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Thermodynamics of InterfacesPowerPoint Presentation

Thermodynamics of Interfaces

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### Thermodynamics of Interfaces

And you thought this was just for the chemists...

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)

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’

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]

Fundamental Differential Forms

- We have a fundamental differential form (balance of energy) for each phase
- TdSg = dUg + PgdVg - •dNg (gas) [2.22]
- TdSl = dUl + PldVl - •dNl (liquid) [2.23]
- TdSa = dUa - d (interface) [2.24]

- The total energy and entropy of system is sum of components
- S = Sa + Sg + Sl [2.25]
- U = Ua + Ug + Ul [2.26]

How many angels on a pin head?

- 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 = Nl + Ng [2.27]

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]

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)
SadT = 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.

Laplace’s Equation from Droplet in Space

- Now consider the effect of a curved air-water interface.
- Pg and Pl are not equal
- g = l =

- Fundamental differential form for system
TdS = dU + PgdVg + PldVl - (dNg+dNl ) - d [2.31]

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
TdS = dU + PgdVg + PldVl - (dNg+dNl) - d [2.31]

- Holding the total volume of the system constant, [2.31] becomes
- (Pl - Pg)dV - d = 0 [2.32]

Droplet in space (cont.)

- where Pd = Pl - Pg
- We can calculate the differential noting that for a sphere V = (4r3/3) and = 4r2
- [2.34]
- which is Laplace's equation for the pressure across a curved interface where the two characteristic radii are equal (see [2.18]).

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 r2 at pressure Pd, while the length of surface applying this pressure is 2r at tension
Pd r2 = 2r [2.35]

- so Pd =2s/r, as expected

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

- dl = dg [2.38]

Fundamental differential forms

As before, we have one for each bulk phase

- TdSg = dUg + PgdVg - gdNg (gas) [2.39]
- TdSl = dUl + PldVl - ldNl (liquid) [2.40]
Gibbs-Duhem Relations:

- SgdT = VgdPg - Ngdg (gas) [2.41]
- SldT = VldPl - Nldl(liquid) [2.42]

Some algebra… Dividing by Ng and Nl and assume T constant v indicates the volume per mole. Use dg = dl [2.38] to find which may be written (with some algebra)

- SgdT = VgdPg - Ngdg (gas) [2.41]
- SldT = VldPl - Nldl(liquid) [2.42]

- vgdPg = dg (gas) [2.43]
- vldPl = dl(liquid) [2.44]

- vgdPg = vldPl [2.45]

Using Laplace’s equation...

- or
- since vl is four orders of magnitude less than vg, so suppose (vg - vl)/vlvg/vl
- Ideal gas, Pgvg = RT, [2.49] becomes

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/vl), and left-hand side is just Pd, the liquid pressure:

Psychrometric equation For instance, at a matric potential of -1,500 J kg-1 (15 bars, the permanent wilting point of many plants), Pg/P is 0.99.

- Allows the determination of very negative pressures through measurement of the vapor pressure of water in porous media.

Measurement of Pg/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

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 4oC!

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