We have [finally] finished transport properties

In Chapter 17 we tackled many various transport properties, essentially extending the topics from pchem I [equilibrium thermodynamics] into time-dependent irreversible thermodynamics. Critical to this endeavor are the concepts of flux, gradient and curvature, each of which adds a unique parameter in which to describe nonequilibrium phenomena.

We first tackled ionic conduction, a logical extension of the electrochemical systems we examined in previous chapters [indeed, membrane potentials pointed the way towards this topic]. We saw how charged particles migrate under the influence of an electric field [an electrical potential gradient] and how Arrhenius and Kohlrausch stumbled onto the discovery that solutions were comprised of electrolytes, leading to mathematical expressions for both strong and weak electrolytes. (Arrhenius, incidentally, subscribed -- at least partially -- to a number of unusual theories). Ostwald contributed the dilution law, a clever method of obtaining dissociation constants by measuring the conductance of a series of dilutions of an electrolyte.

We can further characterize how fast an ion moves through a solvent under the influence of an external electric field from the equation s_i=u_iE, where u_i, the ionic mobility, is a parameter based on the identity of a particular ion (which also tells us something about how that ion interacts with water). Of special note is the Grotthuss mechanism, which helps explain why the mobilities of H⁺ and OH^-. Another helpful parameter is the transport number, or the fraction of the total current carried by a particular ion.

Shifting gears, we next tackled viscosity, the phenomena in which linear momentum is transported between sheets of flowing liquid, first described by Sir Isaac Newton in the 18th century. In fact, Newton's Law of Flow leads directly to the observation that, for laminar flow, the velocity profile through a tube is parabolic (with the greatest velocity at R=0). We can use this result in turn to verify Poiseulle's Law, which associates flow rate dV/dt with the fluid viscosity. Newton's Law will not hold for nonlaminar [aka turbulent] flow, characterized by a high Reynolds number, or for thixotropic and/or dilatant fluids.

Our next transport property was one of enormous chemical and physiological importance, the diffusive motion of matter through a medium due only to thermal fluctuations. Brownian motion was observed shortly after the advent of the microscope and was further characterized by the random-walk theories of Smoluchowski and Einstein. A more sophisticated treatment can be found in Fick's 1st and, especially, 2nd Law of Diffusion. In class we looked at three solutions to the diffusion equation: case 1 [a solution of known concentration diffusing into pure solvent, in which we first encountered the erf function]; case 2 [a layer of particles initially sandwiched on either side by pure solvent]; case 3 [a point source of particles diffusing in 3d through pure solvent].

Before moving on, we pause to note that diffusion and viscosity are related through the shape-dependent frictional coefficient, as described in the Einstein relation: D=kT/ƒ. Earlier, George Stokes, a pioneer of fluid dynamics, had established ƒ for many shapes through theoretical and experimental studies of bodies falling through viscous fluids. Of particular importance is the equation for a sphere, ƒ = 6πηa.

Related to diffusion is the motion of matter due to an external force, such as sedimentation [gravity and centrifugation] and electrokinesis [electric field]. Pioneered by Svedberg, analytical ultracentrifugation can reach simulated gravitational fields of nearly 1,000,000-g and is of considerable importance in polymer and biophysical characterizations of macromolecules. Sedimentation, when combined with diffusion information [often simultaneously in the AU], provides a method to experimentally measure the molar mass of a macromolecule/organelle/icky thing, a nontrivial parameter when dealing with conglomerates of particles with no discernible chemical formula. In addition, calculating its ƒ/ƒ_o (the ratio of the actual frictional coefficient to that if the molecule were spherical) gives us a unique shape parameter to describe its sphericality [or not].

Lastly we examined electrokinetic effects, a brief foray back into the world of electrochemistry. In these phenomena [notably electroosmosis and electrophoresis], the particle size is much larger, giving rise to an interface between the particle and the solution. Two models of this interface were discussed: the Helmholtz model [fixed double layer] and the Gouy-Chapman [diffuse double layer] model (truth in advertising: a third picture, the Stern model, which combines aspects of the Helmholtz and Gouy-Chapman models, is now preferred). Key to the diffuse double layer is the generation of a zeta potential, which is critical to explaining colloid stability and electroosmosis/electrophoresis.

Though not discussed in class, electroosmosis is likely to be of tremendous importance as microfluidic technologies gain maturity. Of course electrophoresis has become entrenched in biochemistry, biophysics and molecular biology, first with the development of the Tiselius tube and then with gel electrophoresis. Separation of macromolecules is possible by electrophoretic means if they possess different isoelectric points [that pH at which the electrophoretic mobility is zero, primarily arising from the balancing of cationic and anionic groups on the molecule].

My hope is that its obvious by now why a good grounding in pchem is so necessary to fully appreciate much of what we take for granted in chemistry and biochemistry [for instance, something as mundane as centrifugation] and even in many of our sister disciplines [like physiology and soil science].

And now, onwards to chemical kinetics, the last topic of this quarter, in which we study how chemical identity changes with time. And though you have studied this before in both general and organic chemistry, this time we will do it correctly... :) [with chewy calculus goodness].

Homework 3a Solutions

now that chemweb is back up, hw.3a solutions have been posted in the usual place

Electrochemical Potentials, Membrane Potentials

From last quarter in thermodynamics and the first half of this quarter, we have underscored the great importance of the chemical potential, noting that matter flows spontaneously from high to low μ. When these particles are charged, however, electrical work is performed in addition to chemical work. To take this into account, we transmogrify the chemical potential into the new-and-improved function, the electrochemical potential μ~ = μ + zFφ or

μ_i~ = μ_i° + RTlna_i + z_iFφ

From this thermodynamic function, we can derive most of the basics of electrochemistry:

• voltaic/galvanic cells use spontaneous redox reactions to generate voltage
• the cathode is the site of reduction and the anode is the site of oxidation
• electrons flow towards cathode making it the positive terminal
• standard reduction potentials use the conventional standard [E°(H⁺=0]
• cell potentials can be calculated by E°=E°_cathode-E°_anode
• cell notation is in form anode half cell || cathode half cell
• nonstandard potentials are calculated using Nernst eqn: E=E°-(RT/νF)lnQ
• pH is defined as -log(a_H+)
• concentration cells generate voltage using same half-cell at different concentrations
  

When a membrane separates two ionic environments, we again see that balancing the electrochemical potential leads to an electrical potential difference [the membrane potential]. Note again that, unfortunately, we are employing the term potential twice, the first being a form of Gibbs energy and the latter being a type of voltage. The membrane potential can be found from:

∆φ = (RT/z_iF)ln(a^α_i/a^β_i)

A more realistic scenario, called the Donnan effect, is scene we consider a membrane in which everything is permeable [ions, solvent] except for a macro-ion M. In addition to the electrochemical potential, the macro-ion will generate an osmotic pressure force on the α side, although this term is typically small enough to be neglected if the macro-ion is sufficiently dilute. With a half-page of work, we find the following relationships:

∆φ = (RT/z₊F)ln(Y)

where Y = m^α₊/m^β₊ (the concentration imbalance)

and, when M is dilute, Y ≅ 1 - z_Mm_M/2m^β

Salting-in and Salting-out

Since activity is a measure of the effective concentration of solute in a solution, it can be used to describe solubilities of slightly soluble solutes in ionic solutions. In other words, we can investigate the effects of ionic strength on solubility.

For example, since copper (II) sulfide is only somewhat soluble in water, we can write its dissolution as an equilibrium:

CuS(s) ↔ Cu²⁺(aq) + S^2-(aq)

The thermodynamically correct equilibrium constant is then

K_sp=a_Cua_S=γ_Cuγ_Sm_Cum_S=γ_±²K_sp^obs

The parameter K^obs is the observed (or apparent) equilibrium constant and will be the one that is measured by standard measurements like titration. Since this is a 1:1 ionic compound, K_sp is related to S, the molal solubility, by:

K_sp = S² and K_sp^obs = (S^obs)²

Combining the two equations above, we can solve for the ratio of solubilities:

S/S^obs = γ_±

Therefore, for mean activity coefficients less than one, we see an increase in solubility with respect to water [salting-in] and a corresponding solubility decrease when γ_± is greater than one [salting-out].

Debye-Hückel Theory

One of the problems that arises when treating the theory of ionic solutions is how exactly to deal with the activities of cations versus anions when both are always found together (all solutions are neutral). Recognizing this fact, we can define a mean activity coefficient in terms of the individual ion activity coefficients:

γ_± = [γ₊^ν+γ₊^ν-]^1/ν

But how do we obtain activity coefficients without going through tedious cryoscopic or electrochemical measurements? This is one of the results that the Debye-Hückel theory seeks to uncover. Before going through the details, we need a convenient way to describe how much a given solution can conduct electricity: the ionic strength I.

I = 1/2 Σ z_i² (m_i/m°) = 1/2 Σ z_i² m_i

Ionic strength is unitless so the last term above is basically a unitless molality.

Debye and Hückel proposed a simple model that they hoped would capture most of the behavior of ionic solutions. The assumptions:

1. an ionic atmosphere is established with many ions with valence z and radius a
2. the solvent is structureless with electric permittivity ε
3. the solution is dilute.

Several results were obtained via mind-bleeding statistical mechanics but among them were the Debye length τ_D, which is a measure of the effective range that ionic charges interact in this solution.

Another important result is the extended Debye-Hückel law:

lnγ_± = - |z₊z_-|αI^1/2/(1 + βaI^1/2)

which can also be written in ion activity form:

lnγ_i = -z_i²|αI^1/2/(1 + βaI^1/2)

As mentioned in class, this formula works quite well for dilute to moderately concentrated solutions (and is fairly bad past 0.25 m). For dilute solutions we can consider the limiting Debye-Hückel law:

lnγ_± = - |z₊z_-|αI^1/2

Since α=1.171 for water at 25°C, we can transmogrify any of these formulas to log₁₀ world. For example, the above equation becomes

logγ_± = -0.509 |z₊z_-|I^1/2 (aqueous solutions, 25°C)

Bjerrum, Mayer and Davies extended the Debye-Hückel formula so that it works for a wide range of ionic strengths (and remains the primary equation used in pchem today):

logγ_± = -0.509 |z₊z_-|(I^1/2/1+I^1/2 - 0.30I) (aqueous solutions, 25°C)

At different temperatures, new values for the Debye-Hückel parameter (the 0.509 value) would need to be calculated through α.

Tomorrow, salting-in and salting-out.

hw.1 corrections

For question 1, I mistakenly chose the chemical potential that was the lowest, not the highest.

Also, for question 12, the hydrogens on each benzoic acid are missing.

Activity Gets Promoted

Before grappling with the chemical potentials of ionic solutions, we took a moment to update the colligative property formulas for ionic solutes. Since these properties depend not on what is dissolved in solution but how much, the fact that salts break into ions needs to be addressed, which is accomplished by i, the van't Hoff factor:

• freezing point depression: ∆T_f=iK_fm₂
• boiling point elevation: ∆T_b=iK_bm₂
• osmotic pressure: Π=iM₂RT

where i = (actual number of free particles)/(number before dissociation).

For very dilute solutions, i is approximately equal to the number of ions in the formula (i=2 for KCl, i=3 for Na₂SO₄, etc.). As the concentration increases, the actual van't Hoff factor will be less (sometimes very much so) than these stoichiometrically derived values. We also saw that we could relate i to the degree of dissociation α by the formula α=(i-1)/(ν-1).

Although earlier we defined the chemical potential of a solution in terms of the Raoult's standard state [purity and mole fraction], it is more convenient to use a standard state based on molality:

For a solute in an ideal solution: μ_i = μ°_i + RT lnm_i
For a solute in a real solution: μ_i = μ°_i + RT lna_i

Henceforth the symbol ° will mean 1 bar for gases and 1 m for solutions. Since we now have six different chemical potential expressions [ideal gas, real gas, ideal solution, real solution, liquid, solid] it is customary to promote the idea of activity so that it covers all possibilities. We will then use a general expression μ_i = μ°_i + RT lna_i with the following mappings:

• ideal gas: a_i = P_i [partial pressure]
  
• real gas: a_i = f_i [fugacity]
  
• ideal soln: a_i = m_i [molality]
  
• real soln: a_i = a_i [activity]
  
• liquid: a_i = 1 [assuming normal pressure]
  
• solid: a_i = 1 [assuming normal pressure]

Now, for any reversible chemical reaction aA( )+bB( ) ↔ cC( )+dD( ) we can construct reaction quotient and equilibrium constant expressions immediately in terms of the relevant activities.

Tomorrow we will go through the powerful Debye-Hückel theory, which gives considerable insight into ionic solutions, including a method to calculate activity coefficients. Also, next hw set..