thermodynamics of interfaces
DESCRIPTION
Thermodynamics of Interfaces. 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/. Thermodynamics. … a unifying theory - PowerPoint PPT PresentationTRANSCRIPT
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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!