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Copyright Brian J. Kirby. With questions, contact Prof. Kirby here. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.

[Return to Table of Contents]


Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

B.3 Chemical reactions, rate constants, and equilibrium [water properties top]

While this text does not consider reaction rates in detail, we discuss chemical reaction rates to establish equilibrium chemical conditions and to consider species in different states (e.g., bound to a wall as compared to in the bulk solution), as these affect interfacial charge and electrokinetic transport.

Chemical reactions have rates that can be defined using rate constants and molar concentration. For reversible reactions, this leads to the definition of an equilibrium constant that describes the equilibrium of the reaction system. Consider the general acid dissociation and recombination reactions:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(B.3)

microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(B.4)

where A refers to any species. The rate for Reaction B.3 isk1[HA] and the rate for reaction B.4 is k-1[H+][A-], where [X] denotes the molar concentration of species X. Since these two rates must be equal when the system is at equilibrium, anequilibrium constant Keq or acid dissociation constant Ka can be defined for Reactions B.3 and B.4:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(B.5)

Here, Keq symbolizes an equilibrium constant and can be used for any reaction, and Ka symbolizes acid dissociation constant, applicable only when the dissociation of an acid is being discussed (as is the case in Reactions B.3 and B.4). Keq can be derived for any reversible reaction by taking the ratio of forward and backward reaction rate coefficients.

B.3.1 pH, pKa, and the Henderson-Hasselbach equation

Given the existence of equilibrium constants for any reversible reaction, we can create a convenient notation system and methodology for analyzing acid-base reactions. These reactions are central to micro/nanoscale fluid flow because they determine the charge of walls and macromolecules in the aqueous systems under study.

The Henderson-Hasselbach equation is a useful way to relate solution pH and acid dissociation constant pKa to the degree to which acids dissociate and the concentrations of both acid and conjugate base. The Henderson-Hasselbach equation thus provides a simple quantitative framework for predicting the charge of macromolecules and surfaces. For acid dissociation reactions, we can derive theHenderson-Hasselbach equation from the equilibrium equation:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(B.6)

Taking logarithms and defining pKa = -logKa and pH = -log[H+], we can rearrange this to obtain

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

This is the Henderson-Hasselbach equation, which shows how the ionization states of weak acids can be predicted by the pKa of the reaction combined with the pH of the solution. If the pKa of the weak acidis known (pKa values are listed for several weak acids in Table B.2), the ion concentrations can be readily calculated. For example, when the pH of the solution is equal to the reaction pKa, the weak acid is 50% dissociated. If the pH is more acidic, say one or two pH units below the pKa, then the fraction of the acid in a dissociated state (A-) is reduced (to 9% and 0.9%, respectively). If the pH is more basic, say one or two pH units above the pKa, then the fraction of the acid in a dissociated state is increased (to 91% and 99%, respectively). These results are shown graphically in Figure B.2. Acids and bases are termed strongwhen they are completely ionized upon dissolution in water. Stated mathematically, strong acids and bases have infinite dissociation constants.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure B.2: Fraction of molecules in the form of weak acid (HA) versus conjugate base (A-) as a function of pH.



microfluidics textbook nanofluidics textbook Brian Kirby Cornell


B.3.2 Conjugate acids and bases; buffers

Many definitions of acids and bases are used; for our purposes here, the most useful one is the Brønsted definition, which defines acids as species that donate protons and bases as species that accept protons. When an acid loses a proton, the resulting soluteis its conjugate base, and when a base gains a proton, the resulting solute isits conjugate acid.

Buffersconsist of mixtures of weak acids and their conjugate base, and are so named because the pH of a solution that contains these components changes less upon addition of H+ or OH-than a solution without these components. The reason why the presence of a weak acid buffers the system against perturbation by H+ or OH- is that the equilibrium relation between the weak acid and its conjugate base adjusts in response to these perturbations, largely canceling out the change in H+ or OH-. Thus, in a buffered system, addition of H+ or OH- has little effect on the pH, but a large effect on the relative concentrations of weak acid and conjugate base. Since pH controls the charge states of ions and surfaces that we care about, while the concentrations of weak acid and conjugate base do not, the use of a buffer in an electrolyte solution makes its electrokinetic performance (and thus its fluid flow properties) insensitive to perturbations caused, for example, by passing current through the system. Buffers are thus critical to obtaining reliable flow results in electrokinetically-driven microfluidic systems.

B.3.3 Ionization of water

By writing an acid dissociation equation for water and calculating the concentrations of H+ and OH-, we can derive a simple relation for pH andpOH. For water, we can write an acid dissociation reaction:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(B.10)

The form that protons take in an aqueous solution is H3O+, but we use H+ and H3O+ interchangeably in this text. For this reaction, theequilibrium constant is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(B.11)

For this reaction at 25C, Keq = 1.8×10-16molL. If we assume that [H2O] is constant, then [H2O] can be calculated from the density of water and its molar mass, leading to

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where pOH is the negative logarithm of [OH-]. This relation links the concentration of H+ and OH- in aqueous systems at room temperature, and defines a neutral pH at pH = 7.


microfluidics textbook nanofluidics textbook Brian Kirby Cornell


B.3.4 Solubility product of weakly-soluble salts

As was the case for dissociation of weak acids, dissociation of weakly soluble salts leads to a simple relation between the concentrations of the dissolved ions. For a salt, we can write a dissociation reaction:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(B.14)

If we assume that ample solid is available, the concentration of the solid-phase ions is a constant, and the equilibrium can be described in terms ofa solubility product Ksp = [B+][A-]. This is often written an terms of the negative logarithm of the concentrations:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where pA and pB is the negative logarithm of [A-] and [B+]. This relation links the concentration of the ions to one fundamental property of the salt. For example, the pKsp of AgI at room temperature is 16.5.

B.3.5 Ideal solution limit and activity

An ideal solution is one in which solutes can be assumed to interact exclusively with the solvent—no solute-solute interactions affect the solution properties. In the ideal solutionlimit, the properties of ions and molecules are not affected by their concentration. In real solutions, however, finite solute concentrations lead to a certain amount of solute-solute interactions. These interactions affect the chemical interactions of the solute and reduce the apparent molarity of the solute for chemical equilibrium. We thus define the activity(a) as the effective or apparent molarity of a solute. When studying chemical equilibrium in real solutions, we use the activity rather than the concentration to account for these effects.

B.3.6 Electrochemical potentials

The molar chemical potential (or partial molar Gibbs free energy) of a species [Jmol] is defined in the ideal (dilute) solution limit as:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where gi is the chemical potentialat the reference concentration ci. For real solutions, a must be used instead of c.

Equilibrium between systems that involve electrical potential drops and chemical reaction is defined in the ideal solution limit using the electrochemicalpotential gi:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Again, a must be used instead of c when studying real solutions. We use chemical and electrochemical potentials to treat surface adsorption and ion distributions in an electric field.




Weak Acid pK a


H3PO4 2.1
H2PO4- 7.2
HPO4-2 12.3
Tris+ ((CH3OH)3CNH3+) 8.3
HEPES+ 7.5
TAPS+ 8.4
borate 9.24
citrate 3.06
citrate 4.74
citrate 5.40
ACES+ 6.9
PIPES+ 6.8
acetic acid 4.7



Table B.2: List of weak acidsand corresponding pKa’s.

[Return to Table of Contents]



Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

Copyright Brian J. Kirby. Please contact Prof. Kirby here with questions or corrections. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.


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