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# Thermodynamics of Interfaces - PowerPoint PPT Presentation

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

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

• 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)

• 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’

• 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]

• 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]

• 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]

• 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]

• 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.

• 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]

• 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]

• 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]).

• 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

• 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]

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]

• 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)

• 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

• 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:

• 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 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

• 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!