HYDROGEN-ION CONCENTRATION FROM THE THERMODYNAMIC AND ELECTROCHEMICAL POINT OF VIEW Theoretical Discussion Sorenson's method of expressing the concentration of hydrogen ions as pH, or the logarithm of the reciprocal of the actual concentration of hydrogen ions, has proved to be a convenient system to designate the strength of weak acids and bases, and also a very useful means of studying the properties of industrial and physiological solutions. The principles involved in expressing the concentration of hydrogen ions in pH and converting the latter back into numbers of normality may be briefly illustrated by the following examples:'
- - -= 2N
2
1000
5w
pH = log 2m,
laN 1,000,000
10'
x
lo-' = 0.002 N;
1
=
10" log -= log lo5 - log 2 = 3 2
18 X lo-' = 1.8 X
loS
= log-= log 106 1.8 X 1.8 To convert pH back into normality:
PH = log
PH
=
2 = log
- 0.301 = 2.699
-x1 =-log
x, log r = -2, 1
Again pH = 2.699 = log-x = -log
I.,
iog 1.8 = 4.7447.
z = 10-9 = 0.01. log x = -2.699,
but the abscissa of the logarithm is always positive, therefore we suljtract the- decimal part from 1 and add to the characteristic -1, then log z or 0.002. For ordinary purposes the pH values of =3.301; x = 2 X solutions may be approximately determined by the use of proper indicators.' In spite of its convenience, this twisted method of expression would perhaps lose much weight for its justification if it were not for the energy relation which it indicates clearly. In explaining this relation from thermodynamic and electrochemical principles, the treatment of the subject, while far from being rigorous or exhaustive, is believed to be comprehensive enough to convey a fair understanding of the ionic functions of solutions 1 R.H. Ashley, "Hydrogen-Ion Concentration," THIS JOWAL, 5,1647-58 (Dec.. 1928).
and dispel a good deal of mystery with which this branch of study is surrounded in the minds of students. Gaseous elements that ionize in a solution behave like metals. Hydrogen gas is an example. If we dip a plate of platinized platinum (having a spongy surface) in a solution containing hydrogen ions and bubble a steady current of hydrogen gas under one atmospheric pressure around the Hz plate, hydrogen gas will be adsorbed on the rough surface of the plate until the latter is saturated. Under this condition the plate will act as a hydrogen electrode: either atoms of hydrogen will . . - ~ from the plate into the solution as ions, leaving .-P. . -..- pass electrons on the plate, if the solution is weak with .. . - -. . @ .y .; : respect to hydrogen ions; or the ions will deposit on . ..- . -.- -~ - ---. . . - -. . the plate, rendering the latter positively charged, if the solution is strong with respect to hydrogen FIG& 1 ions. In either case there will be a potential difference between the plate and the solution which may be measured as a single electrode potential. This potential difference or electrical energy may be calculated tbeoretically from thermodynamic considerations. The fundamental principle is that the ions in dilute solutions obey the gas law, acting as if a gaseous mass were compressed into the volume of the solution:
-
PV
=
RT,
(1)
P means ionic pressure or concentration, V = the volume of the solution, T = absolute temperature, and R = gas constant. Let us imagine that a limited number of hydrogen ions pass from the electrode into the solution. The volume of the solution will be practically the same, but the pressure (or concentration) of the ions will be increased by an infinitesimal amount, which may be indicated by the &pression dP. The ions passing into the solution (or into the atmosphere in a gaseous state, as the case may be) do a certain amount of work dW. This change in energy will be: dW = VdP
(2)
but from the gas law we have v = -RT P
(3)
Substituting the value of V in (2) we get RT dW=-dP=RTP
dP P
(4)
Evidently this &pression lies within the domain of simple differential calculus, which may be integrated within definite limits.
HYDROGEN-ION CONCENTRATION
VOL.6. NO. 10
1661
Those who are not familiar with differential and integral calculus naturally feel handicapped when they see such expressions, but they may take for granted certain conventions in the study of the elements of physical chemistry.= Infinitesimal quantities that approach certain limits are marked by d before the symbol of quantity, and the expressions such as d P are called differentials. The differentials dE, dP, etc., may he integrated to their proper quantities according to certain rules. When a differential expression is to be integrated, it is marked with a symbol C (a constant) ; when it is to be integrated of integration as f dE = E between desired limits the process is called definite integration and the limits are marked above and below the integration symbbl.
+
Coming back to o w equation (5) we see that our differential is divided by its own whole quantity dP/P. The integral of such expressions is always equal to the natural logarithm or 2.3026 times the common logarithm of the quantity, thus:
J!!P
=
+
I ~ P C (a constant)
The expression, ln, stands for natural or Naperian logarithm and C for some constant dependent on the case. But the integration in our case is within definitelimits, namely, within the limits of the initial and finalconcentration. The symbol of integration shows the limits
/s.$,
-. PI
where ~ ~ a n d ~ ~ r e p r e s e n t
~
respectively the strong and weak solutions. In the process of definite integration, the constant C is cancelled. It follows that the definite integration of (5) will give W
= R T Y P$
PI dP = RT (InP, - In?'*) = RTlnP P,
(6)
The osmotic pressure of the hydrogen ions is determined by their concentrations; therefore PI and Pzrepresenting the pressure of the ions, may be substituted by C, and Cz representing their respective concentrations: a Those without adequate preparation in mathematics are referred to "Mathematical Preparation for Physical Chemistry," Farrington Daniels, McGraw-Hill & Co., 1928.
If the stronger solution is a normal solution (CI = I), our equation will be further simplified, 1 W = RT(8) C2
As we usually deal with common logarithms, we may convert the natural log into common log by multiplying the expression with the constant 2.3026, thus: 1 W = RT 2.3026 log (9) C*
But by definition log l/Ca = pH, accordingly equation (9) becomes This means that the energy of the system may be measured and pH or log l/x (where x = concentration of the unknown solution) may be determined directly. W = electrical work = volts X current. I n the transfer of one gram atom of hydrogen ions one Faraday (or 96,500 coulombs) of electric current is used, therefore, W = volts X 96,500 = RT 2.3026 pH (11) Electromotive force (potential difference) or volts = e. m. f. = E and
the number of volts we read on the potenfiometer; T is the absolute temperature under which the experiment is performed. As long as the measurement is taken in electrical units, R must be converted into joules (mechanical equivalent of electrical energy). This is done from our fundamental equation,
P is the pressure in atmospheres, and V is the volume in cubic centimeters, molar volume being 22,400 cc. One atmosphere pressure is equal to the weight of a mercury column 76 cm. high and 1 sq. cm. in cross-section, the specific gravity of mercury being 13.596. One atmosphere = 76 X 13.596 = 1033.3 grams. Convert this into energy units, ergs, by multiplying it with the gravitational constant, 1033.3 X 980.6 = 1,013,254 ergs, and finally convert it into joules by dividing by 10,000,000 (number of ergs in one joule). Now taking the value of PVIT,
Therefore the electromotive force or the voltage, E, between the limits of our concentration a t 25' will be:
1 If CI = 0.1 normal, pH = log - = log 10 0.1
=
1
Then the voltage between N / 1 0 and N limits will be E = 0.059 X 1 = 0.059
(14)
This means that for each ten-fold change in the concentration of a solution, change in energy will be 0.059 volt a t 25OC. If the concentration is unknown, the potential differencebetween the hydrogen electrode and the solution may be measured as E = 0.059 pH. Now E being known PH=-=
0.059
E X 16.9
(15)
If E is 0.2 volt, pH = 0.2 X 16.9 = 3.38, and if we wish to convert this value to figures of normality we have pH = log 1 / x = 3.38, -log x = 3.38, log x = -3.38; subtract 0.38 abscissa from 1 and add -1 to -3, then log x = 3.62 = 6.62 - 10; x = antilog 3.62 = 4.169 X lo-&normal. The Method of Measurement To measure the single potential between an electrode and the solution of its ions involves difficulties and requires great skill. Comparatively, it will be a simple matter to balance an unknown solution against a standard solution of hydrogen ions, say, a normal one. Let each solution have its hydrogen electrode in the form of platinsaturated with ized h~hogengas under atmospheric pressure; the two solutions may be connected through a partition or by a bridge or inverted ~ - containing ~ ~ one of the solutions with cotton
HZ
?!I A
6
Froma 2.-S~ownrc METHODOP MEAS&MENT OP HYDROGEN-ION CONCBNTRATION A-Reference solution and spongy platinum electrode; B-unknown solution and its electrode; V-voltmeter. The bridge connecting two solutions may contain one of the solutions with cotton plugs at the two openings, or may be prepared irom a satub rated potassium ~ chloride solution and about six per cent agar while hot, which solution on cooling will be rigid.
plugs to prevent convection or siphoning. The electrodes may be connected externally through a voltmeter. Under these conditions there will be a potential difference recorded, which is the result of the algebraic sum of the potential difference between each electrode and its solution and also a t the liquid junction between the two solutions. The potential at the liquid junction, being very small, may be neglected for our purpose. We may write our two potentials separately as: RT F RT E. = --lnC
El
= -- lnC, = 0 . 0 5 9 log C,, = -0.059
F
log Ca
(1) (2)
El is the potential of the reference solution CI and Eiis that of the unknown solutiou Cz. Subtract (1)from (2).
E2
=
c, + 0.059 log -
EL
c 2
(4)
If C1 is normal, its potential is 0.27 volt. We read the value of E2 - El or 0.059 log CJC2 on the voltmeter, and then the potential of the unknown solution may be determined by adding the potential of the standard or reference solution (0.27 volt in.case of one normal in hydrogen ions) to our reading. It is almost impossibb to prepare a reference solution exactly normal in hydrogen ions. Various stable buffer solutions are prepared and their hydrogen-ion concentration is determined, which may serve as a reference solution, but all of them need checking against the most stable calomel cell. For the most exact measurement a potentiometer is used instead of a voltmeter. Any ordinary variable dry or wet battery with a rheostat is used to balance against a standard Weston cell, in order to secure a reliable direct reading on the potentiometer. After this balancing, the switch is thrown from the Weston cell to hydrogen-calomel cells without changing the position of the battery and rheostat, and a direct reading of E2 - El is obtained. 1
Now EX - El = 0.059 log - = 0.059 p H
c,
but the value El for saturated calomel cell is known as 0.246. Subtract this value from the potentiometer reading and divide by 0.059 to get the pH of the unknown solution as
VOL. 6, No. 10
HYDROGEN-ION CONC~NTRATION
1665
The following diagram shows the make-up of the calomel cell and the setting of the apparatus for measurement: +-
*-
potentiometers have a C-Calomel cell special place for galvan&-Mercury b-Calomel paste ometer) c,d-SaturatedKClsolution R-Rheostat e-Unknown solution with G-Galvanometer platinum-hydrogen elec- B-Dry or storage haltttedes S-Switch trode P-Potentiometer (some W-Standard Weston cell *
a'-Mercury b'-Paste of cadmium and mercurous sulfate 6'-CdHg amalgam d'-Small crystals of CdSOl e ' S a t u r a t e d solution of C ~ S Oand I large crystals of the same
If the reading of the potentiometer is 0.480 volt, the value of pH is 0.480 = 0.059 pH. P H = -0'234 - 3.967
- 0.246
0.059
The hydrogen electrode is somewhat tedious to manipulate and offen fails to work satisfactorily for various reasons. The platinum must be platinized first by electrolysis in a solution of platinic chloride for about 20 seconds. After this the electrode must be washed and saturated with hydrogen by electrolysis in a dilute sulfuric acid solution for about 20 minutes. After washing again it is ready to use. It is claimed that the electrode can be used for about 20 measurements. All this trouble and waste of time and material may be eliminated by using a quinhydrone electrode, which is much simpler and also reliable.% In this case, instead of using hydrogen gas a pinch of quinhydrone is used around the platinum electrode, and it is not necessary to platinize the lata Above about pH 8 the quinhydrone electrode is not so reliable: the first reading on the potentiometer is probably nearest to the potential of the system.
ter. The equilibrium between hydrogen of quinhydrone and hydrogen ion of the solution is reached in a very shorttime and stays fairly constant. The equilibrium is illustrated as follows:
E:i\,
C-OH
&O Quinone
Hydroquinoue
A strong acid solution will shift the equilibrium toward the right. Even the calomel half cell as a reference electrode may be eliminated, and in its place a M/20 solution of potassium acid phthalate may be used with a platinum electrode and a little qninhydrone. The pH value of the latter solution is definitely known as 3.98 at 25'C. The set? ,060 ting of apparatus for quinhydrone electrode is the same as given "i.02: above, with the elimination of hy,060 .loo drogen gas. In case acid phtha,140 late is used as a reference solution, ,180 the bridge joining this with the ,220 .ZGO I-1 unknown solution may contain 1 2 3 4 5 0 7 8 half saturated potassium chloride pH solution, . prepared with about five . FIGURE 4 per cent agar, which, on cooling, Ordinat-Readingson the potentiometer. Abscissa-pH of solutions determined will gelatinize the content of the aeainst M / 2 0 acid ohthalate usine -auinhv. . b r i d ~ eand make it a zood condkne around the pl&inum electrodes. Gold ductor, ~h~ bridge must be kept may be used instead of platinum. Above 0 volts and below pH 3.98 the in a saturated potassium chloride potentialis positive, below 0 volts and above solution, pH 3.98 it is negative. The method of calculation of pH of an unknown solution against acid phthalate solution at 25' is asfollows: Reading on the potentiometer, E = 0.059 X 3.98 = -0.059 pH (unknown). Suppose the reading is 0.070 volt (unknown being on the negative side): -0.070-0.059 X 3.98 = -0.059 pH
-
pH
=
-
305 - = 5.17 59
If the unknown solution is positive against acid phthalate solution (which means greater hydrogen-ion concentration in the unknown), then our equation will be:
Vol.. 6, No. 10
H~ROGEN-ION CONCENTRATION
1667
If the concentration of hydrogen ions in the unknown solution is the same as in the reference solution of M/20 acid phthalate, the potential difference between the two electrodes will he zero, because the number of hydrogen atoms passing from each electrode into the surrounding solution as ions will he the same, and therefore the number of electrons given to each electrode will be the same. But if one of the solutions is weaker, in a given time more hydrogen atoms will pass in ionic state into that solution, causing more electrons to accumulate on its electrode; the pressure of electrons being greater on this electrode, the negative electricity (electrons) will flow from it through the wire and potentiometer into the opposite electrode, and the negative ions from the strong solution will migrate toward the electrode of the weak solution. This means that the weaker solution will have the negative electrode while the stronger will have the p o s i t i ~ e . ~ The curve given below is plotted from the potential difference between M / 2 0 potassium acid phthalate and solutions of various hydrogen-ion concentration. pH values may be read directly against volts or e.m. f. readings of a potentiometer, whether negative or positive. According to the old idea positive electricity entered from the electrode into the weaker solution, hence the more negatively charged electrdie was called anode, while the other was named cathode.
