Thermodynamics of Interfaces

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Thermodynamics of Interfaces

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

Courtesy of John Selker (Oregon State University)
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Terms
Key Concept two kinds of variables Intensive do
not depend upon the amount (e.g.,
density) Extensive depend on the amount (e.g.,
mass)

• Extensive Variables
• S entropy
• U internal energy
• N number of atoms
• V volume
• ?? Surface area

• Intensive Variables
• P pressure
• ?? Surface tension
• T Temperature (constant)
• ?? Chemical potential

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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
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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

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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

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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

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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 FDFs
• TdS dU PdV - ?dN - ?d? (system) 2.27

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Gibbs-Duhem relationship

• For an exact differential, the differentiation
  may be shifted from the extensive to intensive
  variables maintaining equality.
• TdS dU PdV - ?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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Laplaces 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 - ??(dNgdNl ) - ?d?
  2.31

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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 - ??(dNgdNl) - ?d?
  2.31
• 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.)

• 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).

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Simple way to obtain La Places 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

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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

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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

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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)

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Using Laplaces 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

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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

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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.

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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

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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!