Problems from the last lecture• Problem 2.5 Use the Kelvin equation to calculate the radius of an open-

ended tube within which capillary condensation of nitrogen at 77 K and a relative pressure of 0.75 might be expected. Assume that prior adsorption of nitrogen has formed a layer 0.9 nm thick coating the inside of the tube. For nitrogen at 77 K the surface tension is 8.85 mN m-1 and the molar volume is 34.7 cm3 mol-1. – Answer: 2.57nm

• Problem 2.6 A jet aircraft is flying through a region where the air is 10% supersaturated with water vapour (i.e. the relative humidity is 110%). After cooling, the solid smoke particles emitted by the jet engines adsorb water vapour and can then be considered as minute spherical droplets. What is the minimum radius of these droplets if condensation is to occur on them and a “vapour trail” form? Data: γ(H2O) = 75.2 mN m-1, M(H2O) = 0.018 kg mol-1, ρ(H2O) = 1030 kg m-3, T = 275 K.– Answer: 12nm

Lecture 2

Adsorption and the thermodynamics of surfacesAdsorption at gas liquid interface

Adsorption

• Adsorption - a tendency of one component to have a higher or lower concentration at the interface in comparison to adjacent phase.

Models of the interface

• A system containing an interface can be divided into 3 regions

for an extensive property B:

B B B Bα β σ= + +

1. Surface phase approach• interfacial region has finite thickness;• properties in the bulk phases are uniform

Models of the interface

• interface is infinitely thin;• properties of bulk phases assumed to extend till

the interface;• difference between the real property B and the

model value is called excess

1. Surface excess properties approach

excess real modelB B B Bσ= = −

, 0V V V Vα β σ= + =

extensive property:

intensive property:

/b B V=

Example: concentration

( ),modeli i i

i i i i

n c V c V

n n c V c V

α α β β

σ α α β β

= +

= − +excess

Adsorption

• the value of adsorption depends of the interface position!so, we need to agree on a convention….

( )/

i i i i

i i

n n c V c V

n A

σ α α β β

σ

= − +

Γ =

surface excess:

adsorption

Gibbs convention:• adsorption of a major component is zero:

/ 0A An AσΓ = =

Adsorption• relative adsorption

( ) ( )( )

i i i i i i i i

A A A A A

n n c V c V n c V c c V

n n c V c c V

σ α α β β α α β β

σ α α β β

= − + = − + −

= − + −

eliminating Vb:

A i ii i A

A A

c cc c

α β

α β

−Γ = Γ −Γ

−

0, ( )AA i i GibbsΓ = Γ = Γ

Gibbs convention:

• adsorption isotherm: dependence of the adsorption vs. bulk concentration at constant temperature

Thermodynamic properties of interfaces

• Other thermodynamics excess properties can be defined similarly

mech

i ii

w P dV P dV dA

w P dV P dV dA dn

α α β β

α α β β

δ γ

δ γ µ

= − − +

= − − + +∑

, , ,

,

,

i

i ii

i ii

i ii

T P P n

dU q w TdS P dV P dV dA dn

H U PV dH TdS P dV P dV dA dn

G H TS dG SdT P dV P dV dA dn

GA α β

α α β β

α α β β

α α β β

δ δ γ µ

γ µ

γ µ

γ

= + = − − + +

= + = + + + +

= − = − + + + +

∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠

∑

∑

∑

Thermodynamic properties of interfaces

• surface excessi i

idG SdT P dV P dV dA dnα α β β γ µ= − + + + +∑

i ii

i ii

i ii

i ii

i ii

dG dG dG dG

dG S dT P dV dn

dG S dT P dV dn

dG S dT dA dn

G dA

U H TS dA A A

α β σ

αα α α α α

ββ β β β β

σσ σ σ

σσσ

σ σ σσσ

µ

µ

γ µ

γ µ

γ µ

= + +

= − + +

= − + +

= − + +

= + Γ

= = + + Γ

∑

∑

∑

∑

∑

Measurement of adsorption

• adsorption from concentration change( )0 0

i i i i in n n c c Vσ = − = −

• adsorption from surface analysis (sampling at the surface)

( ),

, ,model

L L S S G Gi real i i i

S L Si i real i i i

n c V c V c V

n n n c c Vσ

= + +

= − = −

• adsorption from surface tension change

( )ln( )

( )

i ii

BB

B

d d aRT

a dRT d a

γ

γ

−= Γ

Γ =

∑

for 2 component system:

Gibbs equation:

Probelms

• Problem 3.2 From surface tension measurements on aqueous solutions of butanol, it was found that the slope of the graph of surface tension against concentration was –0.156 mN m2 mol-1 at a bulk concentration of 6.40 mol m-3. Calculate the adsorption at this concentration and state any assumptions made.

• Problem 3.3 An aqueous solution of acetic acid (50 cm3) was shaken with charcoal (2.5 g) until equilibrium was reached. The concentration of the original solution was 0.010 mol dm-3, while after adsorption the concentration had fallen to 0.006 mol dm-3. Calculate the amount of acetic acid adsorbed on 1.0 g of charcoal.

Adsorption at Gas-Liquid interface

• Measurements of equilibrium adsorption– surface tension measurements (Wilhelmy plate)– surface analysis

• radio-labelled solutes• neutron reflectometry (deuterated solutes)• X-ray reflectometry• formation and collection of foam

Adsorption at Gas-Liquid interface

• Observation of adsorption kinetics– adsorption at freshly formed

interfaces– surface waves:

• transverse capillary waves (ripples generated by an oscillating hydrophobic knife edge, f=100-300 Hz). Energy dissipation caused by compression-expansion cycle.

• longitudinal waves (horizontal movement of barrier, <0.1 Hz)

Adsorption of non-electrolyte solutes

experimental: surface tension vs. butanol concentration in aqueous solution

calculated: adsorption( )

BB

B

a dRT d a

γΓ =

Adsorption of non-electrolyte solutes

• Szyszkowski-Langmuir adsorption

* * ln(1 / )A A Bb c aγ γ γ= − +

constants

* /( ) 1 / 1

1

B B A B BB

B B B

B B

B B B

a a b a cdRT d a RT c a c

c c

γ αγα

α

∞

∞ ∞

ΓΓ = = =

+ +

= +Γ Γ Γ

• Szyszkowski-Langmuir isotherm

valid at low concentrations

Adsorption of non-electrolyte solutes• Equation of state (Schonfield and Rideal)

0

solvent

ˆ ˆ( ) ,ˆwhere surface pressure = , 1/ ,

measure of the affinity of the adsorbed moleculessolution i i

A A qkT

Aq

γ γ

Π − =

Π − = Γ−

• Negative adsorptionobserved in dilute aqueous solutions of e.g. glycine and sucrose

• Kinetics of adsorbtionin case of no stirring and no energy barrier:

1/ 2 1/ 21/ 2 , 2d D Dct ct

dt π π−Γ ⎛ ⎞ ⎛ ⎞= Γ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

Adsorption of ionized solute

• in case ionized solute the Gibbs equation must include contribution of all ions:

ln ln

and

12

M M X X

M X M X

d d c d cRT

c c c

dRT dc

γ

γ

+ + − −

+ − + −

−= Γ + Γ

= = Γ = Γ = Γ

−Γ =

from electro neutrality:

Absorption of surfactants

• surfactants (stands for: surface active agents) belong to a class of amphiphiles

• Gibbs monolayers

Absorption of surfactants

• Surfactants can be:– anionic,– cationic,– non-ionic

Absorption of surfactants• Surface tension of surfactant usually falls to a lower limit and

becomes a constant after

saturationadsorption

Absorption of surfactants

• Gibbs equation occasionaly reveals discrepancies with experimentally measured values.– surface hydrolysis: one of the ions is not adsorbed by the surface and

replaced by H+.

ln( ) ln( ) ln( )M M X X H H

d d c d c d cRTγ

+ + − − + +− = Γ +Γ +Γ

=0 =const

ln( )X X

d d cRTγ

− −− = Γ

– indifferent electrolyte: same situation can be created artificially to avoid ambiguities that might arise due to surface hydrolysis by adding large amounts of M+Y-:

ln( ) ln( ) ln( )M M X X Y Y

d d c d c d cRTγ

+ + − − − −− = Γ +Γ +Γ

=const =0

ln( )X X

d d cRTγ

− −− = Γ

Micelles• above certain critical

concentration the surface tension becomes independent of concentration: critical micelle concentration (cmc).

ln( ) ln( )S S M MX X X X

d d c d cRTγ

− = Γ +Γ

not all amphiphiles form micelles.

• at cmc other properties show distinct changes as well (e.g. osmotic pressure indicate that the number of “solute particles” stays the same above cmc).

• formation of micelles means decrease in G, primarily due to large increase in entropy: hydrophobic interaction.

Micelles• solubility of surfactants: there is a

sharp rise in solubility of ionic surfactant above a certain temperature: Krafft temperature

• for non-ionic surfactants raising temperature appears to decrease solubility, so above cloud pointlarge aggregates of surfactant appear

Micelle structure

• at low concentrations micelles are spherical with diameter slightly less than twice the length of the molecule

• at higher concentration, more complex structures are formed

• if organic solvent are used, reverse (inverted) micelles will be formed

Micelle structure

Solubilization

• micelles can increase solubility of otherwise sparingly soluble substances, as the centre of a micelle is a liquid hydrocarbon

Factors affecting cmc

• cmc is affected by factors changing solubility and interaction in the solution:– Amphiphile chain length (l): longer chain means lower solubility.

ln( )cmc a bλ= −

– Salt concentration: as ionic strength increase the interaction between the charged head groups of ionic surfactants will decrease, i.e. cmc will decrease.

– Head group structure: cmc can be affected (but not much)– Structure of the alkyl chain: increase in bulkiness raises cmc, addition

of benzene ring lowers cmc– Other surfactants– Polar additives: long-chain materials with polar group lower cmc and

incorporate in the micelles.

Impurity effects• impurities can lead to a minimum on surface tension vs.

concentration curve.

• another (better) test for impurities is based on variation of surface tension when area of the surface is changed.

Application of surfactants• Wetting: contact angle is reduced.• Detergency:

– detergent: mixture of surfactants with other additives for effective cleansing (builders, brighteners and bleaches, electrolyte filler)

– cleansing efficiency grows with the concentration up to cmc

• Water repulsion– hydrophilic surface can be made hydrophobic

• Emulsification• Froth flotation in ore treatment:

– particles with hydrophobic surface stick to bubbles and are carried upwards when foam is formed (lead and copper sulfide ores, oxides, coal etc.)

• Oil recovery• Membrane disruption absorption

on the interface

emulsifying soil

Films and Foams

• when a foam thickness decreases due to liquid drainage it forms black films: common black film (liquid core) and Newton black film (no liquid core)

Structure of a foam layer:

Foam layer has a surface tension, therefore foam film tries to minimize the area:

Laplace equation is applicable as well:

Films and Foams

• Permeability to gases

monolayer permeation: m mJ k c= ∆

diffusion through the solution: b

b

DHJ cz

= ∆

1 1b

m m

zJ ck DH k

⎛ ⎞+ + = ∆⎜ ⎟

⎝ ⎠

ik

Films and Foams

• Foam formation (by bubbling)

Films and Foams

• the Plateau border– as the curvature radius is smaller at the triple point, the pressure will be

lower and the liquid will flow there dirupting the foam– foam can be stabilized by repulsive pressure between the adsorbed

layers: disjoining pressure

Films and Foams

• Tears of wine

Films and Foams

• The Marangoni effect

buoyancy-driven convection presence of surfactant is impeding convection

Aerosols

• Formation:– dispersion method– aggregation methods

monodisperse size distribution can be created by controlled nucleation

• Precipitation, e.g. electrostatic

Problems

• 3.2 From surface tension measurements on aqueous solutions of butanol, it was found that the slope of the graph of surface tension against concentration was –0.156 mN m2 mol-1

at a bulk concentration of 6.40 mol m-3. Calculate the adsorption at this concentration and state any assumptions made.

• 4.1 A surfactant solution of sodium dodecyl sulfonate(concentration 1.7 mmol kg-1) is found to have a surface tension of 63 mN m-1 at 25ºC. Calculate the adsorption of the surfactant at the air/solution interface and state the two assumptions that are required. The surface tension of pure water at this temperature is 72.0 mN m-1.