thermodynamics of interfaces

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Thermodynamics of Thermodynamics of InterfacesInterfaces

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

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/

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ThermodynamicsThermodynamics

… a unifying theory

• Mineral dissolution – precipitation

• Microbial activity

• Surface tension

• Vapor Pressure

… a unifying theory

• Mineral dissolution – precipitation

• Microbial activity

• Surface tension

• Vapor Pressure

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TermsTerms

Intensive VariablesP: pressure Surface tensionT: Temperature (constant) Chemical potential

Intensive VariablesP: pressure Surface tensionT: Temperature (constant) Chemical potential

Extensive VariablesS: entropyU: internal energyN: number of atomsV: volume Surface area

Extensive VariablesS: entropyU: internal energyN: number of atomsV: 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)

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DefinitionsDefinitions

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

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

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Definitions (continued)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

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

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Phases in the systemPhases in the systemThree phases liquid; gaseous; taut interface

Subscripts ‘•’ indicates constant intensive parameter‘g’; ‘l’; ‘a’; indicate gas, liquid, and interface

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’

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Chemical PotentialChemical 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]

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]

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Fundamental Differential FormsFundamental 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 componentsS = Sa + Sg + Sl [2.25]U = Ua + Ug + Ul [2.26]

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 componentsS = Sa + Sg + Sl [2.25]U = Ua + Ug + Ul [2.26]

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Inter-phase surfaceInter-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 = Nl + Ng [2.27]

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]

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FDF for flat interface systemFDF 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]

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]

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Gibbs-Duhem relationshipGibbs-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.

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.

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Laplace’s Equation from Droplet in SpaceLaplace’s Equation from Droplet in Space

Now consider the effect of a curved air-water interface.

Pg and Pl are not equalg = l =

Fundamental differential form for system

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

Now consider the effect of a curved air-water interface.

Pg and Pl are not equalg = l =

Fundamental differential form for system

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

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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 oppositethe 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

(Pl - Pg)dV - d = 0 [2.32]

Considering an infinitesimally small spontaneous transfer, dV, between the gas and liquid phases

chemical potential terms equal and oppositethe 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

(Pl - Pg)dV - d = 0 [2.32]

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Droplet in space (cont.)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]).

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]).

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Simple way to obtain La Place’s eq....Simple way to obtain La Place’s eq....Pressure balance across droplet middleSurface 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 =2/r, as expected

Pressure balance across droplet middleSurface 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 =2/r, as expected

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Vapor Pressure at Curved InterfacesVapor Pressure at Curved Interfaces

Curved interface also affects the vapor pressure

Spherical water droplet in a fixed volumeThe chemical potential in gas and liquid equal

l = g [2.37]

and remain equal through any reversible process

dl = dg [2.38]

Curved interface also affects the vapor pressure

Spherical water droplet in a fixed volumeThe chemical potential in gas and liquid equal

l = g [2.37]

and remain equal through any reversible process

dl = dg [2.38]

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Fundamental differential formsFundamental 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]

As before, we have one for each bulk phase

TdSg = dUg + PgdVg - gdNg (gas)[2.39]

TdSl = dUl + PldVl - ldNl (liquid)[2.40]

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Gibbs-Duhem relationGibbs-Duhem relation

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

Dividing by Ng and Nl and assume T constantvgdPg = dg (gas) [2.43]vldPl = dl (liquid) [2.44]

v indicates the volume per mole. Use [2.38] to findvgdPg = vldPl [2.45]

which may be written (with some algebra)

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

Dividing by Ng and Nl and assume T constantvgdPg = dg (gas) [2.43]vldPl = dl (liquid) [2.44]

v indicates the volume per mole. Use [2.38] to findvgdPg = vldPl [2.45]

which may be written (with some algebra)

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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

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

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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:

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:

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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.

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.

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Measurement of PMeasurement of Pgg/P/PA 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

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

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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!

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!