The DebyeHuckel Theory: Calculating the Thermodynamic Properties of an Ionic Solution McQuarrie, D'A

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The Debye-Huckel TheoryCalculating the
Thermodynamic Properties of an Ionic
SolutionMcQuarrie, D.A. Statistical Mechanics,
University Science Books CA 2000, chapter 15

• An Thien Ngo
• Chemistry 2440
• April 21, 2003

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Ideal vs. real ionic solutions

• Ideal solution Cations and anions do not
  interact.
• Real solution Electrostatic interactions.
• Ions of opposite charge are more likely to be
  near each other.

ideal
Shaded region Solvation cage
real
After Atkins, Physical Chemistry. 6th ed. Figure
10.2
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Debye-Huckel Theory
Potential on one ion due to all the other ions
around it.
Take the canonical ensemble average
Total electrostatic potential of the system.
From this we can calculate
Effects of electrostatic interaction on many
thermodynamic properties.
Helmholtz free energy
Gibbs free energy
Chemical potential
and more
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Setup

• ?(r) Electrostatic potential at some point r,
  due to the surrounding ions (i) at positions ri
  with charge qi
• UN,elec Total electrostatic potential of
  system.
• 1lt?1( r,r1 )gt Canonical avg. of ?(r). Gives
  potential at point r in a system where particle 1
  is fixed at position r1.
• 1lt?( r )gt Average electrostatic potential
  acting on particle 1 fixed at r1.

(McQuarrie 15.19)
(15.14)
(15.20)
(15.22)
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Debye-Huckel approximations
(15.26)

• Potential of mean force (w1s) force acting on
  particle 1 of type s
• Substitute into radial distribution function,
  obtain Poisson-Boltzmann (PB) equation (15-27)
• Approximate RHS of PB equation (linearize by
  expanding exponents), obtain 15-28.

For r gt a where atomic radius a/2. Vanishes
for r lt a. Writing ??1( r ) for 1lt?( r )gt
(15.27)
(15.28)
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Results Basic equations of DH Theory

• Linear Poisson-Boltzmann equation (for distances
  r gt a)
• Laplaces equation (for distances 0ltr?lta)
• Solve these equations for ??(r), using boundary
  conditions illustrated in Fig. 15-1

(15.29)
where
(15.30)
??2 is related to ionic strength
(15.31)
r gt a
(15.40)
0ltr?lt/ a
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Total potential

• lt?1gt potential on particle 1. Derive from
  15-22, 15-23.
• lt?jgt Average electrostatic potential acting on
  the jth ion fixed at the origin, due to all the
  other ions in solution.

(15.41)
Generalize
(15.43)
Use 15-43 to derive the electrostatic
contributions to thermodynamic properties
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Deriving electrostatic contribution to
Helmholtz free energy
charge on each ion
potential on each ion due to all the charges
around it, when each ion has charge lqj
(15-53)

• Rewrite the integral using 15-43 where ?2 is
  given by 15-30 evaluate to get Ael
• Only use limit as ka?0 (theory not valid
  otherwise)

(15-55)
where ??(??a) is given by 15-56
(15-64)
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Deriving electrostatic contribution to Chemical
potential Gibbs free E

• Electrostatic chem potential of jth ion
• See previous slide plug in 15.55 for A_el
• Calculate Gibbs free energy from chemical
  potential

(15.58)
(15.55)
(15.59,15.67)
(15.60)
(15.65)
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So where did DH Theory come from?(one
explanation)

• McMillan Mayer Osmotic pressure of a solution
  of nonelectrolytes is analogous to the pressure
  of an imperfect gas. (15-5)
• Mayer (1950) Expanded this theory for solutions
  of electrolytes.
• Defined a summation function (S) (eqn.15-82)
• related to virial coefficients, from which
  pressure and activity coefficient could be
  obtained.
• Graph theory only included cyclic sums (Fig.
  15-5)
• Expression for cyclic sums ? basic equations for
  DHT

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References

• Atkins, P.W. Physical Chemistry, 6th edition,
  Oxford U. Press Oxford 1999, chapter 10.
• McQuarrie, D.A. Statistical Mechanics. University
  Science Books CA 2000, chapter 15
• Mortimer, R.G. Mathematics for Physical
  Chemistry, Macmillan Publishing Co. Inc NY 1981