Properties of Solutions

Concentration Terms

Dilute - not a lot of solute. Concentrated - a large amount of solute. Concentration can be expressed

quantitatively is many ways:• Molarity

• Molality

• Percentage

• Mole fraction

Molarity and Molality

The molarity is the number of moles of solute in 1 litre of solution.• M = moles of solute / V sol’n (litres)

The molality is the number of moles of solute in 1 kg of solvent.• M = moles of solute / kg solvent

Conversion between the two requires the solutions density.

Partial Molar Thermodynamic Properties

Define a partial molar thermodynamic property as

Euler’s Theorem

'n,P,TJJ n

YY

J

JJYnY

The Chemical Potential

We define the chemical potential of a substance as

'n,P,TJJ n

G

The Wider Significance of

Shows how all the extensive thermodynamic properties depend on system composition

'n,V,SJJ n

U

'n,P,SJJ n

H

'n,V,TJJ n

A

Thermodynamics of Mixing

Spontaneous mixing of two or more substances to form solutions

Gibbs energy of the solution must be less than G(pure components)

The Gibbs Energy of Mixing

J

JJmix XlnXnRTG

The Enthalpy and Entropy

lnmixJ J

JP

mix

GnR X X

T

S

2

mixmix

P

GHT

T T

The Ideal Solution

TmixS/n

TmixG/n

TmixH/n0

kJ/mol

XA

The Volume and Internal Energy of Mixing

VPG

mixT

mix

VPHU mixmixmix

Ideal Solution Def’n

For an ideal solution

0H;0V mixmix

0

VPHU mixmixmix

Raoult’s Law

Consider the following system

Raoult’s Law #2

The chemical potential expressions

AO

AA

A*AA

plnRTvapvap

XlnRTliqliq

Raoult’s Law: Depression of Vapour pressure

VP of solution relates to VP of pure solvent

PA = XAP*A

Solutions that obey Raoult’s law are called ideal solutions.

Raoult’s Law Example

The total vapour pressure and partial vapour pressures of an ideal binary mixture

Dependence of the vp on mole fractions of the components.

An Ideal Solution

Benzene and toluene behave almost ideally

Follow Raoult’s Law over the entire composition range.

Henry’s Law

Henry’s law relates the vapour pressure of the solute above an ideally dilute solution to composition.

The Ideal Dilute Solution

Ideal Dilute Solution• Solvent obeys

Raoult’s Law

• Solute obeys Henry’s Law

Henry’s Law #2

The chemical potential expressions

JO(H)

is the Henry’s law standard state.

It is the chemical potential of J in the vapour when PJ = kJ.

( )' ln

ln

O HJ J J

OJ J J

sol n liq RT X

vap vap RT p

Henry’s Law #3

The Standard State Chemical potential for Henry’s Law

When the system is in equilibrium

The chemical potential expressions reduce to Henry’s Law

J

oJ

oH,J

klnRT

vap

vapn'sol JJ

JJJ XkP

Henry’s Law in terms of molalities

The Standard State Chemical potential for Henry’s Law

When the system is in equilibrium

The chemical potential expressions reduce to Henry’s Law in terms of molalities

oJoHJ

omJ mMRT ln,,

vapn'sol JJ

J

mJJ mkP

Chemical Potentials in terms of the Molality

The chemical potential expressions

o

JmJJ m

mRTnsol ln' ,

oJ,m = chemical potential of the solute in an

ideal 1 molal solution

The Gibbs-Duhem Equation

The Gibbs-Duhem gives us an interrelationship amongst all partial molar quantities in a mixture

J

JJdYn0

Colligative Properties

Colligative Properties

All colligative properties• Depend on the number and not the nature

of the solute molecules Due to reduction in chemical potential

in solution vs. that of the pure solvent• Freezing point depression

• Boiling Point Elevation

• Osmotic Pressure

Boiling Point Elevation

Examine the chemical potential expressions involved

vapliq JJ

J*JJ XlnRTliqvap

G

XRTliqvap

vap

JJJ

ln*

Boiling Point Elevation #2

The boiling point elevation

B

vap

bb X

JHJRT

T2

*

BbJvap

Jbb mKm

JHMRT

T2

*

Freezing Point Depression

Examine the chemical potential expressions involved

sliq *JJ

J*JJ XlnRTliqliq

G

XlnRTliqs

fus

J*J

*J

Freezing Point Depression #2

Define the freezing point depression

B

fus

ff X

JHJRT

T2

*

BfJ

fus

ff mKm

JHJMRT

T2

*

Osmosis

Osmosis

The movement of water through a semi-permeable membrane from dilute side to concentrated side• the movement is such that the two sides

might end up with the same concentration

Osmotic pressure: the pressure required to prevent this movement

Osmosis – The Thermodynamic Formulation

Equilibrium is established across membrane under isothermal conditions

PP *JJ

J*JJJ XlnRTPX,P

- the osmotic pressure

The Final Equation

RTMRTVn

BB

The osmotic pressure is related to the solutions molarity as follows

Terminology

Isotonic: having the same osmotic pressure

Hypertonic: having a higher osmotic pressure

Hypotonic: having a lower osmotic pressure

Terminology #2

Hemolysis: the process that ruptures a cell placed in a solution that is hypotonic to the cell’s fluid

Crenation: the opposite effect

The Partial Molar Volume

In a multicomponent system

J

JJVnV

'n,P,TJJ n

VV

Volume Vs. Composition

The partial molar volume of a substance • slope of the variation of the

total sample volume plotted against composition.

PMV’s vary with solution composition

The PMV-Composition Plot

The partial molar volumes of water and ethanol at 25C.

Note the position of the maxima and minima!!

Experimental Determination of PMV’s

Obtain the densities of systems as a function of composition

Inverse of density – specific volume of solution

mLg1

gmLVs

2CmBmAmolmLV

Example with Methanol.

Plot volumes vs. mole fraction of component A or B

Draw a tangent line to the plot of volume vs. mole fraction.

Where the tangent line intersects the axis – partial molar volume of the components at that composition

The Solution Volume vs. Composition

The Mean Molar Volume

Define the mean mixing molar volume as

• V*J – the molar

volume of the pure liquid

• Vm = V/nT

J

*JJmmix VxVV

The Mean Molar Volume Plot

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.00 0.50 1.00

XMeOH

m

ixV

m /

(m

L/m

ol)

VA-VA* VB-VB

*

Infinite Dilution Partial Molar Properties

The value of a partial molar thermodynamic property in the limit of zero volume is its infinite dilution value• E.g., for the volumes

J0xJ VlimVJ

The Definition of the Activity

For any real system, the chemical potential for the solute (or solvent) is given by

Jo

J aRT ln

Activities of Pure Solids/Liquids

The chemical potential is essentially invariant with pressure for condensed phases

ooJ

p

P

Joo

JJ

P

dpVPPo

Pure Solids and Pure Liquids

For a pure solid or a pure liquid at standard to moderately high pressures

JaRT0 ln

or aJ = 1

Activities in Gaseous Systems

The chemical potential of a real gas is written in terms of its fugacity

Jo

J fRT ln

Define the Activity Coefficient

The activity coefficient (J) relates the activity to the concentration terms of interest.

In gaseous systems, we relate the fugacity (or activity) to the ideal pressure of the gas via

JJJ fP

Activities in Solutions

Two conventions Convention I

• Raoult’s Law is applied to both solute and solvent

Convention II• Raoult’s Law is applied to the solvent;

Henry’s Law is applied to the solute

Convention I

We substitute the activity of the solute and solvent into our expressions for Raoult’s Law

*J

IJJ PaP

IJJ

IJ ax

Convention I (cont’d)

Vapour pressure above real solutions is related to its liquid phase mole fraction and the activity coefficient

*JJ

IJJ PxP

Note – as XJ 1J

I 1 and PJ PJid

Convention II

The solvent is treated in the same manner as for Convention I

For the solute, substitute the solute activity into our Henry’s Law expression

JIIJJ kaP

IIJJ

IIJ ax

Convention II (cont’d)

Vapour pressure above real dilute solutions is related to its liquid phase mole fraction and activity coefficient

JJIIJJ kxP

Note – as XJ 0J

II 1 and PJ PJid

Convention II - Molalities

For the solute, we use the molality as our concentration scale

mJJ

mJ am

mJ

moJJ aRT ln

Note – as mJ 0J

(m) 1 and aJ(m) mJ