Thermometric Titration Curves HUBERT J. KEILY Department
and
DAVID N. H U M E
o f Chemistry and
Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge 39, Mass.
Equations are derived for the relationship between temperature change and volume of titrant added in thermometric titrations. The effects of the heat capacity of the apparatus, the change in volume during titration, and nonadiabatic conditions are derived and discussed. It is shown how heats of reaction, solution, and dilution may be estimated accurately from automatically recorded thermometric titration curves by interpretation of slopes; results obtained are compared with literature values.
T
H E automatic recording of thermometric titration curves was described several years ago by Linde, Rogers, and Hume ( 4 ) . Extension of this work to titrations in nonaqueous media and various refinements in measuring techniques have made it of interest to compare theoretical and observed titration curves and to determine the best method for obtaining fundamental thermochemical data from titration curves measured under ordinary experimental conditions. The usual technique of automatic thermometric titration (4)involves addition of titrant, isothermal with the solution to be titrated, a t a constant rate and measurement of any temperature change in the mixture by means of a rapid-response element such as a thermistor. A Dewar flask and appropriate thermal shielding are employed to make
the system as nearly adiabatic as possible. The speeds of reaction and mixing are such that thermal equilibrium is established immediately in the solution, but because a titration requires only a few minutes for performance, true thermal equilibrium is not reached between the solution and its container. This and the fact that the heat capacity of the system is undergoing constant change due to the addition of titrant makes the temperature t's. volume plot nonlinear. It is useful, first, to consider the effect of nonequilibrium with cell and surroundings by examining n hat happens when a small electric heating coil is used to deliver knobvn amounts of heat to a solution in the titration cell. Because no titrant is being added, the heat capacity of the system is constant. The heat added, dt, during an infinitesimal interval is given by the relation
dH = CpdT = i2R(4.185)-1dt
(1)
where i is current in amperes; R, resistance in ohms; t , time in seconds; H , heat in calories; T , centigrade temperature; and C,, the heat capacity, in calories, of the titration cell and its contents. Accordingly, the relationship between temperature change and time is given by
dT _=dt
i2R 4.185Cp
Temperature rise is linear with time if no heat is lost to the environment. Comparison v-ith a typical curve of T us. t obtained this way under ordinary good experimental conditions reveals that the empirical plots are invariably curved because of heat loss (Figure 1). This might a t fiist seem to be a serious deterrent to use of titration curves for measuring heats of reaction, but, as will be seen, it is an effect which can be eliminated readily by taking the initial slope to estimate a value of C, which applies to the system a t the initial temperature, TI, in equilibrium with its surroundings. TITRATION CURVES
Consider a simple titration in which the titrant is isothermal with the initial solution and the only process contributing to temperature change is the chemical reaction. I n any practical automatic thermometric titration, the reaction must be rapid. Then]
-dH
=
AHTldn = -CpdT
(3)
where AHr, is the isothermal heat of reaction at 7'1 in kilocalories per mole and dn is the increment of titrant in millimoles. Because dn = MdV where V is expressed in milliliters and because V is linear with time t,
Figure 1.
I
1
0
60
I
I
120 180 Time in Sec.
I 240
Comparison of curves of temperature us. time
Heat generated electrically at constant rate 1. Recorded Calculated
1. A .
3
The plot of 7' u s . Tr should then be linear if AH and C, are constant. However, the variation in C , due to added titrant is significant, as may be seen in Figure 2. If we consider C, to consist effectively of two terms-the heat capacity of the liquid and the heat capacity of the Dewar flask, stirrers, and the l i k w we have C, = (cd)V
+ C'
(5)
n-here c is the specific heat of the liquid (assumed constant throughout the titration), d is its density, and C' is the heat capacity of the calorimeter-Le., the titration apparatus. Then, integrating Equation 4 between the initial and final temperatures, 11294
1295
V O L U M E 28, N O . 8, A U G U S T 1 9 5 6
1'2 and T2, and the corresponding volumes in the Dewar, VI and we have for the relationship between the total temperature rise and the volume of titrant Vp,
The nonlinearity of the titration curve caused by changing heat capacity introduces no difficulty in locating end point breaks, because these are very clear and sharp for most reactions. The combined effect of nonlinearity due to variable C, and heat loss xould complicate direct calculation of A H from A T , but it is found that the initial portion (approximately 30%) of most titration curves follows very closely to exact linearity. Therefore, by using the initial slope and Equation 4, both difficulties can be circumvented. Three additional advantages are obtained from the use of the initial slope: Curvature in the vicinity of the end point caused by incompleteness of reaction does not affect the result; errors due to a difference in temperature between the titrant reservoir and the titration solution are minimized, inasmuch as the first part of the titration curve comes from reagent in the buret tip vhich has the greatest chance of being in thermal equilibrium with the solution; and the heat capacity of the initial system, which can be measured readily by means of Equation 2, is appropriate for the calculation.
slopes are very close to the theoretical. Table I1 shows the agreement between the heat of reaction estimated from the initial slope of the titration curves and the literature value, which mas obtained by standard calorimetric techniques (1). Under the appropriate experimental conditions (temperature rise small, titration duration short, system nearly adiabatic) an excellent estimate of the heat of the titration reaction can be obtained from the initial slope of the titration curve. However, the system is not strictly adiabatic, the solution does not reach thermal equilibrium with the container, and the titrant may not be exactly a t the same temperature as the solution. Hence, the ordinary approach of measuring total temperature change, correcting for dilution by the method of mixtures, and correcting for the equilibrium heat capacity of the calorimeter, is not as attractive When the temperature changes involved are very small-sometimes only a few hundredths of a degree-the titration method still gives a usable result although the classical method TTith the same apparatus is unreliable.
HEATS OF REACTION
The application of this technique to measurement of heats of reaction is illustrated by the following data on the heat of neutralization of sodium acetate in glacial acetic acid by anhydrous perchloric acid in glacial acetic acid. The titrations n-ere performed using a motor-driven syringe for titrant delivery and a thermistor in a bridge circuit, capable of detecting changes of 0.001' C. The circuit is designed so that the relationship between unbalance potential and temperature change is essentially linear for changes up to 0.50" C. Temperature changes 17-ere recorded on a 2-mv. Speedomas recorder. The effective heat capacity, C', of the Dewar flask, stirrer, and the like was measured previously under the same conditions by determining the temperature rise of a knon.n quantity of ivater n-lien a measured amount of electrical energy was delivered to it in the form of heat.
T',
311. of 0.5 If HClOh 0.a 1.20 2.50 3.50 4.50 .i .50 5.95 0
u
(Glacial acetic acid solvent) AT, C. Obsd., Obsd.. curve 16 Ca1cd.a curve 2 t 0.048
0.047
0.042
0.134 0.213 0.288 0.353 0.418
0.136 0.228 0.314 0.399
0.127 0.208 0.279
0.442
0.483
am
0.343
0.409 0.433
Obsd., 3b 0.043 0.131 0.215 0.293 0.363 0.433
Ca1cd.b 0.043 0.127 0,209 0.292
0.461
0.485
curie
I
I
Table I. Comparison of Observed and Calculated Temperature Rises in Three Titrations of Sodium ilcetate with Perchloric Acid
00
I ! I 20 40 MI Of 0 5 M HC104
Figure 2.
I 60
Titration curves
1. Experimentally determined using continuous strip-chart recorder; points indicated are taken from eontlnuous record a n d presented In Table I 1, A . Calculated
0.371
0.450
Heat capaclty C'indwendently measured as 2.6 cal. per degree. Heat capacity C' independently measured as 4.8 eal. per degree.
Table 11. Estimation of AH of Reaction Between Sodium Acetate and Perchloric Acid in Glacial Acetic Acid from Thermometric Titration Curves dT Curve 1
Table I s h o w experimental data from three separate titration curves (the data in column 2 are from the curve reproduced in Figure 2) and compares observed temperature rise r i t h that cnlculated from the measured initial heat capacity of the system and the literature value ( 1 ) of the heat of reaction. Although deviations occur after the first 30% of the titration, the initial
Initial dV Measured Theoretical0 0,092 0,092
-AH
T.
5.7 5.6
22.6 23.2 22.2
(Eical. per Mole)
C.
SaOAc,
70 Found
99.9 99.3 99.6 Av. 5.7 99 6 Std. Dev. 0.16 0.30 a Calculated from Equation 4 with Jolly's value of 8.7 kcal. per mole for - A H and independently determined heat capacities of solutions and titration cells. 2 3
0.085 0.088
o ,086
0,086
5.9
1296
ANALYTICAL CHEMISTRY
Titrant and Solution Not Isothermal. I n practical work it is often very difficult to adjust and hold the titrant and the solution to the same initial temperature. If the difference is no more than about half the temperature rise expected in the titration, the error in initial slope, under conditions similar to those in which the sodium acetate-perchloric acid were performed, is less than 5%. I n general, if all symbols have the same meaning as previously, the rate of temperature change due to mixing with titrant of a different temperature, 2'8, is given by
(7)
the same temperature. When the solute is added from the buret to a fixed quantity of solvent, dnl = 0 and (9)
The following symbols and terms are then used, which is essentially the terminology of Le\vis and Randall (S), as modified by mot2 ( 2 ) . Subscripts: 1 = solvent; 2
H H" H' L L'
Correspondinglj.,
H*
dAH
where it is assumed that both the specific heat and density are the eame for titrant and solution. It is seen again that the plot of T us. Vis not linear.
(=)
=
solute.
partial molal heat content = partial molal heat content in infinitely dilute solution = molal heat content of pure substance = relative partial molal heat content, H - If" = relative molal heat content of pure substance, =
8"
= differential heat of solution =
H 2
- H: = LZ -
nl
(g)
= differential heat of dilution = H I
- H:
=
t l
nt
If '11 is the molarity of the pure solute, dnl = MdV, and
0.2
c
/
I
then the initial slope has again a special significance. Here, L z = 8 2 - Zf" = 0 because we start with an infinitely dilute solution in the Dewar, and
Thermometric titration thus provides a simple and direct method for measuring differential heat of solution in infinitely dilute solution. I n the reverse situation, where solvent is added gradually to pure liquid solute, dnz = 0 and
dAH =
(g)
dnl = - CpdT
dAH du CP I
I
,
I
1.0
I
I
I
I
I
I
I
50 MI Added from Buret
I
I
J
10.0
Figure 3. Recorded temperature changes on mixing acetic acid or water with their solutions Temperature scale at right applies t o curve 3 only 1. Acetic acid containing < O . O l % water; added from buret to 50.00 ml. of water 2. $cetic acid containing < O . O l % water: added from buret to 50.00 nil. of acetic acid initially containing 9.38% watef 3. Water; added from buret to 50.00 ml. of acetic acid containing
