Exam 3- PChem
(10.1) Discuss how the Debye- Huckel screening length changes as the (a) temperature, (b) solvent dielectric constant, and (c) ionic strength of an electrolyte solution are increased.
(a) Increasing temp increases the screening length. The random thermal motion spreads out the cloud of screening ions (b) Increasing Et. increases the screening length. Increased constant makes the potential attracting the counter ion cloud weaker, causing it to spread out more (c) Increasing ionic strength decreases the screening length. More counter ions available leads to more effective screening, decreasing 1/K.
(10.10) What is the correct order of the following inert electrolytes in their ability to increase the degree of dissociation of acetic acid? (a) 0.001 m NaCl (b)0.001 m KBr (c) 0.10m CuCl2
0.001 m NaCl and 0.001 m KBr will both increase dissociation of HOAc (salting in) - about the same (dilute solution, y(+/-) same according to Debye-Hückel). 0.1 m CuCl2 will decrease dissociation because the solution is concentrated, leading to salting out.
(12.5) What is a Bernoulli Trial?
A Bernoulli Trial is a probability experiment where one of two outcomes can occur. Flipping a coin is an example of a Bernoulli Trial.
(13.1) What is the difference between a configuration and a microstate?
A configuration is a general arrangement of total energy available to the system. A microstate is a specific arrangement of energy in which the energy content of specific oscillators is described.
(12.1) What is the difference between a configuration and a permutation?
A configuration is an unordered arrangement of objects. A permutation is a specific order of an arrangement of objects.
(12.2) What are the elements of a probability model, and how do they differ for continuous and discrete variables?
A probability model consists of a sample space containing the possible values for a variable, and the corresponding probabilities that the variable will assume a value in the sample space. In the discrete case, the sample space consists of a set of specific values a variable can assume, where in the continuous case there is a range of values the variable can assume.
(12.7) Why is normalization of a probability distribution important? What would one have to consider when working with a probability distribution that was not normalized
A variable will always assume some value from the sample set; therefore, normalization of a probability distribution ensures that the sum of probabilities for the variable assuming values contained in the sample set is equal to unity. If the probability distribution is not normalized, then each individual probability should be divided by the sum of all probabilities.
(10.3) Why are activity coefficients calculated using the Debye-Huckel limiting law always less than one?
Activity coefficients calculated using the Debye-Huckel law are always less than one because the net electrostatic interaction among ions surrounding an arbitrarily chosen central ion is attractive rather than repulsive.
(14.1) What is the canonical ensemble? What properties are held constant in this ensemble?
An ensemble is a collection of identical objects or replicas of the system of interest. In the canonical ensemble, V, N, and T are held constant.
(11.14) Why can batteries only be recharged a limited number of times?
As a battery is discharged, reactants are converted into products which in the case of solids leads to a change in the structure at a microscopic level. Cycling between different solid structures with different crystalline structures and densities induces mechanical stress in the electrodes, which over time causes them to disintegrate.
(14.4) For which energetic degrees of freedom are the spacings between energy levels small relative to kT at room temperature?
At room temperature, the spacings between translational energy levels are extremely small relative to kT. The rotational energy-level spacings are also generally small relative to kT, although some species involving light atoms (such as H2 ) may have energy-level spacings that are comparable to kT. Vibrational and electronic energy levels usually have spacings that are greater than kT.
(10.11) How does salting in affect solubility?
At small values of the ionic strength y(+/-) < 1, and the solubility increases as y (+/-) decreases with concentration until the minimum in a plot of y (+/-) versus I is reached.
How is Beta related to temperature? What are the units of kT?
Beta is inversely proportional to temperature, and equal to (kT )^-1. The units of the Boltzmann constant are J K^-1; therefore, the product kT has units of joules, or energy
(15.17) Why should the equilibrium constant be dependent on the difference in Gibbs energy? How is this relationship described using statistical mechanics?
Both the equilibrium constant and the Gibbs energy are dependent on the partition function for the species of interest
(11.11) By convention, the anode of a battery is where oxidation takes place. Is this true when the battery is charged, discharged, or both?
Both, but the relative location of the anode and cathode change in going from the discharge to the charge mode because the direction of current flow in the external circuit changes.
(13.9) What is degeneracy? Can you conceptually relate the expression for the partition function without degeneracy to that with degeneracy?
Degeneracy is the number of states that exist at a certain energy level. The expression for the partition function (q) including degeneracy is: q = SUM g,n*e^-Beta*E,n In this expression, g,n is the degeneracy of a given energy level, E,n is the energy of the level, and the sum extends over all energy levels.
(10.20) It takes considerable energy to dissociate NaCl in the gas phase. Why does this process occur spontaneously in an aqueous solution? Why does it not occur spontaneously in CCl4?
Dissociation takes place readily in water because the energy gain through formation of the solvation shell is greater than the energy needed to dissociate a formula unit of NaCl. It does not occur in CCl4 because CCl4 is not an ionic solvent and is not capable of forming a tightly bound solvation shell
(15.10) Describe the model used to determine the heat capacity of atomic crystals.
Each atom is attached to neighboring atoms through a set of springs which provides a harmonic potential. Therefore, there are 3N harmonic oscillators per atom, and the heat capacity is then determined using this harmonic-oscillator model.
(12.9)What properties of atomic and molecular systems could you imagine describing using probability distributions?
Electron orbital densities, distributions of bond lengths or molecular geometries, locations of particles in space, etc. In quantum mechanics, the square modulus of the wave function is simply a statement of a probability distribution.
(13.8) Why is the probability of observing a configuration of energy different from the Boltzmann distribution vanishingly small?
For a macroscopic system, the large number of atoms/molecules in the system results in the probability of observing anything other than the Boltzmann distribution to be very small. A configuration of energy that is extremely similar to the Boltzmann distribution may have a finite probability, but the macroscopic state of the system corresponding to this distribution will be identical to that corresponding to the Boltzmann distribution.
(15.12) How does the Boltzmann formula provide an understanding of the third law of thermodynamics?
For a perfect crystal at low temperature, only one spatial arrangement of the atoms or molecules will be present so that W = 1 and S = 0.
(11.8) Why is it possible to achieve high-resolution electrochemical machining by applying a voltage pulse rather than a dc voltage to the electrode being machined?
For a tool that is very close to a surface, the RC time constant for charging the double layer depends strongly on location. Thus, pulsed voltage charges will charge the layer in those locations where the tool is closest to the surface- allowing subsequent electrochemical reaction - but not in other locations. Therefore, the reactions are localized. This gives the high spatial selectivity needed for nanomachining.
(12.3) How does Figure 12.2 change if one is concerned with two versus three colored-ball configurations and permutations?
For the case where two balls are chosen from the four-ball set, the number of possible configurations is: C(4,2) = 4!/ 2!2! = 6 Therefore, there will be sic rows in the left column corresponding to the six possible configurations, with each configuration having two associated permutations.
(15.18) For the equilibrium involving the dissociation of a diatomic, what energetic degrees of freedom were considered for the diatomic and for the atomic constituents?
For the diatomic, translational, rotational, vibrational, and electronic degrees of freedom were considered. For atoms only translational and electronic degrees of freedom were considered.
(12.8) When can one make the approximation of treating a probability distribution involving discrete variables as continuous?
If the differences between the discrete values are small relative to the range of values, then a distribution of discrete values can be treated as continuous.
(12.6) What must the outcome of a binomial experiment be if PE = 1?
If the probability of a successful trial is unity, the probability of observing j successful trials out of n total trials is unity. That is, every trial will be successful.
(11.15) Why can more work be extracted from a fuel cell than a combustion engine for the same overall reaction?
In a fuel cell, electrical work is converted to mechanical work. In principle, this conversion can approach 100% efficiency. By contrast, the conversion of heat into work in a heat engine is limited by the second law of thermo- dynamics.
(11.1) To determine standard cell potentials, measurements are carried out in very dilute solutions rather than at unit activity. Why is this the case?
In order to determine the standard potential, the mean ionic activity must be known. It can be calculated in dilute solutions using the Debye-Hückel limiting law, but there is no reliable way to calculate the activity coefficient near unit activity.
(10.9) Why does an increase in ionic strength in the range where the Debye-Huckel law is valid lead to an increase in the solubility of a weakly soluble salt?
In this regime, ln y (+/-) decreases with increasing I (as sqrt I ). The chemical potential of the electrolyte in solution becomes lower with increasing ionic strength, thus increasing solubility. Physically, increasing I leads to increased screening of repulsion between like-charged solute ions.
(14.5) For the translational and rotational degrees of freedom, evaluation of the partition function involved replacement of the summation by integration. Why could integration be performed? How does this relate back to the discussion of probability distributions of discrete variables treated as continuous?
Integration can be performed in evaluating q for the translational and rotational degrees of freedom because the energy-level spacings along these degrees of freedom are small relative to kT. Since there are numerous states within the energy range of interest (given by kT), it is reasonable to treat the variable of interest (i.e., the number of states) as continuous.
(10.19) How do you expect S for an ion in solution to change as the charge increases at constant ionic radius?
It decreases because the solvation shell is more tightly bound for the ion with the greater charge.
(10.18) How do you expect Sm, 0 for an ion in solution to change as the ionic radius increases at constant charge?
It increases because the solvation shell is more tightly bound for the smaller ion.
(10.7) Why is it not possible to measure the activity coefficient of Na+(aq)?
It is impossible to create a solution of pure Na+ in water. A counter ion is always required to give a solution that is electrically neutral, and this anion will always affect the measurement.
(11.17) How does the emf of an electrochemical cell change if you increase the temperature?
It is positive (negative) if the reaction entropy is positive (negative).
(11.10) What is the voltage between the terminals of a battery in which the contents are in chemical equilibrium?
It is zero because only for this value is there no driving force for change.
(14.6) How many rotational degrees of freedom are there for linear and nonlinear molecules?
Linear molecules have two moments of inertia; therefore, they have two rotational degrees of free- dom. Nonlinear molecules have three moments of inertia; therefore, they have three rotational degrees of freedom.
(15.5) Write down the contribution to the constant volume heat capacity from translations and rotations for an ideal monatomic, diatomic, and nonlinear polyatomic gas, assuming that the high-temperature limit is appropriate for the rotational degrees of freedom.
Monatomic: Translations 3/2 R Rotations 0 Diatomic: Translations 3/2 R, Rotaions R Nonlinear Polyatomic: Translations 3/2 R Rotations 3/2 R
(11.13) If you double all the coefficients in the overall chemical reaction in an electrochemical cell, the equilibrium constant changes. Does the emf change? Explain your answer.
No, the emf will not change because it is an intensive quantity.
(10.15) What can you conclude about the interaction between ions in an electrolyte solution if the mean ionic activity coefficient is greater than one?
Overall repulsive interactions must more than compensate any favorable attractive interactions between electrolyte solute and solvent.
(14.2) What is the relationship between Q and q? How does this relationship differ if the particles of interest are distinguishable versus indistinguishable?
Q is the canonical partition function, while q is the molecular partition function, or the partition func- tion that describes an individual member of the ensemble. The relationship between Q and q depends on whether the ensemble members are distinguishable or indistinguishable. For N members or objects in the ensemble: Q= q^N (distinguishable) Q-(q^N)/N! (indistinguishable)
(15.13) For carbon monoxide, the calculated molar entropy was more negative than the experimental value. Why?
Residual entropy due to orientational disorder in crystalline CO
(15.11) What is the Boltzmann formula, and how can it be used to predict residual entropy?
S k ln(W), where S is entropy, W is weight or number of microstates associated with a specific configuration, and k is Boltzmann's constant. This equation can be used to determine the residual entropy in a crystal at 0 K associated with the number of spatial arrangements available to the system.
(12.4) What is Stirling's approximation? Why is it useful? When is it applicable?
Stirling's approximation provides a way to evaluate the factorials of large numbers (N), and is given by: ln N! = N ln N -N Most calculators cannot calculate factorials beyond N = 300; therefore, this approximation provides a convenient way to evaluate values larger than this. It is applicable when N is large.
(13.7) Explain the significance of the Boltzmann distribution. What does this distribution describe?
The Boltzmann distribution is the energy distribution associated with the dominant configuration of energy. It provides a quantitative description of the probability of a given object occupying a certain energy level. The Boltzmann distribution also represents the energy distribution associated with a chemical system at equilibrium
(15.15) Which thermodynamic quantity is used to derive the ideal gas law for a monatomic gas? What molecular partition function is employed in this derivation? Why?
The Helmholtz energy can be used to derive the ideal gas law for a monatomic gas. The translational partition function is employed in the derivation since one is dealing with a monatomic gas for which rotations and vibrations are not present. It should be noted that the derivation can be performed for a molecular system, and will also result in the ideal gas law (see Problem P31.34 of this chapter).
(10.16) Why is the inequality y (+/-) < 1 always satisfied in dilute electrolyte solutions?
The activity coefficients are less than one because the charge on an electrolyte lowers the chemical potential of the electrolyte when compared with an analogous solution of uncharged solute molecules. This occurs because the charges interact attractively with the solvent, lowering the energy. At high concentrations, repulsions between solvated ions become important and y(+/-) increases with concentration.
(12.10) What is the difference between average and root-mean-squared?
The average values correspond to the numerical average of a distribution, or the sum of all values divided by the number of values. The average is also the first moment of a distribution. The root-mean-squared is the square root of the average of the values squared. The root-mean-squared value is the square root of the second moment of the distribution.
(11.4) How can one conclude from Figure 11.20 that Cu atoms can diffuse rapidly over a well-ordered Au electrode in an electrochemical cell?
The copper atoms would be dispersed randomly over the Au surface if they could not diffuse laterally. Instead, they are seen only in islands and on edges, where they are tightly bound. They must have diffused there freely over the surface.
(15.20) Assume you have an equilibrium expression that involves monatomic species only. What difference in energy between reactants and products would you use in the expression for KP?
The difference in energy would be zero; therefore, only the ratio of translational and electronic partition functions would be relevant when calculating the equilibrium constant.
(10.8) Why is it possible to formulate a general theory for the activity coefficient for electrolyte solutions, but not for non-electrolyte solutions?
The dissolved species in electrolyte solutions have a universal form for their dominant interaction with the solvent- the Coulomb interaction. Non-electrolytes interact with much weaker, system specific potentials. The universality of the electrolyte-electrolyte interaction allows a general theory to be developed.
(13.4) Describe what is meant by the phrase "the dominant configuration."
The dominant configuration is the configuration with the largest weight, dictating that this will be the most likely configuration to be observed
(10.12) Why is it not appropriate to use ionic radii from crystal structures to calculate Delta G0 of ions using the Born model?
The effective cavity that the ion occupies in the solvent medium will not have the same radius as the lattice-based radius. Better results can be obtained using the ion-water distance (to the center of charge), a more realistic estimate of the cavity radius r in the Born model.
(15.9) Why do electronic degrees of freedom generally not contribute to the constant volume heat capacity?
The electronic degrees of freedom do not generally contribute to the constant volume heat capacity since the electronic energy-level spacings are generally quite large compared to kT. Therefore, the higher electronic energy levels are not readily accessible so that the contribution to the heat capacity is minimal.
(11.9) Why is it not necessary to know absolute half-cell potentials to determine the emf of an electrochemical cell?
The emf of a cell depends on the difference between the half-cell emfs. Therefore, the common reference level of the half-cells cancels out.
(15.1) What is the relationship between ensemble energy and the thermodynamic concept of internal energy?
The ensemble energy is equal to the difference in internal energy at some finite temperature and that present at 0 K.
(15.14) What thermodynamic property of what particular system does the Sackur-Tetrode equation describe?
The entropy of an ideal monatomic gas.
(15.19) The statistical mechanical expression for KP consisted of two general parts. What are these parts, and what energetic degrees of freedom do they refer to?
The first part is a ratio of partition functions corresponding to the energy levels of the reactants and products. The second term corresponds to the difference in the "zero" of energy between the reactants and products.
(15.8) The molar constant volume heat capacity of N2 is 20.8 J mol1 K1. What is this value in terms of R? Can you make sense of this value?
The heat capacity is equal to 5/2 R. Translational degrees of freedom provide 3/2 R to this heat capacity, and the two rotational degrees of freedom provide a total of R.
(14.9) What is the high-T approximation for rotations and vibrations? For which of these two degrees of freedom do you expect this approximation to be generally valid at room temperature?
The high-temperature approximation is when the product kT is significantly greater than the energy-level spacings. This will be true for translations at all but the lowest temperatures. For rotations, the high-temperature limit is appropriate with T >/=10ThetaR, where ThetaR is the rotational temperature equal to hcB/k, where B is the rotational constant. The high-temperature approximation will generally be valid for rotations at room temperature. For vibrations, the high-temperature limit is appropriate when T >/= 10ThetaV , where ThetaV is the vibrational temperature equal to hcv~/k, where v~ is the vibrational frequency (in cm1 ).
(15.4) List the energetic degrees of freedom for which the contribution to the internal energy determined by statistical mechanics is equal to the prediction of the equipartition theorem at 298 K
The high-temperature approximation will generally be applicable to translations and rotations at 298 K; therefore, these degrees of freedom will make contributions to the internal energy in accord with the equipartition theorem.
(14.7) Assuming 19 F2 and 35 Cl2 have the same bond length, which molecule do you expect to have the largest rotational constant?
The molecule with the smaller reduced mass (u) will have the larger rotational constant. Therefore, 19F is expected to have the largest rotational constant.
(13.3) How does one calculate the number of microstates associated with a given configuration?
The number of microstates associated with a given configuration is known as the weight of the configuration (W), and is given by: W= N!/ PI a,n! where N is the number of objects, and an (referred to as the occupation number)
(13.5) What is an occupation number? How is this number used to describe energy distributions?
The occupation number represents the number of objects occupying a given energy level. The distribution of energy over a collection of objects can be specified as the number of objects occupying a given energy level.
(14.8) Consider the rotational partition function for a polyatomic molecule. Can you describe the origin of each term in the partition function, and why the partition function involves a product of terms?
The partition function involves a product of terms each of which corresponds to one of the three moments of iner- tia. Since there are three moments of inertia, there are three rotational energetic degrees of freedom, and the corre- sponding partition function is the product of partition functions for each degree of freedom.
(13.6) What does a partition function represent? Can you describe this term using concepts from probability theory?
The partition function is the normalization factor for the probability distribution. For molecular systems we are interested in the distribution of energy over the available energy levels. The partition function consists of the sum of weights associated with these different energy levels.
(11.6) Why is the capacitance of an electrolytic capacitor so high compared with conventional capacitors?
The positive and negative plates of an electrolytic capacitor are separated by an extremely small distance (the thickness of the electrical double layer, essentially). And so, the electric field is very strong and the capacitance large.
(15.7) When are rotational degrees of freedom expected to contribute R or 3/2 R (linear and nonlinear, respectively) to the molar constant volume heat capacity? When will a vibrational degree of freedom contribute R to the molar heat capacity?
The rotational degrees of freedom will contribute R or 3/2 R to the molar constant volume heat capacity when the high-temperature limit is valid, defined as when T > 10TheatR , where ThetaR is the rotational temperature, defined as hcB/k. A vibrational degree of freedom will contribute R to the molar heat capacity when the high-temperature limit is applicable, defined as when T >10ThetaV , where ThetaV is the vibrational temperature, defined as hcv~/k.
(11.16) What is the function of a salt bridge in an electrochemical cell?
The salt bridge allows current flow in the internal circuit while preventing the mixing of reactants and products in the two half-cells.
(10.2) Why is it not possible to measure the Gibbs energy of Cl- directly?
The sum of Gibbs energies of solvation of H+ and Cl- can be related to the Gibbs energy for the reaction 1/2 H2 +1/2 Cl2 -> H+ +Cl- , which can be determined experimentally by DeltaG0. However, the Gibbs energies of solvation of Cl- cannot be determined individually.
(13.11) How would you expect the partition function to vary with temperature? For example, what should the value of a partition function be at 0 K?
The value for the partition function should decrease with temperature, since fewer energy levels are accessible as the temperature is reduced. At 0K, only the lowest-energy state will be populated, and q will equal the degeneracy of this lowest-energy state (in particular, if this state is non-degenerate, then q = 1).
(13.2) What is meant by the "weight" of a configuration?
The weight of a configuration is equal to the number of microstates associated with the configuration.
(15.16) What is the definition of "zero" energy employed in constructing the statistical mechanical expression for the equilibrium constant? Why was this definition necessary?
The zero of energy is the dissociation energy for each molecule. It is employed to establish a common energetic reference state for all species involved in the reaction of interest.
(14.10) In constructing the vibrational partition function, we found that the definition depended on whether zero-point energy was included in the description of the energy levels. However, the expression for the probability of occupy- ing a specific vibrational energy level was independent of zero-point energy. Why?
The zero-point energy can be thought of as a constant energy offset that is applied to all of the vibrational energy levels. Since the partition function involves summation over all of the energy states, the specific energy of these states must be considered (including zero-point energy). However, when discussing the probability of occupying a specific vibrational energy level ( pn ), only the relative differences in energy between levels are relevant, and the zero-point energy factors out of the expression for P,n.
(15.2) Why is the contribution of translational motion to the internal energy 3/2RT at 298 K?
There are three translational degrees of freedom, all of which are at the high-T limit at 298 K. Each degree of freedom contributes 1/2RT to the internal energy.
(10.6) Tabulated values of standard entropies of some aqueous ionic species are negative. Why is this statement not inconsistent with the third law of thermodynamics?
This is possible due to the choice of standard state as Sm, 0 (H+, aq) = 0. Although absolute entropies of m neutral species can be determined, this is not possible for ionic species because the solution must be electrically neutral. Therefore, it is necessary to choose a reference value, but as generally only differences in entropies (and other thermodynamic quantities) are desired, it is not necessary to know the absolute value.
(10.17) Under what conditions does y (+/-) -> 1 for electrolyte solutions?
This is the case as m --> 0.
(10.13) Why do deviations from ideal behavior occur at lower concentrations for electrolyte solutions than for solutions in which the solute species are uncharged?
This is the case because the electrostatic interactions between ions in solution are long range interactions, whereas the corresponding interactions for neutral species have a much shorter range.
(15.6) Q15.6 The constant volume heat capacity for all monatomic gases is 12.48 J mol-1 K-1. Why?
This value is equal to 3/2 R, and corresponds to each of the three translational degrees of freedom contributing 1/2 R to the heat capacity, and since the gas is monatomic only translations contribute to the heat capacity (assuming electronic energy levels are not accessible at 298 K).
(11.12) You wish to maximize the emf of an electrochemical cell. To do so, should the concentrations of the products in the overall reaction be high or low relative to those of the reactants? Explain your answer.
To maximize the emf, the concentration-dependent terms in the Nernst equation should be positive. This is the case if Q < 1 or if the concentration of products is low compared to the concentration of reactants
(15.3) List the energetic degrees of freedom expected to contribute to the internal energy at 298 K for a dia- tomic molecule. Given this list, what spectroscopic information do you need to numerically determine the internal energy?
Translational, rotational, and vibrational energetic degrees of freedom are expected to contribute to the internal energy at 298 K. Translational degrees of freedom will be in the high-temperature limit; therefore, a molar contribution of 1/2RT per degree of freedom is expected. Rotations are also expected to be in the high-temperature limit. Since a diatomic has two nonvanishing moments of inertia, a molar contribution of RT is expected. Finally, one needs to know the vibrational frequency to determine if the high-temperature limit is appropriate, and if not, to determine the vibrational contribution to the internal energy.
(14.3) List the atomic and/or molecular energetic degrees of freedom discussed in this chapter. For each energetic degree of freedom, briefly summarize the corresponding quantum mechanical model
Translational: Free particle, Particle in a box Vibrational: Harmonic oscillatorRotational: Rigid rotor Electronic: Hydrogen-atom model, MO theory
(10.14) Why is the value for the dielectric constant for water in the solvation shell around ions less than that for bulk water?
Water molecules immediately around the ion are ordered more than those in the bulk because of the attractive forces between them and the ion. Thus, they are oriented in space around the ion and cannot respond to the electric field due to solvated ions. Therefore, they are less effective in screening the electric field due to solvated ions, leading to a smaller value of Et.
(12.11) When is the higher moment of a probability distribution more useful as a benchmark value as opposed to simply using the mean of the distribution?
When the spread or width of the distribution is of interest in addition to the average value. Particle velocity distributions serve as a good example of this issue. Consider motion in a single dimension. Since particles are just as likely to be moving in the positive and negative direction, the average velocity is equal to zero (see Problem P12.23); however, the width of the velocity distribution will be finite, as can be judged by considering higher moments of the distribution
(10.5) How is the chemical potential of a solute related to its activity?
u(+)= u0(+) + RTlna(+) and u(-)= u0(-) + RTlna(-)
(11.7) What is the difference in the chemical potential and the electrochemical potential for an ion and for a neutral species in solution? Under what conditions is the electrochemical potential equal to the chemical potential for an ion?
u,~ = u + z(phi).For a neutral species, the chemical potential and electrochemical potential have the same value. They differ by z(phi) for an ion. The electrochemical potential is equal to the chemical potential for an ion if (phi) = 0.
(10.4) How is the mean ionic chemical potential of a solute related to the chemical potentials of the anion and cation produced when the solute is dissolved in water?
u= usolute/v = {v(+)u(+) + v(-)u(-) }/ v
