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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.

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thermodynamics of interfaces

Thermodynamics of Interfaces

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

terms
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
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
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
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
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
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
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
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
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
Droplet in space (cont.)
  • where Pd = Pl - Pg
  • We can calculate the differential noting that for a sphere V = (4r3/3) and  = 4r2
  • [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
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 2r at tension 

Pd r2 = 2r [2.35]

  • so Pd =2s/r, as expected
vapor pressure at curved interfaces
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]
fundamental differential forms1
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 - Ngdg (gas) [2.41]
  • SldT = VldPl - Nldl(liquid) [2.42]
some algebra
Some algebra…
    • SgdT = VgdPg - Ngdg (gas) [2.41]
    • SldT = VldPl - Nldl(liquid) [2.42]
  • Dividing by Ng and Nl and assume T constant
    • vgdPg = dg (gas) [2.43]
    • vldPl = dl(liquid) [2.44]
  • v indicates the volume per mole. Use dg = dl [2.38] to find
    • vgdPg = vldPl [2.45]
  • which may be written (with some algebra)
using laplace s equation
Using Laplace’s equation...
  • or
  • since vl is four orders of magnitude less than vg, so suppose (vg - vl)/vlvg/vl
  • Ideal gas, Pgvg = RT, [2.49] becomes
continuing
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
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), Pg/P is 0.99.
measurement of p g p
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
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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