pH APPROXIMATIONS ARTHUR J. MeBAY Massachusetts College of Pharmacy, Boston, Massuchusetts
THEpH
of solutions may be approximated by calculations. The data necessary for many of these calculations are given in the tables included in this paper. Such calculations are not to be used in place of actual pH determinations, but they may lead to an TABLE A Dissociation Values of Acids
A A' 2.4 .
Acetin
Hypophosphor-
A A' 1.0 . . .
tlorir Caeodylic Camphoric Caproir Caprylic Carbonic Chlorie Cinnarnir Diethvlharbituric, ~iph&ylscetic Formic Gallic Glutamie Glutwic Glvcine. Glycolic Hioouric ~ikkline Hyd~iodic Hydrahromic livdrochloric
1.5 2.2
2.2 4.9 2.2
. .. . .. . ..
C 3.1 4.9 . . . 1.9 . . .
Propionic Sacchan'n Salicylic Stsnnic Succinic Sulfnnilic Sulfuric
understanding of some of the problems of pH, and may sometimes help in the solution of these problems. In general, it may be stated that the formulas presented in this paper do not apply to extremely dilute or extremely concentrated solutions. In the former the dissociation of water is a major factor, and as the concentration becomes less than molar, the pH approaches 7.0. In the latter the activities of the ions, which are usually negligible, have a great influence on the pH. The approximations are very good for pure chemicals if the molar concentration is not greater than one molar or less than 1 0 P molar or if the pH calculation lies between 2.0 and 12.0. Many of the compounds on the market contain free acid or free base, with the result that they may have a more desirable pH or that they may be more stable. Since the theoretical pH is for the pure chemical, preparations that contain free acid or free base will have a pH different from that obtained by calculation based on a pure substance. Sodium salicylate, a salt of a weak acid and a strong base, should theoretically have a pH greater than 7.0. The United States Pharmacopeia states that this salt is neutral or slightly acid to litmus; and the pH is given as 5.0 to 6.0. The calculated pH of a 0.1 molar solution of sodium salicylate is 8.0. The measured pH of a 0.1 molar solution of a sample labeled Sodium Salicylate U.S.P. was 6.7. DEFINITIONS
As an aid in the understanding of the formulas of this paper the following definitions and symbols are presented a t the beginning. = 'l%pK, = -I/. log K,, where K. is the dissociation constant for the first stage of ionmation of the acid. K.' and K." indicate. resoectivelv. the second and the third &gm of hnization oi the'aeid. A ' and A " designate, respectively, the A value of K.' and K.". B = 'hpKa = -'/r log Kb, where Ks is the ionization constant of the base. C = '/2pc = - - ' / a log c, where e is the molar concentration. c., cs, and c. designate, respectively, the molar concentration of the acid, base, and salt. C, Ca, and Cs.designate, respectively, the C value of the acid, base, and salt,
A
".
for 1st stage; A' for 2nd stage and A" for 3rd
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OCTOBER, 1952
Therefore,
TABLE B Dissociation Values of Bases 8
B
B
-4eetanilid Aeonitine Alanine Aminopyrine. Ammonium hydroxide Aniline .4pomorphine Arsenoua oxide Atropine Barium hydroxide Caffeine Cinchmidine Cinchonine Calcium hydroxide Cocaine Codeine Colehieine Coniine Emetine Ephedrine E~erina
6.7 2.9 5.6 4.6 2.4 4.7 3.5 2.0 2.2 C 1.7 2.9 2.9 C 2.8 3.0 6.2 1.5
Ethylenediamine Ethylmorphine Glutamio acid Glyeine Histidine Hydrastine Hydroquinone L&d hydroxide
2.0 3.1 5.8 5.8 4.0 3.9 2.7 1.5
B
=
'/%
Leucine Lithium hydroxide Magnesium hydroxide Mor~hine
5.8 C
C
3.1
eon
Piperidine Piperine Procaine Pyridine Potrts~iumhydroxide Quinidine Quinine Quinoline Silver hydroxide Sodium hydroxide Sparkine Strontium hydroxide Strychnine Theohromine Thiaeole Thiourea. Urea Veratrine Xanthine Zinc hydroxide
3.6 1.4 7.0 2.6 4.4
C
2.7 3.0 1.6 2.0
C
1.1
C
3.0 6.7 5.7 7.5 6.9 2.6 6.7 2.2
pK, = -lag K,, where K, is the ionic product of water. pH = 14 - pOH = -log ent, where cnf is the molar concentration of the hydrogen ions. pOH = -log con-, where con- is the molar concentration of the hydroxyl ions. pI = Isoeleet,ric point.
Weak acids and weak bases do not dissociate completely. I n order to approximate the pH of these the dissociation constant must be known and must be considered in the calculation. Tables A and B list respectively the A and B values of some of the important acids and bases. These values may be substituted directly into the formulas. The dissociation values of chemicals not listed in these tables may be obtained from Handbooks and "International Critical Tables" ( 1 ) . If the values in the literature are given in the form -log K or pK, they may be converted to A or B values by dividing them by two. If the values are given as exponential expressions, Table C is used t o convert them to A or B values. The conversion factors given in Table C may be used to convert exponential expressions to the form required in the formulas. The basic formulasused in the development of the simplified formulav presented in this paper may be found in. textbooks (3,3). The simplified formulas are summarized in Table D. Weak Acids. For a weak acid the hydrogen ion concentration and the pH may be calculated from the following equations: en+ = (eK.)'A
Taking negative logs, -1
pH = C
+A
=
0.5
+ 4.6 = 5.1
Values: U. S. P., 5.1
Measured, 5.4
Weak Bases. Similarly wit,h a weak hme it can be shown:
pKb for the hase.
-log ca+ =
Example: The pH of 0.1 M boric arid is calculated:
1. log c - 'Izlog K o
-log
=
(rKa)'/z
COE-
pOH = C
=
-I/%
log c -
I/.
log K I
+B
pH=14-B-C
Example: The pH of 0.1 M ammonia water is calculated: pH = 14 - B - C = 14 Values:
- 2.4 - 0.5
U. S. P., 11.3
=
11.1
Measured, 11.1
Strong Acids. To approximate the pH of solutions of strong acids or of strong bases, the dissociation may be considered to be complete. The hydrogen ion concentration of a strong acid is considered to be equal to the molar concentration of that acid. The pH of the acid is equal to the negative logarithm of the hydrogen ion concentration. pH
=
-log cn+
For 0.1 M hydrochloric acid CH+ = 1 X 10-I This calculation may be accomplished in another manner. Since, -'/zlog c = C, then -log c = 2C. The formula for the pH of a weak acid given previously could also be used. pH=C+A
TABLE C Conversion Factors
This table mny he used to convert c to C, K. to A, and Ks to B. Example: 0.0024 = 2 . 4 X lo-=. Read across the row to 2 . 0 3 . 1 , then down this column t o the row designated lo-'. The answer is 1.3.
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JOURNAL OF CHEMICAL EDUCATION
If with a strong acid the A in this formula equals C, then the formula for a weak acid may be used. The A values of strong acids in Table A are given as equal to C. For the nroblem above refer to Table A: the value for hydrochoric acid is given as C. By referring to Table C, 1 X lo-' is seen to have a numerical value of 0.5. By substitution then, pH = C
+A
C
=
+ C or 2C = 2(0.5) = 1.0
Values: U. S. P., 1.0
The calculation for the hydrochloric acid problem above is simple enough to be easily handled without resorting to the formula or the tables. Assume that the acid is 0.16 M then the c,+ is.l.6 X lo-', pH = -log (1.6) -log (lo-') = -0.2 1.0 = 0.8. From Table A, A equals C, and from Table C, 1.6 X lo-' equals 0.4. Then
+
pH = 2C = Z(0.4) = 0.8 Value:
pH = -log (K,'K,')'/z pH = A'
+ A"
~ ~~h~ p~ ~of 0.1 M disodium ~ ~ l hydrogen phosphate is calculated: pH = A'
+ A" = 3.6 + 6.0 = 9.8
Values: U. S. P., 9.2
Measured, 1.2
+
Similarly an acid salt of the type MzHA
Measured, 1.0
a
Measured, 9.1
Salt of a Strong Base and a Weak Acid. The salts of strong bases and weak acids hydrolyze to give a basic solution. The reaction of the anion of a weak acid with water to form the undissociated acid and hydroxyl ion makes this solution basic. pH = '/rpK, pH = 7.0
+ l/npK. + '1%log c
+A - C
Example: The pH of 0.1 M potassium cyanide is calculated:
Strong Bases. For strong bases it follows that p~~ = - log coa-
=
The formula for weak bases may be used instead of the above formulas by making B in Table B equal to C. Example: The pH of Milk of Magnesia. U. S. P. is cdculated: For pH approximationthis suspension msy be considered a saturated solution of magnesium hydroxide. The solubility of magnesium hydroxideis 1.5 X lo-' males per liter. From Table B,R equals Cand 1.5 X 10-'is found to beeaual to 1.9fromTeble C pH = 14 - B - C = 14
- C - C = 14 - 1.9 - 1.9 = 10.2
Values: U. S. P.. 10.6
Measured, 10.1
Salts. The approximation of the pH of the end point in a neutralization titration is one of the important applications of these formulas. The pH of a salt that is formed in a neutralization is the same as the pH value of the end point of the titration. Salts of strong acids and strong bases do not undergo hydrolysis. The pH of solutions of such salts may thus be considered to be 7.0. The solutions may be referred to as "neutral." Most other salts, when dissolved, are ionized and dissociated to some degree. The pH produced as a result of hydrolysis may be approximated. The method of making the approximate calculation is given in a later section. Acid Salts. For an acid salt of the type MHA and MHzA pH = -log (K.K,')'/z pH = -'/s pH = A
lag K.
+ A'
-
log K*'
+ A'
Salt of a Weak Base and a Strong Acid. The salts of weak bases and strong acids hydrolyze to give an acidic solution. The reaction of the cation of the weak base with water to form the undissociated base and the hydronium (hydrogen) ion makes this solution acidic.
=
3.2
Values: U. S. P., 8.2
+ 5.1 = 8.3 Measured, 8.2
- '/*pKs -B +C
pH = '/rpIZ, pH = 7.0
log c
Example: The pH of 0.1 M ammonium chloride is calculated: =
7.0 - B
+ C = 7.0 - 2.4 + 0.5 = 5.1
Values: U. S. P., 4.6
Measured, 5.4
Salt of a Weak Base and a Weak Acid. A solution of a salt of a weak acid and a weak base will be considerably hydrolyzed; and if the acid is stronger than the base, the solution will be acidic. Conversely, if the base is strongsr than the acid, the solution will be basic. Amino acids and other ampholytes may be considered to be salts of weak acids and weak bases. For amino acids, the pH of the pure amino acid is also the isoelectric point, the p H a t which the basic and acidic ionizations of an ampholyte take place to the same extent.
+ '/.pK.
pH = ' / % p K , pH = 7.0
- '/zpKs = pI
+ A - B = pI
Example: The pH of 0.1 M ammonium acetate is calculated.: pH = 7.0
Example: The pH of 0.1 M sodium bicarbonate is calculated : pH = A
Vslue: Measured, 10.8
2C thus pH = 14 - 2C
+ A - B = 7.0 + 2.4 - 2.4 = 7.0 Value:
Measured, 6.8
Example: The pH of 0.1 M glycine is calculated: pH = i.0
+A
- B = 7.0
Values: Handbook, 6.1
+ 4.0 - 5.8 = 0.1 Measured, 6.5
~
529
OCTOBER, 1952
Buffers. A buffer solut,ion is one which resists a change of pH upon the addition of acid or alkali. Buffer solutions consist of a mixture of weak acid and its salt, or of a weak base and its salt, or of an acid salt, or of a salt of a weak acid and a weak base. Approximations of the pH of the last two types of salts have been discussed previously. Weak Acid and Its Salt. pH = pK. pH = 2A
+ log(c,/e.) = pK. + log + 2C,
cs
- log c.
- 2C:
Example: The pH of a solution 0.1 M sodium acetate in 0.05 M acetic acid is calculated: pH = 2A - 2C.
+ 2C.
Value:
=
4.8 - 1.0
+ 1.2 = 5.0
Measured, 5.0
-
TABLE D F o m u l a s for pH Calculations
1. Acids 2. Rases
pH=A+C pH=14-B-C
Salts Acid MHAorMHaA Acid MzHA Strong base-wesk acid Strong aeid-weak base Weak bsse-weak acid Buffers 8. Week acid and its salt 9. Weak base and its salt
3. 4. 5. 6. 7.
pH = log Ks
pH=A+At pH = A ' + A' pH=7.0+A-C pH = 7 . 0 - B C pH = 7 . 0 + A - B = PI
+
+
pH = 2A - 2C8 ZC, pH = 14 - 2B - 2Cs t- 2C,
+ log cs - log K,
- log. c.
- 2Ca + 14 + 2C.
pH = -28
pH = 14 - 2B - 2Ca
+ 2C,
Example: The pH of U. S. P. Phosphate Buffer pH 8.0 is calculated:
Example: The pH of a solution of 0.1 M ammonium chloride in 0.05 M ammonium hydroxide is calculated:
By calculation a liter of this buffersolution is found to oontain 3.2 X lo-' moles KHaP04. Thib is considered to be the "acid," (H?PO$, with a C. value of 1.7 and an A value of 3.6. A liter of the buffer is caloulated also to oontain the equivalent of 2.2 X moles NaKHPO,. This is considered to be the salt with a. C, value of 1.3.
pH = 14 - 2B - 2Cs
pH = 2A
+ 2C. - 2C.
Weak Base and Its Salt.
=
7.2
+ 3.4 - 2.6
=
8.0
+ 2C,
Value:
=
14 - 4.8 - 1.2
+ 1.0 = 9.0
Measured, 9.1
LITERATURE CITED (1) "International Critical Tables," vol. 6, p. 259. (2) GLASSTONE, S., "The Elements of Physical Chemist,ry," D. Van Nostrand, Inc., New York, 1949, p. 495. (3) KOLTHOFF, I. M., AND H. A. LAITINEN,"pH and Electro Titrations," 2nd ed., John Wilry & Sons Ino., Ken York, 1941, p. 1.
