avayty
`aT
`
`ALFRED MARTIN
`
`pd —)an~l a] FX)ga© aa [—)Pitasoe)—_=
`
`FRESENIUS EXHIBIT
`1057 Page 1 of 81
`
`

`

`FOURTH EDITION
`
`Physical Pharmacy-
`
`PHYSrCAL CHEMICAL PRINCIPLES IN THE PHARMACEUTICAL SCIENCES
`
`Alfred Martin, Ph.D.
`Emeritus Coult.er R. Sublett Professqr
`Drug Dynamics lnBtitute,
`College of Plw:rmacy,
`University of T~as
`
`with the 'JKl,rtici'JKl,tion of
`PILAR BUSTAMANTE, Ph.D.
`Titular Professqr
`Department.of Plw:rmacy
`and Pkaf"l'll,(J,C6'Utical Technology,
`University Alcala de Henares,
`Madrid, Spain
`
`and with illustrations by
`A.H. C. CHUN, Ph.D.
`Associate Reaearch. Fellow
`Pharmaceutical Prod'U,CtB DiviBian,
`Abbott Labcrratmes
`
`.I.
`
`B. I. Waverly Pvt Ltd
`New Delhi
`
`FRESENIUS EXHIBIT 1057
`Page 2 of 81
`
`

`

`B. L Waverly Pvt Lid
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`
`Indian Reprint 1994
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`
`ISBN 81-743UI01-0
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`prioled al Uailed ladla Pr-. New Delhi.
`
`FRESENIUS EXHIBIT 1057
`Page 3 of 81
`
`

`

`6
`Solutions of Electrolytes
`
`I
`
`Properties of Solutions of Electrolytes
`Arrhenius Theory of Electrolytic Dissociation
`Theory of Strong Electrolytes
`
`Coefficients for Expressing Colligative
`Properties
`
`The first satisfactory theory of ionic solutions was
`that proposed by Arrhenius in 1887. The theory was
`based largely on studies of electric conductance by
`Kohlrausch, colligative properties by van't Hoff, and
`chemical properties such as heats of neutralization by
`Thomsen. Arrhenius1 was able to bring tojrether the
`results of these diverse investigations into a broad
`generalization known as the theory of electrolytic
`dissociation.
`Although the theory proved quite useful for describ(cid:173)
`ing weak electrolytes, it was soon found unsatisfactory
`for strong and moderately strong electrolytes. Accord(cid:173)
`ingly, many attempts were made to modify or replace
`Arrhenius's ideas with better ones, and finally, in 1923,
`Debye and Hdckel put forth a new theory. It is based on
`the principles that strong electrolytes are completely
`dissociated into ions in solutions of moderate concentra(cid:173)
`tion and that any deviation from complete dissociation
`is due to interionic attractions. Debye and Hdckel
`expressed the deviations in terms of activitieS', activity
`c:oefficients, and ionic strengths of electrolytic solu(cid:173)
`tions. These quantities, which had been introduced
`earlier by Lewis, are discussed in this chapter together
`with the theory of interionic attraction. Other aspects
`of modem ionic theory and the relationships between
`electricity and chemical phenomena are considered in
`following chapters.
`We begin with a discussion of some of the properties
`of ionic solutions that led to Arrhenius theory of
`electrolytic dissociation.
`
`PROPERTIES OF SOLUTIONS OF ELECTROLYTES
`Electrolysis. When, under a potential of several volts,
`a direct electric current (de) flows through an electro(cid:173)
`lytic cell (Figure 6-1), a chemical reaction occurs. The
`_process is known as electrolysis. Electrons enter the
`
`cell from the battery ~r generator at the catkade (road
`down); they combine with positive ions or cations, in
`the solution, and the cations are accordi_ngly reduced.
`The l).egative ions, or anions, carry electrons through
`the solution and discharge them at the anode (road up),
`and the anions are accordingly oxidized. Reduction is
`the addition of electrons to a chemical species, and
`oxidation is removal of electrons from a species. The
`clll'l'ent in a solution consists of. a flow of positive and
`negative ions toward the electrodes, whereas the
`current in a metallic conductor consists of a· flow of free
`electrons migrating through a crystal lattice of fixed
`positive ions. Reduction occurs at the cathode, where
`electrons enter from the external circuit and are added
`to a chemical species in solution. Oxidation occurs at the
`anode where the electrons are removed from a chemical
`species in solution and go into the external circuit.
`
`_ Electrons 7l
`
`Anode +
`(oxldatl >-
`
`0
`
`0
`O ~Escapln1
`02
`so.=
`Anions(-)
`Fe+++
`Cations(+)
`
`-Current direction-
`
`Fla. 8-1. Electrolysis in an electrolytic cell.
`
`FRESENIUS EXHIBIT 1057
`Page 4 of 81
`
`

`

`126 Physical Pharmacy
`
`In the electrolysis of a solution of fenic sulfate in a
`cell containing platinum electrodes, a fenic ion mi(cid:173)
`grates to the cathode where it picks up an electron and
`is reduced:
`
`(6-1)
`
`Fe3+ + e = Fe2+
`The sulfate ion canies the current through the solution
`to the anode, but it is. not easily .oxiqized;· therefore,
`hydroxyl ions of the water are converted into molecular
`oxygen, which escapes at the anode, and sulfuric acid is
`found in the solution around the electrode. The oxida(cid:173)
`tion reaction at the anode is
`oH- = ~ + ½H20 + e
`(6-2)
`Platinum electrodes are used here since they do not
`pass into solution to any extent. When attackable
`metals, such as copper or zinc, are used as the anode,
`their atoms tend to lose electrons, and the metal passes
`into solution as the positively charged ion.
`In the electrolysis of cupric chloride between plati(cid:173)
`m.un electrodes, the reaction at the cathode is
`½cu2+ + e = ½cu"
`(6-3)
`2
`2
`while at the anode, chloride and hydroxyl ions are
`converted respectively into gaseous molecules of chlo(cid:173)
`rine and oxygen, which then, escape. In each of these
`two examples, the net result is the transfer of one
`electron from the cathode to the aJ)ode.
`Transference Numbers. It should be noted that the
`flow of electrons through the solution from right to left
`in Figure 6-1 is accomplished by the movement of
`cations to the right as well as anions to the left. The
`fraction
`of total current carried by the cations or by the anions
`is known as the trar,,sport or trar,,sference number
`t+ or t_.
`
`t _ ·current canied by cations
`+ -
`total current
`t = current carried by anions
`-
`total current
`
`(6-4)
`
`(6-5)
`
`The sum of the two transference numbers is obviously
`equal to unity:
`
`(6-6)
`
`t+ + L = 1
`The transference numbers are related to the veloci(cid:173)
`ties of the ions, the faster-moving ion carrying the
`greater fraction of current. The velocities of the ions in
`turn depend on hydration as well as ion size and charge.
`Hence, the speed and the transference numbers are not
`necessarily the same for positive and for negative ions.
`For example, t~e transference number of the sodium
`ion in a 0.10-M solution of NaCl is 0.385. Because it is
`greatly hydrated, the lithium ion in a 0.10-M solution of
`LiCl moves slower than the sodium ion and hence has a
`lower transference number, viz., 0.317.
`Electrical Units. According to Ohm's law, the strength
`of an electric current I in amperes flowing through a
`
`metallic conductor is related to the difference in applied
`potential or voltage E and the resistance R in ohms, as
`follows:
`
`E
`l=-
`R
`The current strength I is the rate of flow of current or
`the quantity Q of electricity (electronic charge) in
`coulombs flowing per unit time:
`
`(6-7)
`
`(6-8)
`
`and
`Quantity of electric charge, Q
`= current, Ix time, t (6-9)
`The quantity of electric charge is expressed in coulombs
`(1 coul = 3 x 109 electrostatic units of charge, or esu),
`the current in amperes, and the electric potential in
`volts.
`Electric energy consists of an intensity factor, elec(cid:173)
`tromotive force or voltage, and a quantity factor,
`coulombs.
`
`Electric energy = E x Q
`(6-10)
`Faraday's Laws. In 1833 and 1834, Michael Faraday
`announced his famous laws of electricity, which may be
`summarized in the statement, the passO{Je of 96,500
`coulomf!..s of electricity through a cor,,d,uctivi~y _cell
`produces a chemical change of 1 tpam ·equivalent
`weight of any substance. The quantity 96,500 is known
`as the faraday, F. The best estimate of the value today
`is 9.648456 x 104 coulombs per gram equivalent.
`A univalent negative ion is an atom to which a
`valence electron has been added; a univalent positive
`ion is an atom from which an electron has been
`removed. Each gram equivalent of ions of any electro(cid:173)
`lyte canies Avogadro's number (6.02 x 1<>28) of positive
`or negative charges. Hence, from Faraday's laws, the
`passage of 96,500 coulombs of electricity results in the
`transport of 6.02 x 1<>28 electrons in the cell. A f~aday
`is an Avogadro's number of el~ctrons, correspondmg to
`the mole which is an Avogadro's number of molecules.
`The p~age of 1 faraday of electricity causes the
`electrolytic deposition of the following number of gram
`atoms or "moles" of various ions: lAg+, lCu\ ~u2+,
`!Fe2+, !Fe3+. Thus, the number of positive charges
`~ed ty 1 gram equivalent of Fe3+ is 6.02 x 1<>28, but
`the number of positive charge~ canied by 1 gram atom
`or 1 mole of fenic ions is 3 x 6.02 x 1<>28.
`Faraday's laws can be used to compute the charge on
`an electron in the following way. Since 6.02 x 1<>28
`electrons are associated with 96,500 coulombs of elec(cid:173)
`tricity, each electron has a charge e of
`e = 96,500 coulombs
`6.02 x 1<>28 electrons
`= 1.6 x 10-19 coulombs/electron (6-11)
`
`FRESENIUS EXHIBIT 1057
`Page 5 of 81
`
`

`

`and since 1 coulomb= 3 x 109 esu
`, = 4.8 x 10-10 electrostatic units
`of charge/electron (6-12)
`Electrolytic Conductance. The resistance R in ohms of
`any unifonn metallic or electrolytic conductor is di(cid:173)
`rectly proportional to its length l in cm and inversely
`.
`, proportional to its cross-sectional area A in cm2,
`R = p ¾
`
`(6-13)
`
`in .which p is the resistance between opposite faces of a
`1-cm cube of the conductor and is known as the specific
`resistance.
`The conductance C is the reciprocal of resistance,
`1 C=(cid:173)
`R
`· and hence can be considered as a measure of the ease
`with which current can pass through the conductor. It
`is expressed in reciprocal ohms or mhos. From equation
`(6-13),
`
`(6-14)
`
`lA
`1
`C=-=--
`R pl
`The specific ~uctance K is the reciprocal of specific
`resistance and is expressed in mhos/cm.
`
`(6-15)
`
`(6-16)
`
`It is the conductance of a solution confined in a cube 1
`cm on an edge as seen in Figure 6-2. The relationship
`between specific conductance and conductance or resis(cid:173)
`tance is obta::~d by combining equations (6-15) and
`(6-16).
`
`1 l
`l
`K=C-=--
`A RA
`Measurina the Conductance of Solutions. The Wheat(cid:173)
`stone bridge assembly for measuring the conductance of
`a solution is shown in Figure 6-3. The. solution of
`unknown rest.st~ .. -;:, Rx is placed in the cell and
`
`(6-17)
`
`e
`
`Specific conductance (1<)
`
`FIi, 6-2. Relationship between specific conductance and equivalent
`conductance.
`
`Ch.apter 6 • Solutions of Electrolytes 127
`
`-~ Yariabla cond, ... r
`
`Ce l l~
`b ~!!!!!9,U:.,,.,..,,__-::--➔ C
`
`r
`
`Fig. 6-3. Wheatstone bridge for conductance measurements.
`
`connected in the circuit. The contact point is moved
`along the slide wire be until at some point, say d, no cur(cid:173)
`rent from the source of alternating current (oscillator)
`.flows through the detector (earphones or oscilloscope).
`When the bridge is balanced the potential at a is equal
`to that at d, the sound in the earphones or the
`oscillating patte~ on the oscilloscope is at a minimum,
`and the resistances R8 , Ri, and R2 are read. In the
`balar,iced state, the resistance of the solution R:e is
`obtained from the equation
`
`(6-18)
`
`R1
`R:e = Ra R2
`The variable condenser across resistance R. is used to
`produce a sharper balance. Some conductivit~. ~ridges
`are calibrated in conductance as well as resistance
`values. The electrodes in the cell are platinized with
`platinum black by electrolytic depositio~ so that caW(cid:173)
`ysis of the reaction will occur at the platmum surfaces,
`and formation of a nonconducting gaseous film will not
`occur on the electrodes.
`Water that is carefully purified by redistillation in the
`presence of a little permanganate is used to prepare the
`solutions. Conductivity water, as it is called, has a
`specific conductance of about 0.05 x 10-6 mho/cm at
`18° C whereas ordinary distilled water has a value
`some;hat over 1 x 10-6 mho/cm. For most conductiv(cid:173)
`ity studies, "equilibrium water" containing_ CO2 from
`the atmosphere is satisfactory. It 'has a specific conduc(cid:173)
`tance of about 0.8 x 10-6 mho/cm.
`The specific conductance K is computed from ~e
`resistance R or conductance C by use of equation
`(6-17). The ;uantity l!A, th'e ratio of distance betw~:i:i
`electrodes to the area of the electrode, has a definite
`value for each conductance cell; it is known as the cell
`const,a,nt, K. Equation (6-17) thus can be written
`K =KC= KIR
`(6-19)
`(The subscript x is no longer needed on R and is
`therefore dropped.) It would be difficult to measure l
`and A, but it is a simple matter to determine the cell
`constant experimentally. The specific conduc~ce ?f
`several standard solutions has been detenmned m
`carefully calibrated cells. For example, a solution
`
`FRESENIUS EXHIBIT 1057
`Page 6 of 81
`
`

`

`128 Pkyaical PIUJ:rmacy
`
`containing 7.45263 g of potassium chloride in 1000 g of
`water has a specific conductance of 0.012856 mho/cm at
`25° C. A solution of this concentration contains 0.1 mole
`of salt per cubic decimeter (100 cm3) of water and is
`known as a 0.1 demal solution. When such a solution is
`placed in a cell and the resistance is measured, the cell
`constant can be determined by use of equation (6-19).
`
`,_,,,,,. 6- 1. A 0.1-demal solution of KCl was placed in a cell
`whose constant K was desired. The resistance R was found to be
`34.69 ohms at 25° C.
`K = KR = 0.012856 mho/em x 34.69 ohms
`= 0.4460 cm- 1
`Eample 6-2. When the cell described in EmmpltJ 6-1 was filled
`with a 0.01-N N-.SO, solution, it had a resistance of 397 ohms. What
`"is the specific conductance?
`K = j = 0· : = 1.1234 x 10-3 mho/cm
`Equivalent Conductance. To study the ~tion of
`molecules into ions, independent of the concentr.ation'1d'
`the electrolyte, it is convenient to use equivalent
`conductance rather than specific conductance. All sol-
`- utes of equal normality produce the same number of
`ions when completely dissociated, and equivalent con(cid:173)
`ductance measures the current-carrying capacity of this
`given number of ions. Specific conductance, on the
`other hand, measures the current-carrying capacity of
`all ions in a unit volume of solution and accordingly
`varies with concentration.
`Equiwumt conductance A is defined as the conduc(cid:173)
`tance of a solution of sufficient volume to contain 1 gram
`equivalent of the solute when measured in a cell in
`which the electrodes are spaced 1 cm apart. The
`equivalent conductance A,; at a concentration o( c gram
`equivalents per liter is calculated from the product of
`the specific conductance K and the volume Vin ems that
`contains 1 gram equivalent of solute. The cell may be
`imagined as having electrodes 1 cm apart and to be of
`sufficient area so that it can contain the solution. The
`~ ;
`cell is shown in Figure· 6-2.
`s
`;
`V = 1000 cm lliter = 1000 cms/E ;
`·. ~ .,
`c Eqlliter
`c
`The equivalent conductance is obtained when K, the
`conductance per ems of solupon (i.e., the specific
`conductance), is multiplied by V, the v1>lume in cm8 that
`oontains 1 gram equivalent weight of solute. Hence, the
`equivalent conductance Ac, expressed in units of mho
`cm2/Eq, is given by the expression
`Ac=KXV
`1000 K
`= - - mho cm /Eq
`2
`C
`
`(6_ 20)
`
`(6-21)
`
`If the solution is 0.1 N in concentration, then the
`volum~ containing 1 gram equivalent of the solute will
`be 10,000 cm8, and, according to equation (6-21), the
`equivalent conductance will be 10,000 times as great as
`the specific conductance. This is seen in E:x:ampk 6-1.
`
`balnpla 1-3. The measured conductance of a 0.1-N solution of a
`drug is 0.0563 mho at 25° C. The cell constant at 25° C is 0.600 cm-1•
`What is the specific conductance and what is the equivalent conduc(cid:173)
`tance of the solution at this concentration?
`K = 0.0563 x 0.620 • O.cm93 mho/em
`he= 0.cm93 X 1000/0.1
`= 293 mho cm1/Eq
`Equivalent Conductance of Stran1 and Weak Electrolytes.
`As the solution of a strong electrolyte is diluted, the
`specific conductance K decreases because the number of
`ions per unit volume of solution is reduced. (It some(cid:173)
`times goes through a maximum before decreasing.)
`Conversely, the equivalent conductance A of a solution
`of a strong electrolyte steadily increases on dilution.
`The increase in A with dilution is explained as follows.
`The quantity of electrolyte remains constant at 1 gram
`equivalent according to the definition of equivalent
`conductance; however, the ions are hindered less by
`their neighbors in the more dilute solution and hence
`can move faster. The equivalent conductance of a weak
`electrolyte also increases on dilution, but not as rapidly
`at first.
`Kohlrausch was one of the first investigators to study
`this phenomenon. He found that the equivalent conduc(cid:173)
`tance was a linear functjon of the square root of the
`concentration for strong electrolytes in dilute solutions,
`as illustrated in Figure 6-4. The expression for A,;, the
`equivalent conductance at a C(?ncentration c (Eq/L), is
`Ac = Ao - bVc
`(6-22)
`in which Ao is the intercept on the vertical axis and is
`known as the equivalent conductance at infinite dilu(cid:173)
`ticm. The constant b is the slope of the line for the
`strong electrolytes shown in Figure 6-4.
`When the equivalent conductance of a weak electro(cid:173)
`lyte is plotted against the square root of the concentra-
`
`1 4 0 . . - - - - - - - - - - - - - - - . AHc1
`440
`
`100
`
`80
`A
`
`60
`
`40
`
`20
`
`420
`
`400
`
`380
`
`360
`
`340
`
`320
`
`00
`
`0.1
`
`0.2
`,Jc
`FIi, 8-4. Equivalent conduetance of strong and weak elect.rolytea.
`
`0.3
`
`FRESENIUS EXHIBIT 1057
`Page 7 of 81
`
`

`

`tion, as shown ·for acetic acid in Figure 6-4, the curve
`cannot be extrapolated to a limiting value, and A0 must
`be obtained by a method such as is described in the
`following paragraph. The steeply rising curve for acetic
`acid results from the fact that the dissociation of weak
`electrolytes increases on dilution, with a large increase
`in the number of ions capable of carrying the current.
`Kohlrausch concluded that the ions of all electrolytes
`begin to migrate independently as the solution is
`diluted; the ions in dilute solutions are so far apart that
`they do not interact in any. way. Under these condi(cid:173)
`tions, Ao is the sum of the equivalent conductances of
`the cations le° and the anions la. 0 at infinite dilution
`Ao = lc0 + la.0
`(6-23)
`Based on this law, the known Ao values for certain
`electrolytes can be added and subtracted to yield Ao for
`the desired weak electrolyte. The method is illustrated
`in the following example.
`Eumple 6-4. What is the equivalent conductance at infinite
`dilution of the weak acid phenobarbital.? The A,, of the strong
`electrolytes, HCl, sodium phenobarbital (NaP), and NaCl are ob(cid:173)
`tained from the experimental results shown in Figure 6-,4. The
`values are A,,Hci = 426.2, AoNaP = 73.5, and i\.Naci = 126.5 mho
`cm•/Eq.
`Now, by Kohlrausch's law of the independent migration of ions,
`AoHP = tft + + ti-
`
`and
`A..HCI + A..NaP - AoNaCJ = lft+ + lf:i- + l!lia+. + ti- - t!lia+ - lf:i(cid:173)
`which, on simplifying the right-hand side of the equation, becomes
`AoHCI + AoNaP - AoNaCI = lft+ + IJ-
`
`Therefore,
`
`AoHP = AoHCI + AoffaP - AoJ.iaCI
`
`and
`
`AoHP = 426.2 + 73.5 - 126.5
`= 373.2 mho cm1/Eq
`Colliptm Properties of Electrolytic Solutions and Con(cid:173)
`centrated Solutions of Nonelectrolytes. AB stated in the
`previous chapter, van't Hoff observed that the osmotic
`pressure of dilute solutions of nonelectrolytes, such as
`sucrose and urea, could be expressed satisfactorily by
`the equation, 'II' = RTc, equation (5-34), page 118, in
`which R is the gas constant, T is the absolute temper(cid:173)
`ature, and c is the conee~tration in moles per liter.
`Van't Hoff found, ho~ever, that solutions of electro(cid:173)
`lytes gave osmotic pressures approximately two, three,
`and more times larger than expected from this equa(cid:173)
`tion, depending on the electrolyte investigated. By
`introducing a correction factor i to account for the
`irrational behavior of ionic solutions, he wrote
`'II' = iRTc
`(6-24)
`By the use of this equation, van't Hoff was able to
`obtapi calculated values that compared favorably with
`the Jxperimental results of osmotic pressure. V an't
`Hoff recognized that i approached the number of ions
`into which the molecule dissociated as the solution was
`made increasingly dilute.
`·
`
`Cha:pt.er 6 • Sotuticma of Electrolytu DI
`
`The factor i may also be considered to express the
`departure of concentrated solutions of nonelectrolytes
`from the laws of ideal solutions. The deviations of
`concentrated solutions of nonelectrolytes can be ex(cid:173)
`plained on the same basis as ,deviations of real solutions
`from Raoult's law, considered in the preceding chapter.
`They included differences of internal pressures of the
`solute and solvent, polarity, compound formation or
`complexation, and association of either the solute or
`solvent. The departure of electrolytic SQlutions from the
`colligative effects ~ ideal solutions of nonelectrolytes
`may be attributed-in addition to the factors just
`enumerated-to dissociation of weak electrolytes and
`to interaction of the ions of strong electrolytes. Hence,
`the van't Hoff factor i accounts for the deviations of real
`solutions of nonelectrolytes and electro]ytes, regardless
`of the reason for the discrepancies.
`The i factor is plotted against the molal concentration
`of both electrolytes and nonelectrolytes in Figure 6-'-6.
`For nonelectrolytes, it is seen to approach unity, and
`for strong electrolytes, it tends toward a v~ue equal to
`the number of ions formed upon dissociation. For
`exall)ple, i approaches the value of 2 for solutes such as
`NaCl and CaSO4, 3 for Ka8O4 and CaCia, and 4 for
`KaFe(C)6 and. FeC13•
`The van't Hoff factor can also be expressed as the
`ratio of any colligative property of a real 1JOlution to that
`of an ideal solution of a nonelectrolyte, since i repre(cid:173)
`sents the number of times greater that the colligative
`effect is for a real solution (electrolyte ·or nonelectro-
`lyte) than .for an ideal nonelectrolyte.
`·
`The colligative properties in dilute solutions of elec(cid:173)
`trolytes- are expressed on the molal scale by the
`equations
`
`Ap = 0.018ipi°m
`'II'= iRTm
`AT1 = iK1m
`A.T1, = iK.1,m
`
`(6-25)
`
`(6-26)
`
`(6-27)
`
`(6-2.8)
`
`5
`
`•
`
`2
`
`1
`
`NaCl
`
`SucroM
`
`00
`
`1
`
`2
`MolalltJ
`flt. 8-5. V an't Hoff i faeto.r of repreeentative compounds.
`
`3
`
`FRESENIUS EXHIBIT 1057
`Page 8 of 81
`
`

`

`130 Physical Pharmacy
`
`Equation (6-26) applies only to aqueous solutions,
`whereas (6-26), (6-27), and (6-28) are independent of
`the solvent used.
`Example 6-5. What is the osmotic pressure of a 2.0-m solution of
`sodium chloride at 20° C?
`The i factor for a 2.0-m solution of sodium chloride as observed in
`Figure 6-5 is about 1.9.
`1f = 1.9 X 0.082 X 293 X 2.0 = 91.3 atm
`
`ARRHENIUS THEORY OF ELECTROLYTIC DISSOCIATION
`During the period in which van't Hoff was developing
`the solution laws, the Swedish chemist Svante Arrhe(cid:173)
`nius was preparing his doctoral thesis on the properties
`of electrolytes at the University of Uppsala in Sweden.
`In 1887, he published the results of his investigations
`and proposed the now classic theory of dissociation. 1
`The new theory resolved many of the anomalies
`encountered in the earlier interpretations of electrolytic
`solutions. Although the theory was viewed with disfa(cid:173)
`vor by some influential scientists of the nineteenth
`century, Arrhenius's basic principles of electrolytic
`dissociation were gradually accepted and are still
`considered valid today. The theory of the existence of
`ions in solutions of electrolytes even at ordinary
`temperatures remains intact, aside from some modifi(cid:173)
`cations and elaborations that have been made through
`the years to bring it into line with certain stubborn
`experimental facts.
`The original Arrhenius theory, together with the
`alterations that have come about as a result of the
`intensive research on electrolytes, is summarized as
`follows. When electrolytes are dissolved "in water, the
`solute exists in the form or ions in the solution, as seen
`in the following equations
`H2O + Na+c1-
`- Na+ + c1- + H2O
`[Ionic compound]
`
`.[Strong electrolyt!i!]
`- Hao+ + c1-
`
`H2O + HCl
`[Covaient
`compound]
`
`[Strong electrolyte]
`H2O + CHaCOOH ~ Hao+ + CHaCOO(cid:173)
`[Covaient
`compound]
`
`(6-29)
`
`(6-30)
`
`(6-31)
`[Weak electrolyte]
`The solid form of sodium chloride is. marked with +
`and - signs in reaction (6-29) to indicate that sodium
`chloride exists as ions even in the crystalline state. If
`electrodes are connected to a source of current ana are
`placed in a mass of fused sodium chloride, the molten
`compoUJ1d will conduct the electric current, since the
`crystal lattice of the pure salt consists of ions. The
`addition of water to the solid dissolves the crystal and
`separates the ions in solution.
`
`Hydrogen chloride exists essentially as neutral mol(cid:173)
`ecules rather than as ions in the pure form, and does not
`conduct electricity. When it reacts with water, how(cid:173)
`ever, it ionizes according to reaction (6-30). H3O+ is
`the modern representation of the hydrogen ion in water
`and is known as the hydronium or oxonium ion. In
`addition to H3O+, other hydrated species of the proton
`probably exist in solution, but they need not be
`considered here. 2
`Sodium chloride and hydrochloric acid are strong
`electrolytes because they exist almost completely in the
`ionic form in moderately concentrated aqueous solu(cid:173)
`tions. Inorganic acids such as iICl, HNO3, H2SO4, and
`HI; inorganic bases as NaOH and KOH of the alkali
`metal family and Ba(OH)2 and Ca(OH)2 of the alkaline
`earth group; and most inorganic and organic salts are
`highly ionized and belong to the class of strong
`electrolytes.
`Acetic acid is a weak electrolyte, the oppositely
`directed arrows in equation (6-31) indicating that an
`equilibrium between the molecules and ions is estab(cid:173)
`lished. Most organic acids and bases and some inorganic
`compounds, such as HaBOa, H2CO3, and NH4OH,
`belong to the class of weak electrolytes. Even some
`salts (lead acetate, HgC12, Hgl, and HgBr) and the
`complex ions Hg(NH3)2 +; Cu(NH3)l+, and Fe(CN)&3-
`are weak electrolytes.
`Faraday applied the term ion (Greek: wanderer) to
`these species of electrolytes and recognized that the
`cations (positive,ly charged ions) and anions (negatively
`charged ions) were responsible for conducting the
`electric current. Before the· time of Arrhenius's publi(cid:173)
`cations, it was believed that a solute was not spontane(cid:173)
`ously decomposed in water, but rather dissociated
`appreciably into ions only when an electric current was
`passed through the solution.
`Drugs and Ionization. Some drugs, such as anionic and
`cationic antibacterial and antiprotozoal agents, are
`more active when in the ionic state. Other compounds,
`such as the hydroxybenzoate esters (parabens) and
`many general anesthetics, bring about their biologic
`effects as nonelectrolytes. Still other compounds, such
`as the sulfonamides, are thought to exert their drug
`action both as ions and as neutral molecules. 3
`Degree of Dissociation. Arrhenius did not originally
`consider strong electrolytes to be ionized completely
`except in extremely dilute solutions. He differentiated
`between strong and weak electrolytes by the fraction of
`the molecules ionized: the degree of dissociation a. A
`strong electrolyte was one that dissociated into ions to
`a high degree and a weak electrolyte one that dissoci(cid:173)
`ated into ions to a low degree.
`Arrhenius determined the degree of dissociation
`directly from conductance measurements. He ·recog(cid:173)
`nized that the equivalent conductance at infinite dilu(cid:173)
`tion A0 was a measure of the complete dissociation of
`the solute into its ions and that Ac represented the
`number of solute particles present as ions at a concen(cid:173)
`tration c. Hence, the fraction of solute molecules
`
`FRESENIUS EXHIBIT 1057
`Page 9 of 81
`
`

`

`ionized, or the degree of dissociation, was expressed by
`the equation•
`
`(6-32)
`
`in which A,/A0 is known as the conducf,ance ratio.
`Example B-B. The equivalent conductance of acetic acid at 25° C
`and at infinite dilution is 390. 7 mho cm2/Eq. The equivalent conduc(cid:173)
`tance of a 5.9 x 10-3 M solution of acetic acid is 14.4 mho cm2/Eq.
`What is the degree of dissociation of acetic acid at this concentration?
`a= 3~~ = 0.037 or 3.7%
`The van't Hoff factor i can be connected with the
`degree of dissociation a in the following way. The i
`factor equals unity for an ideal solution of a nonelectro(cid:173)
`lyte; however, a term must be added to account for the
`particles produced when a molecule of an electrolyte
`dissociates. For 1 mole of calcium chloride, which yields
`3 ions per molecule, the van't Hoff factor is given by
`i = 1 + a(3 - 1)
`(6-33)
`or, in general, for an electrolyte yielding v ions,
`i = 1 + a(v - 1)
`(6-34)
`from which is obtained an expression for the degree of
`dissociation,
`
`i-1
`a= - -
`v-1
`The cryoscopic method is used to determine i from the
`expression
`
`(6-35)
`
`(6-36)
`
`or
`
`(6-37)
`
`ll.Tt
`.
`t= - -
`K1m
`Eamp/1 B- 7. The freezing point of a 0.10-m solution of acetic acid
`is -0.188" C. Calculate the degree of ionization of acetic acid at this
`concentration. Acetic acid dissociates into two ions, that is, v = 2.
`i =
`0.188 = 1.011
`1.86 X 0.10
`a = i - 1 = 1.011 - 1 = 0.0ll
`2-1
`v-1
`In other words, according to the result of E:eample
`6- 7 the fraction of acetic acid present as free ions in a
`0.10-m solution is 0.011. Stated in percentage terms,
`acetic acid in 0.1 m concentration is ionized to the
`extent of about 1 %.
`
`THEORY OF STRONG. ELECTROLYTES
`Arrhenius used a to express the degree of dissocia(cid:173)
`tion of both strong and weak electrolytes, and van't
`Hoff introduced the factor i to account for the deviation
`of strong and weak electrolytes and nonelectrolytes
`
`Chapter 6 • SolutionB of Electrolytes 131
`
`from the ideal laws of the colligative properties,
`regardless of the nature of these discrepancies. Accord(cid:173)
`ing to the early ionic theory, the degree· of dissociation
`of ammonium chloride, a strong electrolyte, was calcu(cid:173)
`lated in the same manner as that of a weak electrolyte.
`
`Example B-8. The freezing point depression for a 0.01-m solution of
`ammonium chloride is 0.0367° C. Calculate the "degree of dissocia(cid:173)
`tion" of this electrolyte.
`
`0.0367° = 1.97
`i = l:!.Tt =
`1.86 x 0.010
`K1m
`a = 1.97 - 1 = 0.97
`2 - 1
`The Arrhenius theory is now accepted for describing
`the behavior only of weak electrolytes. The degree of
`dissociation of a weak electrolyte can be calculated
`satisfactorily from the conductance ratio A,/ A0 or
`obtained from the van't Hoff i factor.
`Many inconsistencies arise, however, when an at(cid:173)
`tempt is made to apply the theory to solutions of strong
`electrolytes. In dilu~ and moderately concentrated
`solutions, they dissociate almost completely into ions,
`and it is not satisfactory to write an equilibrium
`expression relating the concentration of the ions an~
`the minute amount of undissociated molecules, as is
`done 1br weak electrolytes (Chapter 7). Moreover, a
`discrepancy exists between a calculated from the i
`value and a calculated from the conductivity ratio for
`strong electrolytes in aqueous solutions having concen(cid:173)
`trations greater than about 0.5 M.
`For these reasons, one ·does not account for the
`deviation of a strong electrolyte from ideal nonelectro(cid:173)
`lyte behavior by calculating a degree of dissociation. It
`is more convenient to c1Jnsider a strong electrolyte ·as
`completely ionized and to introduce a factor that
`expresses the deviation of the solute from 10()% ioniza(cid:173)
`tion. The activity and osm.otic coefficient, discussed in