolyte Solutons, Ideal, unts o concentraton: C D ν ν ν ν C ν D ν D ν ν C D K C C D C C C C C a = = C C C C C Non-electrolyte Solutons, Ideal, unts o molalty:

6 νc ν D ν ν C D K m C D m m m m m a = = m m m m m Non-electrolyte Solutons, Real, unts o molalty: ν C ν D ν ν m Cm C Dm D m m K m C D γ γ γ γ γ a = = m m m m m where all actvtes correspond to equlbrum condtons.