1
Thermodynamics of
Interfaces
And you thought this was just
for the chemists...
Williams, 2002 http://www.its.uidaho.edu/AgE558
Modified after Selker, 2000 http://bioe.orst.edu/vzp/
2
Thermodynamics
… a unifying theory
•
Mineral dissolution
–
precipitation
•
Microbial activity
•
Surface tension
•
Vapor Pressure
3
Terms
Intensive Variables
P: pressure
Surface tension
T: Temperature
(constant)
Chemical potential
Extensive Variables
S: entropy
U: internal energy
N: number of atoms
V: volume
Surface area
Key Concept: two kinds of variables
Intensive
: do not depend upon the amount (e.g., density)
Extensive
: depend on the amount (e.g., mass)
4
Definitions
Internal Energy (U)
–
The change in
internal energy is the sum of the
change of the heat absorbed by a
system and the change of the work
done on a system.
First Law
:
dU = dq + dw
5
Definitions (continued)
Entropy (S)
–
The change in entropy
is the change in heat absorbed by a
system per temperature, in a
reversible process
Second Law:
dS = dq / T
where q is reversible
Entropy always increases for
spontaneous processes
6
Phases in the system
Three phases
liquid; gaseous; taut interface
Subscripts
‘•’ indicates constant intensive parameter
‘g’; ‘l’; ‘a’; indicate gas, liquid, and interface
Gaseous phase ‘g’
Interface phase ‘a’
Liquid phase ‘l’
7
Chemical
Potential
refers to the per molecule energy due
to chemical bonds.
Since there is no barrier between
phases, the chemical potential is uniform
g
=
a
=
l
=
•
[2.21]
8
Fundamental
Differential
Forms
We have a fundamental differential form
(balance of energy) for each phase
TdS
g
= dU
g
+ P
g
dV
g

•
dN
g
(gas)
[2.22]
TdS
l
= dU
l
+ P
l
dV
l

•
dN
l
(liquid)
[2.23]
TdS
a
= dU
a

d
(interface)
[2.24]
The total energy and entropy of system is
sum of components
S = S
a
+ S
g
+ S
l
[2.25]
U = U
a
+ U
g
+ U
l
[2.26]
9
Inter

phase
surface
The inter

phase surface is two

dimensional,
The number of atoms in surface is
zero in comparison to the atoms in
the three

dimensional volumes of gas
and liquid:
N = N
l
+ N
g
[2.27]
10
FDF
for
flat
interface
system
If we take the system to have a flat interface
between phases, the pressure will be the
same in all phases (ignoring gravity), which
we denote P
•
The FDF for the system is then the sum of
the three FDF’s
TdS = dU + P
•
dV

•
dN

d
(system)
[2.27]
11
Gibbs

Duhem
relationship
For an exact differential, the differentiation may
be shifted from the extensive to intensive
variables maintaining equality).
TdS = dU + P
•
dV

•
dN

d
(system)
S
a
dT =
d
[2.29]
or
Equation of state for the surface phase
(analogous to Pv = nRT). Relates temperature
dependence of surface tension to the magnitude
of the entropy of the surface.
12
Laplace’s
Equation
from
Droplet
in
Space
Now consider the effect of a curved air

water
interface.
P
g
and P
l
are not equal
g
=
l
=
Fundamental differential form for system
TdS = dU + P
g
dV
g
+ P
l
dV
l

(dN
g

dN
l
)

d
[2.31]
13
Curved interface Thermo, cont.
Considering an infinitesimally small
spontaneous transfer, dV, between the gas
and liquid phases
chemical potential terms equal and opposite
the total change in energy in the system is
unchanged (we are doing no work on the system)
the entropy constant
Holding the total volume of the system constant,
[2.31] becomes
(P
l

P
g
)dV

d
= 0
[2.32]
14
Droplet
in
space
(cont
.
)
where P
d
= P
l

P
g
We can calculate the differential noting that for a
sphere V = (4
r
3
/3) and
= 4
r
2
[2.34]
which is Laplace's equation for the pressure
across a curved interface where the two
characteristic radii are equal (see [2.18]).
15
Simple
way
to
obtain
La
Place’s
eq
....
Pressure balance across droplet middle
Surface tension of the water about the center
of the droplet must equal the pressure exerted
across the area of the droplet by the liquid
The area of the droplet at its midpoint is
r
2
at
pressure P
d
, while the length of surface
applying this pressure is 2
r at tension
P
d
r
2
= 2
r
[2.35]
so P
d
=2
/r, as expected
16
Vapor
Pressure
at
Curved
Interfaces
Curved interface also affects the vapor
pressure
Spherical water droplet in a fixed volume
The chemical potential in gas and liquid equal
l
=
g
[2.37]
and remain equal through any reversible process
d
l
= d
g
[2.38]
17
Fundamental
differential
forms
As before, we have one for each bulk phase
TdS
g
= dU
g
+ P
g
dV
g

g
dN
g
(gas)
[2.39]
TdS
l
= dU
l
+ P
l
dV
l

l
dN
l
(liquid)
[2.40]
18
Gibbs

Duhem
relation
S
g
dT = V
g
dP
g

N
g
d
g
(gas)
[2.41]
S
l
dT = V
l
dP
l

N
l
d
l
(liquid)
[2.42]
Dividing by N
g
and N
l
and assume T constant
v
g
dP
g
= d
g
(gas)
[2.43]
v
l
dP
l
= d
l
(liquid)
[2.44]
v indicates the volume per mole. Use [2.38] to
find
v
g
dP
g
= v
l
dP
l
[2.45]
which may be written (with some algebra)
19
Using
Laplace’s
equation
...
or
since v
l
is four orders of magnitude less
than v
g
, so suppose (v
g

v
l
)/v
l
v
g
/v
l
Ideal gas, P
g
v
g
= RT, [2.49] becomes
20
Continuing...
Integrated from a flat interface (r =
) to
that with radius r to obtain
where P
is the vapor pressure of water
at temperature T. Using the specific gas
constant for water (i.e., = R/v
l
), and left

hand side is just P
d
, the liquid pressure:
21
Psychrometric
equation
Allows the determination of very
negative pressures through
measurement of the vapor pressure
of water in porous media.
For instance, at a matric potential of

1,500 J kg

1
(15 bars, the permanent
wilting point of many plants), P
g
/P
is
0.99.
22
Measurement
of
P
g
/P
A thermocouple is cooled while its
temperature is read with a second
thermocouple.
At the dew point vapor, the temperature
decline sharply reduces due to the energy of
condensation of water.
Knowing the dew point T, it is straightforward
to obtain the relative humidity
see Rawlins and Campbell in the Methods of
Soil Analysis, Part 1. ASA Monograph #9,
1986
23
Temperature Dependence of
Often overlooked that all the measurements
we take regarding water/media interactions are
strongly temperature

dependent.
Surface tension decreases at approximately
one percent per 4
o
C!
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