Thermodynamics of Interfaces

bronzekerplunkMechanics

Oct 27, 2013 (3 years and 11 months ago)

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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!