Chem131_Chapter9

7/28/2019 Chem131_Chapter9

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CHAPTER 9Solutions


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SOLUTIONS

Solution = Homogeneous mixture

one phase(s, l, g)

Molecular interactions play a central role

more than 1 component

Solution composition

mole fraction:

molar concentration:

mass concentration:

molality:

weight percent:

xi =

ni

ntot

ci =ni

V

i =mi

V

mi =i

nsolvMsolv

wi

wtot 100%


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SOLUTIONS

Partial molar quantities

separated components mixture

1, 2, . . . , r

Vm,1

, Vm,2

, . . . , Vm,r

V

= n1V

m,1+ n

2V

m,2+ + n

rV

m,r


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SOLUTIONS



1, 2, . . . , r

Vm,1

, Vm,2

, . . . , Vm,r

V

=

r

i=1

niV

m,r

V = V

because ofinteraction differences,

rearrangements,...


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SOLUTIONS



1, 2, . . . , r

Vm,1

, Vm,2

, . . . , Vm,r

V

=

r

i=1

niV

m,r

V = V

U = U H = H

S= S

G = G A = A


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SOLUTIONS


mixture

V= V(T,P,n1, n2, . . . , nr)

U= U(T,P,n1, n2, . . . , nr)

similarly for H, S, G, A, etc.

dV=

V

T

P,ni

dT+

V

P

T,ni

dP+

V

ni

P,T,nj=i

dni + +

V

nr

P,T,nj=r

dnr


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SOLUTIONS


mixture

X= V, U, H, etc.

Xi = X

m,iFor a pure substance:

In a mixture: Xi = X

m,i(except ideal gases)

Xi =

X

ni

P,T,nj=i


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SOLUTIONS


mixture

X= V, U, H, etc.

X Xi ?

Xi =

X

ni

P,T,nj=i

X= nf(T,P,x1, x2, . . . , xr)

n =

r

i=1

ni


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SOLUTIONS


V= nf(T,P,x1, x2, . . . , xr)

T, P and xi constant: dV= f(T,P,x1, x2, . . . , xr)dn

but

dV=

V

T

P,ni

dT+

V

P

T,ni

dP+

ri=1

Vidni

dV=

r

i=1

xiVidn f(T,P,x1, x2, . . . , xr)dn =r

i=1

xiVidn

dni = xidn+ ndxi


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SOLUTIONS


f(T,P,x1, x2, . . . , xr)dn =

r

i=1

xiVidn

f(T,P,x1, x2, . . . , xr) =r

i=1

xiVi

V=

r

i=1

nxiVi =

r

i=1

niVi


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SOLUTIONS



mixV= V V

mixV=

i

niVi

i

niV

m,i =

i

ni(Vi V

m,i)


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SOLUTIONS



mixX= XX =

i

ni(Xi X

m,i)

Due to molecular interactions and entropy changes


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SOLUTIONS


Gi =

G

ni

T,P,nj=i

= i

T and P constant: G =

i

nii

fundamental quantitythat leads to all otherthermodynamicproperties of a solution

G = H TS

Gni

T,P,nj=i

= i = Gi = Hi TSi


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SOLUTIONS


GiT

P,nj=i

=i

TP,nj=i

=

Si

i = Gi = Hi TSi

Gi

P

T,nj=i

=

i

P

T,nj=i

= Vi


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SOLUTIONS

Mixing quantities

mixG

P

T,nj

= Vmix

mixG

T

P,nj

= Smix

mixG = mixH TSmix

the changes are due todifferences in the interactions


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SOLUTIONS

How to determine partial molar quantities

Partial molar volume

Slope method

From mixV

Partial molar enthalpy, entropy and Gibbs energy

Hmix =i

ni(HiH

m,i) =nA(HAH

m,A) +nB(

HBH

m,B)

It is only possible to determine the enthalpy of a solution relative to the enthalpyof the pure components

Similar equations for S and G


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SOLUTIONS


Integral heat of solution per mole of B

Heat adsorbed by the system when 1 mole of pure B is added at constant T and Pto pure A to produce a solution of the desired mole fraction xB

Integral heat of solution ad infinite dilution

Hint,B =mixH

nB

H

int,B = limxA1

Hint,B


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SOLUTIONS


Differential heat of solution per mole of B

Enthalpy change per mole of added B when B is added at constant T and P to asolution of fixed composition xB

Differential heat of solution ad infinite dilution

Hdiff,B = HB H

m,B

H

diff,B = limxA1

Hdiff,B

H

diff,B = H

int,B


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SOLUTIONS

Ideal solutions

Solution where the molecules of the various species are so similar to one anotherthat replacing molecules of one species with molecules of another species will notchange the spatial structure of the intermolecular interaction energy in the solution

mixV = mixU = mixH = 0

like ideal gases, but broader definition

constant T & P


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SOLUTIONS

Ideal solutions



mixS=?

constant T & P


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SOLUTIONS

Ideal solutions



mixS= 0

constant T & P


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SOLUTIONS

Ideal solutions


mixV = mixU = mixH = 0constant T & P

mixG = RT

i

ni lnxi

mixS= 0

from experiments

mixS= Ri

ni lnxi


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SOLUTIONS

Ideal solutions

Lets find the equivalent thermodynamic definition (constant T & P)

mixG = RT

i

ni lnxi mixG =

i

ni(Gi G

m,i)

RT

i

ni lnxi =

i

ni(i

i)

i =

i(T, P) + RT lnxi

SO O S


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SOLUTIONS

Ideal solutions

Thermodynamic properties

standard state: pure liquid or solid at temperature T and pressure P of the solution

i=

i(T, P)

i =

i(T, P) + RT lnxi

SOLUTIONS


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SOLUTIONS

Ideal solutions


mixing quantities: mixV = mixU = mixH = 0

mixG = RTi

ni lnxi

mixS= R

i

ni lnxi

SOLUTIONS


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SOLUTIONS

Ideal solutions


vapor pressure: decrease P above a solution until it begins to vaporize

liquid

x1, x2, ..., xN

vapor

y1, y2, ..., yN li= v

i

li (P, T) + RT lnxi =

i+ RT ln

Pi

P

pure liquid

pure vapor

Pi*

li (P

i , T) =

i+ RT ln

Pi

P

li (P

i , T) = v

i(Pi , T)

P = P1 + P2 + ...+ PN

SOLUTIONS


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SOLUTIONS

Ideal solutions



l

i (P, T) + RT lnxi =

i + RT ln

Pi

P

li

(Pi, T) =

i+ RT ln

Pi

P

li (T, P) l

i(T, Pi ) + RT lnxi = RT ln

Pi

P

i

SOLUTIONS


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SOLUTIONS

Ideal solutions



li (T, P) l

i(T, Pi )for condensed phase systems:

Pi = xiP

i

l

i (T, P)

l

i (T, P

i ) + RT lnxi = RT ln

Pi

Pi

Raoults law

SOLUTIONS


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SOLUTIONS

Ideal solutions

liquid

xA & xB

vapor

P = PA + PB

PB

PA

P

0 1

PA*

PB*

xBliq

P* are related to the volatility of a substance

if PA* > PB*, then A is more volatile than B

SOLUTIONS


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SOLUTIONS

Ideally dilute solutions

Solute molecules (B) dont see each other

xA 1

B

B

A and B interact (different from ideal solutions!)

A - A

A - B

B - B

Valid assumption only for nonelectrolytes... why?

SOLUTIONS


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SOLUTIONS


B

B PA

PB

0 1

PA*

xAliq

A =

A(T, P) + RT lnxAFor the solvent:

For the solute: B = fB(T, P) + RT lnxB

?

SOLUTIONS


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SOLUTIONS


A =

A(T, P) + RT lnxAFor the solvent:

For the solute: B = fB(T, P) + RT lnxB

A =

A(T, P)

A(T, P

A)

Standard state

A =

A +RT lnxA

B =

B+RT lnxB Holds only near xB 0

Bdefines a hypothetical state at T and P in which B keeps the same properties

as in the very dilute solution although xB

1. It depends on the solvent!

SOLUTIONS


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SOLUTIONS


Vapor pressure

liquid

xA, x1, ..., xN

vapor

yA, y1, ..., yN

xA 1

xi 0

T and P constant

liqi =

vapi

,liqi +RT lnxi =

,vapi +RT ln

Pi

P

,liqi

,vapi

RT = ln

Pi

xiP

SOLUTIONS


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SOLUTIONS


Vapor pressure

liquid

xA, x1, ..., xN

vapor

yA, y1, ..., yN

,liqi

,vapi

RT= ln

Pi

xiP

Pi

xiP= e

,liqi

,vapi

RT

Ki(T, P) = Pe

,liqi

,vapi

RT

Henrys constanttabulated at 1 bardepends on the solvent

Pi = xiKi(T, P)

SOLUTIONS


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SOLUTIONS


Vapor pressure

Pi = xiKi(T, P)

PA = xAP

ASolvent

Raoults law

SolutesHenrys lawPB

PA

P

0 1xB

liq

KB

Gas dissolved in liquids are usually a good

approximation of ideally dilute solutions

SOLUTIONS


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SOLUTIONS



i =

i+RT lnxi

i

T

P,nj=i

=Si

i

PT,nj=i

= Vi

Ideal solutions

i=

i(T, P

i)

i(T)

Si = S

m,i R lnxi

Vi = V

m,i

Hi = H

m,i

mixV = mixH = 0

mixS= R

i

ni lnxi

SOLUTIONS


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SOLUTIONS


Ideal solutions

i=

i(T, P

i)

i(T)

Si = S

m,i R lnxi

Vi = V

m,i

Hi = H

m,i

mixV = mixH = 0

mixS= R

i

ni lnxi


still true for the solvent

SOLUTIONS


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SOLUTIONS


Ideal solutions

i=

i(T, P

i)

i(T)

Si= S

m,iR lnx

i

Vi = V

m,i

Hi = H

m,i

mixV = mixH = 0

mixS= R

i

ni lnxi


solvent solute

i= hypothetical state

Vi = V

i= V

i

Hi = H

i= H

i

o different fromideal solutions

Si = S

R lnxi

mixV = 0

mixH = 0

mix

S=different from

ideal solutions

SOLUTIONS


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SOLUTIONS


Reaction equilibrium

Kx =

i

(xi,eq)i

G

= RT lnKx

must be interpreted correctly

in terms of the appropriatechemical potentials

i

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Chem131_Chapter9