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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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