Art of Calorim

8/3/2019 Art of Calorim

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Thermochimica Acta 417 (2004) 257273

The art of calorimetry

Lee D. Hansen , Roger M. Hart Department of Chemistry and Biochemistry, Brigham Young University, C100 BNSN, Provo, UT 84602, USA

Received 20 June 2003; accepted 11 July 2003

Available online 10 February 2004

Abstract

An introduction to the various types of calorimeters is given. The requirements for precise temperature measurements with thermistors arederived.Methods and equationsfor accurateheat exchangecorrections for isoperibol temperature-change calorimetry undervariousconditionsare derived. The characteristics and design principles for constant temperature baths are discussed. The construction of devices for additionof reagents, of stirrers, and of calibration heaters is described. 2004 Elsevier B.V. All rights reserved.

Keywords: Calorimetry; Temperature; Thermistor; Heat exchange corrections; Constant temperature baths; Stirrers; Heaters; Calibration

1. Introduction

The science of calorimetry consists of only two laws; thelaw of conservation of energy and the law of heat trans-

fer. The art of calorimetry is the large, and still growing,body of knowledge on the craft required to make accuratemeasurements of heat and rate of heat exchange.

There are only three methods for measuring heat: (1)measurement of a temperature change which is then mul-tiplied by a thermal equivalent (the apparent heat capacity)determined in a calibration experiment; (2) measurementof the power required to maintain isothermal conditions,with the power being supplied either through an electronictemperature controller or by an isothermal phase changein a substance in contact with the calorimeter; and (3)measurement of a temperature difference across a path of xed thermal conductivity, the thermal conductivity beingdetermined in a separate calibration experiment. All of themethods are based on measurement of relative temperaturesexcept the method which uses a phase change, in whichcase the measurement consists of determining the amountof phase change that has occurred and multiplying by theenthalpy change for the phase change. All calorimetric mea-surements thus require at least two separate experiments,

Corresponding author. Tel.: + 1-801-378-2040;fax: + 1-801-422-0153.

E-mail address: Lee [email protected] (L.D. Hansen).

one for the measurement and one for calibration, and mayrequire a third for a baseline (zero) determination.

A reaction calorimeter always has three identiable com-ponents that directly affect the quality of the data: (1) the

calorimeter vessel, including the means for heat measure-ment; (2) the immediate surroundings of the reaction ves-sel which may range from a high precision liquid bath ortemperature-controlled metal block to simply the laboratoryatmosphere; and (3) a means for initiating the reaction whichmay again be complex or may be simply some means forinsertion of a sample. The design of the calorimetric vesseland reaction initiation method are largely determined by thenature of the reaction and the physical properties of the re-actants and products. The choice of a method for heat mea-surement is largely dictated by the time resolution desiredin the measurement. In general, the temperature rise methodhas a shorter time constant than power compensation, whichin turn has a shorter time constant than the heat conduc-tion method. The calorimeter may contain duplicate reactionvessels (twins) with one vessel acting as a blank referencefor the measurements. The blank or reference vessel servesto subtract extraneous effects from the measurements. Thesurroundings may be either isoperibol (i.e. constant T and p) or adiabatic (i.e. controlled to be at the same T as thereaction vessel). Reaction initiation may be done by chang-ing the temperature, pressure, or volume (for example, byscanning the temperature), or by changing the concentrationof a reactant or catalyst. Concentration changes may be ac-complished by mixing the reactants all at once (batch reac-

0040-6031/$ see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.tca.2003.07.023


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258 L.D. Hansen, R.M. Hart / Thermochimica Acta 417 (2004) 257273

tor), by incremental or continuous titration, by mixing in acontinuous ow reactor, or by changing the partial pressureof a gas in the calorimeter vessel. Reactants may be com-pletely contained within the calorimetric vessel at the be-ginning of an experiment or may be added from outside thevessel.

2. Nomenclature

Calorimeter designs have traditionally been named foreither the developer or some unique characteristic, e.g. theBunsen ice calorimeter, the Parr oxygen bomb calorimeter,the Tian-Calvet calorimeter, a titration calorimeter, and a dif-ferential scanning calorimeter. Such names were appropriatebefore the advent of commercially available calorimeterswhich are often acquired and operated by people with littlehistorical knowledge of the eld of calorimetry, but contin-ued use of such nomenclature other than for historical pur-poses is detrimental to further development of calorimetry.A simple and widely acceptable systematic nomenclaturefor calorimeters such as suggested in [1] would help bothnovices and long-time devotees of calorimetry to better orga-nize their knowledge of and access to the subject. The scien-tic name of a calorimeter should consist of four parts, withthe rst three describing one of the necessary componentsof the calorimeter. These are, in order, (a) the surroundings(either isoperibol or adiabatic), (b) the principle of heat mea-surement (either temperature-change, power-compensation,or heat-conduction), (c) the method of reaction initiation(e.g. batch, incremental titration, continuous titration, ow,

temperature scanning, pressure scanning), and (d) other de-scriptors. The fact that many calorimeters can be operatedin more than one way has given rise to glaring misnomerslike isothermal DSC (which literally means isothermaldifferential temperature-scanning calorimeter). In such acase the calorimeter should be named according to the wayit is operated in a given application.

3. Temperature measurements with thermisters

Accurate measurement of heat invariably involves controland/or measurement of temperature, but the absolute temper-ature is only required for establishing the temperature of themeasurement. Only relative temperature need be measuredfor a heat determination, but the sensitivity of the heat mea-surement can be no better than the sensitivity of the temper-ature measurement. The thermometer must be very stable,but only for a time period exceeding the period of the exper-iment. Long-term stability is a convenience rather than a re-quirement. Although other sensors can be used, thermistorsare particularly useful for this application for two reasons;high sensitivity thermometers are easily constructed fromvery simple and inexpensive components, and because of the very small mass of the sensor, the time constant can also

Fig. 1. Wheatstone-type thermistor bridge for high-sensitivity measure-ments of changes in relative temperature.

be very small. In practice, most other sensors have a lowersensitivity and/or a longer time constant than a thermistor.

The major disadvantage of thermistors is that they areself-heated. This is important in some applications becausethe temperature measured by a self-heated sensor dependson the power generated in the sensor and the thermal con-ductivity between the sensor and surroundings. The powergenerated in the sensor is xed by the electrical power, butthermal conductivity to the surroundings may be altered bychanges in the surroundings. For example, a change in thewetting properties or a change in stirring pattern or veloc-ity of the liquid the sensor is immersed in or formation of a precipitate or bubble around the sensor can cause largechanges in the sensor temperature that are not reective of temperature changes in the medium.

Although there are several thermistor circuits thatcan be used for temperature measurement [2], a simple

Wheatstone-type bridge as shown in Fig. 1 is suitable formost applications. Properly designed, such a bridge with asensitivity of tens of mV K 1 , a noise level less than 1 K,a time constant of less than 0.1 s, an output very close to lin-ear in temperature over more than 1K, and stable over morethan a day can be constructed in a few minutes and at a verynominal cost. The problem of designing such a bridge con-sists of choosing components and constructing the bridgeto minimize noise sources and optimize sensitivity. Anyother resistance type sensor may replace the thermistor inthe following discussion, only the details will differ.

The bridge should be designed to obtain the required tem-perature resolution without an amplier if possible and withan impedance 1000 times less than the output detector. Mostdigital devices currently available have input impedances inthe G range, but strip chart recorders and some older am-pliers and digital devices still in use have input impedancesin the M range. To avoid loading a bridge connected to alower impedance device, the sum of R1 + RT or of RR + R2must be less than about 10 k . In addition, as a practicalmatter, each of the resistive components of the bridge shouldbe below 50k to avoid problems associated with leakagecurrents. Also, thermistors with room temperature resistanceabove 50 k are generally made from materials that are notas stable over a long time period. Most detectors have a


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L.D. Hansen, R.M. Hart / Thermochimica Acta 417 (2004) 257273 259

natural detection limit, i.e. without additional amplication,of about 1 V. Therefore, the bridge sensitivity should besuch that 1 V is equal to the desired detection limit in T .

The output voltage of a bridge as shown in Fig. 1 operatedin a near balance condition is given by Eq. (1).

V o = V BR2

R1 + R2

RRRR + RT (1)

The sensitivity S of the bridge is obtained by taking thetemperature derivative of Eq. (1) as given in Eq. (2)

S = r( 1 r)V B (2)

where is the relative temperature dependence of the sensor(equal to (d RT /dT )/ RT with units of reciprocal temperature),r is the ratio RR /(R R + RT) and V B is the voltage acrossthe bridge. Because the voltage drop across the thermistorV T is equal to (1 r)V B, S is also given by Eq. (3)

S = rV T (3)

Note that S is independent of the value of R1 + R2 and thatthe bridge does not have to have equal arms, i.e. RR does nothave to equal RT . What can be said so far about the choiceof components is that the sums R1 + R2 and RR + RT mustbe high enough not to overload the current capability of thepower supply V ps and that S will increase linearly with V B.

The optimum values for RR , RT and V B depend on therelative magnitudes and sources of noise in the measure-ment and on the time constant selected. Three sources of noise must be considered; the thermistor and its immediatesurroundings, the power supply and the remainder of thebridge elements. Because the noise actually seen in the de-tector will be a function of the overall time constant of thethermometer circuit, should be chosen to be the maximumvalue consistent with the measurements to be done with thebridge. The best means for establishing is with a low-passRC lter placed across the input to the detector. To mini-mize the effects of noise, this RC lter should have a timeconstant longer than any other component in the circuit.

Three types of noise in the thermistor itself must be con-sidered, i.e. noise that is independent of V B, noise that in-creases linearly with V B, and noise that increases as somehigher power of V B. Johnson (or white) noise, 1/ f (or pink)noise and shot noise are independent of V B. Shot noise is of such high frequency that it will be ltered out if > 0.5 sand thus will not be considered further here. The 1/ f noisedepends on the surface properties of the semiconductor ma-terial in the thermistor. Because generation and recombina-tion rates of carriers in surface energy states and the densityof surface carriers are important in determining the level of this noise, thermistors from different manufacturers and indifferent resistance ranges, i.e. made of different materials,may have signicantly different 1/ f noise. At the low fre-quencies considered here 1/ f noise will have approximatelythe same dependence on circuit properties as Johnson noise.Johnson noise can be calculated by Eq. (4)

V Jrms = (4kTRT f) 0.5 (4)

where V Jrms is the root mean square of the Johnson noisevoltage, k is Boltzmanns constant, T is the kelvin temper-ature, and f is the noise frequency bandwidth. Assumingthe circuit is equivalent to a simple, one RC circuit, f canbe replaced by 1/4 as shown in Eq. (5)

f = (2 RC) 1

2 = (4RC ) 1

=

14 (5)

Substituting (5) into (4) and combining constants givesEq. (6)

V a =aR T

0.5(6)

which represents the sum of the 1/ f and Johnson noise if ais an empirical constant.

The noise linear in V B results from thermal noise in thematerial in which the thermistor is immersed. The noise volt-age at the detector from this source is described by Eq. (7)

V b = b

0.5 V T (7)

where b is an empirical constant. The magnitude of this noisesource depends on the nature of the material in contact withthe thermistor. For example, a vigorously stirred liquid willhave a higher thermal inhomogeneity than a block of highthermal conductivity metal.

The third type of noise results from self-heating in thethermistor and thus depends on the rate of heat dissipationfrom the thermistor to the surroundings as shown in Eq. (8)

V c =c

0.5 P T

(8)

where c is an empirical constant, P T is the power generatedin the thermistor, and is the thermal conductivity from thethermistor bead to the surroundings.

Combining Eqs. (6)(8) as required by the statistics of independent noise sources and substituting V T2/P T for RTin Eq. (6) results in Eq. (9)

V rmsnoise =V T2

a

P T+ b +

cP TRT2

0.5

(9)

for the root mean square noise expressed as a voltage. Toexpress the noise as the peak-to-peak temperature noise T n ,Eq. (9) is multiplied by 6 and divided by the bridge sensi-tivity from Eq. (3) . The result is Eq. (10)

T n =6V rmsnoise

rV T=

6r 0.5

aP T

+ b +cP T

RT2

0.5(10)

Similar, but not identical, equations have previously beenpublished by others [3,4]. What is clear from both Eq. (10)and the literature equations is that there is a well denedminimum in T n as a function of P T and thus an optimumP T at which a given bridge should be operated.

T n can be determined as a function of P T with an experi-mental system such as that described in Fig. 2. The important


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Fig. 2. Experimental setup for thermistor-bridge noise measurements witha xed time constant of 2 s.

features of the equipment shown in Fig. 2 are (a) the timeconstant of the isothermal vessel must be long compared tothat of the RC lter, 2 s in the case shown, (b) the environ-ment of the thermistor in the isothermal vessel must closelyresemble the actual environment the thermistor will be usedin, (c) the RC lter must determine the time constant of thesystem or must be a constant determined by some othercomponent of the system, and (d) the data collection ratemust be several times faster than . Under these conditions,

measurement of the width of the band of ink on a plot of volt-age versus time is a good measure of the peak-to-peak noiseif the time scale is compressed so that a solid band of ink isproduced. The isothermal vessel need not be truly isothermalso long as the time constant of the drift rate is much smallerthan , and the actual calorimeter reaction vessel will usu-ally serve this purpose. The value of should be reportedwith any description of the bridge. The literature on temper-ature measuring circuits contains many examples of report-edly better, i.e. lower noise, circuits which are in fact poorerdesigns, but which have longer time constants than the cir-cuits used for comparison. Passive noise in the circuit anddetector with V B = 0 should also be measured and reported.

Fig. 3 shows a typical set of T n noise data obtained asa function of P T with = 4%/K. Several different bridge

Fig. 3. Temperature noise data collected in a 25 ml Tronac isoperibol power-compensation calorimeter vessel stirred at 600rpm.

congurations tested at the same value of 2s with thethermistor in stirred water gave very similarly shaped curveswith the minimum at about the same P T value. The minimumin T n establishes the optimum P T value and the detectionlimit for a particular bridge. The branch of the curve at lowP T values is largely determined by the values of a and b in

Eq. (10) and is curved as predicted. The curve at high P Tvalues is essentially linear as predicted by Eq. (10) if thethird term is much larger than the rst term. The slope of thecurve at high P T values is thus a measure of c / RT2. A valuefor can be obtained as the slope of a plot of the apparenttemperature measured against P T as in the example shownin Fig. 4. The data shown in Fig. 5 verify the functionaldependence of T n on P T given in Eq. (10) and show that thethermal inhomogeneities present in the medium surroundingthe thermistor are very important in determining the valueof the minimum in curves such as shown in Fig. 3.

Since P T at the minimum T n value is largely dependenton the surroundings and not on bridge conguration, P T fora given condition may be treated as a constant. Substitutionof the square root of P T RT for V T in Eq. (3) gives Eq. (11)

S = r(P TRT)0.5 (11)

showing that bridge sensitivity is linear with the square rootof RT . Assuming = 0.04 K 1 and P T = 0.25mW thenallows construction of Fig. 6. The large triangle outlined inbold in Fig. 6 denes all of the bridges with S > 50mVK 1

(i.e. 20 K gives 1 V), r 0.9, and RT 62.5 k . Thesmall triangle denes all the low impedance bridges withRT 10 k .

As an example of how a gure like 6 can be used to

design a bridge, assume a thermistor bridge with an outputsensitivity of 100mV K 1 is desired for use in a stirred


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Fig. 4. Sample of data showing how , the thermal conductivity between the thermistor and surroundings, may be determined.

liquid similar to water. Assuming the same values for and P T used to generate Fig. 6, any thermistor resistancebetween 30 and 60 k would be suitable. Selecting 50 k for the thermistor resistance then determines the values of r = 0.7 and V T = 3.6 V. V B therefore equals 12 V for thissensitivity (see the derivation of Eq. (3)).

Fig. 6 was derived on the assumption that P T = 0.25mWgives the minimum noise possible in the system. This valueof P T was based in turn on T n = 20 K peak-to-peak, which

is probably the minimum achievable noise in stirred water.Fig. 5 , however shows that lower noise levels canbe achieved

Fig. 5. Data showing examples of the dependence of T n on and surroundings of a thermistor in a wheatstone-type bridge.

in other media, namely a metal block in the example shown,as a result of a lower value of b in Eq. (10) . Eq. (10) showsthat the value of P T moves to a smaller value as b and c arereduced. According to Fig. 6, achieving higher sensitivityrequires going to higher values of V B, r and RT . There arepractical limits to increasing these parameters, however, andsensitivities above about 150mV K 1 are better achievedby increasing the detection limit of the detector or addingan amplier to the system. With an amplier or detector

capable of resolving 0.1 V, a simple DC thermistor bridgecan detect temperature changes of about 1 K.


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Fig. 6. Plot of bridge sensitivity ( S) as a function of V T , RT , and r assuming P T = 0.25mW and = 0.04 K 1 .

The use of an amplier with a thermistor bridge has otheradvantages. Because the power and hence self-heating in thethermistor can be greatly reduced, thermistors can be used

to measure the temperature of systems where the ow rateor stirring is variable [5]. Ampliers with very low noise ornoise rejection must be used however. On the other end of the scale, using a high self-heating rate makes thermistorsuseful as sensors to measure ow and thermal conductivity.A familiar application is the use of thermistors in thermalconductivity detectors in gas chromatography.

Besides the choice of the operating parameters such asresistances and voltages, the choice of materials and howthey are used also has a signicant effect on performance of thermistor bridges. Resistors should all be metal lm or wirewound and of low temperature coefcient materials. Carbonresistors tend to be noisy and have a sizeable temperaturecoefcient. Any change of resistance of the resistors in thecircuit will be indistinguishable from a change in resistanceof the thermistor and will be interpreted as a temperaturechange. Metal lm resistors typically have a temperature co-efcient of 100ppmK 1, or about 0.25% of the sensitivityof the thermistor. The resistors are also self-heated elementsand subject to the same noise as the thermistor if the sur-roundings are variable. R1 and R2 should be thermally cou-pled to minimize any differential effects of the environmenton these two resistances. The simplest way to do this is touse a single, multi-turn potentiometer for both of the resis-tors as shown in Fig. 1, and heat sink the potentiometer to

the chassis or a metal block. Resistors RR and RV cannot bepaired, so they should be thermostatted or at least thermallyattached to a large metal block. The sensitivity of the bridge

is a linear function of V B and therefore the stability of thebridge will be no better than the stability of the power sup-ply. Power supplies with output voltage regulation to 0.02%are readily available, but it is sometimes necessary to addsome capacitance to the output of the bridge to eliminateany residual ripple in the voltage. An inexpensive alterna-tive is to use batteries. Either primary batteries with a nearlyconstant voltage such as mercury or lithium batteries, or arechargeable battery connected to a trickle charger can beused. Electrical connections of unlike materials should beavoided in construction of bridges to be operated with dcvoltage because such connections will act like thermocou-ples in response to environmental temperature changes. Forexample, use of common Pb/Sn solders on tinned copperwires results in a thermocouple with a Seebeck coefcient of about 5 V K 1 . Since the thermocouple effect is not a func-tion of either bridge voltage or resistance, its signicancecan be reduced by maximizing V B, RT and r , or eliminatedcompletely by using ac voltage and a phase-locked detector.Other than elimination of thermocouple effects, there are noadvantages to using an ac voltage instead of a dc voltage onthermistor bridges. Thermocouple effects can be measuredby setting V B to zero and measuring the change in bridgeoutput when the temperature of the bridge or a componentis changed.


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4. Heat exchange corrections for isoperiboltemperature-change calorimeters

Measurement of heat by measuring the temperature cha-nge in a well-insulated (sometimes called pseudo-adiabatic)reaction vessel is perhaps the most direct and simplest

method of calorimetry, in its simplest form requiring onlya thermometer and a dewar ask. However, doing accuratework with such a calorimeter requires correcting for heatexchange with the surroundings, a procedure that becomesmore and more complex as the physical size of the reactionvessel decreases, as the rate of heat exchange with the sur-roundings increases, as the reaction time increases, and asthe difference in the physical properties of the reactants andproducts increases. The purpose of this section is to discussthe fundamentals of corrections that must be made to obtainan accurate heat of reaction from the temperature changemeasurement and assess the accuracy of these correctionsunder various conditions.

The raw data collected with an isoperibol temperature-change calorimeter may include heat effects from stirring,electrical sensors and heaters, and evaporation and conden-sation of liquids, as well as the effects of heat exchangewith the surroundings. All of these extraneous effects mustbe properly accounted for to obtain an accurate heat forthe process of interest. In the following discussion, the sur-roundings are assumed to be perfectly constant. In reality,some noise will arise from uctuations in the surroundings,and transfer of that noise to the measuring system will de-pend on the noise frequency, time constant, and thermal con-ductivity for heat exchange between the reaction vessel and

surroundings.The following discussion is limited to measurements

made in batch or titration calorimeters. Heat measurementsin isoperibol temperature-change ow calorimeters in whichthe temperature change is measured between reactants andproducts also require correction for heat loss to the environ-ment. However, accurate calculation of heat loss in this caserequires knowing the temperature prole along the lengthof the reaction vessel. Because of the difculty of obtainingand interpreting such data, this type of ow calorimeterhas not yet been proven to be generally useful for ac-curate heat measurements, and is not further consideredhere.

In a typical batch or titration experiment, the data col-lected are separated into three time periods; an initialbaseline collected before the reaction is initiated, the periodduring which reaction occurs, and a nal baseline takenafter the process of interest is nished. The rate of changeof temperature during these three periods is described re-spectively by Eqs. (12)(14) ,

didt

= kii + i (12)

drdt

= krr + r + (13)

df dt

= kf f + f (14)

where is the relative temperature, t is time, k is the heatexchange constant in Newtons law of heat transfer, is therate of temperature change resulting from heat lost or gainedfrom all effects other than exchange with the surroundingsand the process of interest, and is the rate of tempera-ture change resulting from heat from the process being mea-sured. Subscript i indicates the initial period, r the reactionperiod, and f the nal period. The purpose of the experimentis to extract either or its integral from the data by use of Eqs. (12)(14) . How this must be done depends on the prop-erties of the calorimeter and contents during each period of the experiment, on the nature and initiation of the processgenerating , and on how the data are to be interpreted.

4.1. Electrical calibration

In this case ki = kr = kf , i = r = f , is constant,and the problem reduces to solving three equations for threeunknowns once values for i, r , and f , and matching valuesfordi /dt , dr /dt and df /dt have been obtained from the data.Since the value of in units of J s 1 is already known fromdata on the voltage and current in the heater, one purpose fordoing this calculation is to obtain in units of (temperaturechange s 1), and thus determine the thermal equivalent toconvert measured temperature changes in whatever units is measured to Joules. A second reason for doing electricalheating at constant power is to characterize the time constantand heat exchange properties of the calorimeter, see Fig. 7.

If k and the time constant of the calorimeter are bothnegligibly small, the temperaturetime data in each periodwill be linear, the value of in each period is simply themidpoint, and the slopes are easily obtained. Dening whatis meant by a negligibly small time constant is simple,i.e. if the time constant is less than one-sixth of the datainterval, the system can be assumed to be at steady stateand each data point represents the true temperature at thatpoint in time. The time constant of the calorimeter will bedetermined by the time constant of the temperature sensor,the time constant for equilibration of unstirred mass in thereaction vessel, or by the time constant for heat exchangewith the surroundings, whichever is longest. The last isdetermined by the sharpness of the boundary between thesystem and surroundings [6,7]. Dening what is meant bya negligibly small k value is more complex.

The k values of carefully designed and constructed dewarasks have been measured and reported [8]. The minimum k value achievable is about 0.002min 1 , a value that producesa 1% deviation from linearity over a 5 min period, so theeffects of k on d /dt must probably be considered in all prac-tical systems. Thus, the temperaturetime data must be tto a mathematical function and d /dt obtained as the deriva-tive of the function. There are three possible functions thatcan reasonably be t to the temperaturetime data. The data


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Fig. 8. Error in rate of temperature change at the mid-point obtained by tting temperaturetime data to quadratic and linear functions.

for k and , and then in effect use the integral form of Eq. (13) to obtain the integral of as a temperature change, corr . The usual procedure for doing this is to obtain d /dt and values at the midpoint in time of the initial and nalperiods by tting a linear equation to the temperaturetimedata in these periods as described above, calculating a meantemperature m for the middle period, and using these val-ues to calculate corr [9]. This procedure is simple toimplement with a computer program using data collected atequal time intervals t and Eqs. (16)(19)

k =[(di/ dt) (df / dt) ]

f i(16)

m = +b + e

2t [t e t b] 1 (17)

= df dt

+ k( f m ) [t e t b] (18)

corr = e b + (19)

where the is taken from b+ 1 to e 1 and b and e arethe indices on the data points chosen as the beginning andend of the middle period. The thermal equivalent is givenby Eq. (20)

=Q

corr(20)

where Q is the total heat input by the heater, i.e. the productof heater voltage, heater current, and heater-on time.

The beginning point ( t b, b) should be chosen as the lastdata point taken before the heater was turned on and theending point ( t e , e) must be taken at some time after thecalorimeter has reached steadystate in the nal period. Thesepoints are thus at the intersection of the initial and middle

and nal and middle periods. The b and e data at these twopoints may be improved over using a single experimentalvalue by tting the data in the initial and nal periods toEq. (15) and using the values obtained at times t b andt e . Use of a quadratic instead of an exponential functionfor this purpose has been recommended [9], but the readyavailability of commercial software and computers makessuch simplication and the introduced error unnecessary.Use of a linear equation to obtain b and e values resultsin larger errors than use of a quadratic.

When k is large, the temperaturetime data in the initialand nal periods should be tted directly to Eq. (15) to obtaind /dt and values followed by application of Eqs. (16)(20)to obtain a value for . Note that in this case, k is obtainedboth from tting Eq. (15) and from Eq. (16) . These valuesshould agree. There are no published studies that have dealtwith this case, but it may have some interesting uses insituations that dictate a large k or in which data in the nalperiod with = 0 are unobtainable.

4.2. Chemical reactions

Experiments involving only electrical heating are alwayssimpler than experiments in which the heat effects resultfrom a chemical reaction because some physical change usu-ally must be made in the system to initiate the reaction, thereaction results in changes in the physical properties of thesystem, and the reaction may not have a clearly dened end-point in time. Initiating the reaction is usually done in oneof two ways, i.e. by adding material of a known tempera-ture from outside the reaction vessel or by allowing mate-rials already present inside the reaction vessel to come incontact. Either procedure can result in signicant changesin k , , and . Changes resulting from reaction initiation are


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stepwise, i.e. k , and change quickly to a different, butconstant value. The chemical reaction can also change k , and , but the time course of the change to the new valuedepends on the time course of the reaction and on the de-pendence of k , and on the extent of reaction. Possiblechanges in these parameters must be evaluated for each sit-

uation by running electrical calibrations on both the reactantand product systems, i.e. by determining k , and for bothconditions. If these parameters do not change signicantly,then the timetemperature data may be analyzed for or theintegral of in the same manner as for an electrical calibra-tion. Multiplication of or the integral of by then givesrespectively the heat rate or total heat of the reaction.

In all cases of chemical reactions, the effects of thetime constant on the timetemperature data must stillbe considered, and ki = kr = kf , i = r = f , and

i = r = f must be assumed until proven otherwise. If changes signicantly during the reaction, the slopes of thetimetemperature curves in the initial and nal periods arenot directly comparable because the same slope representsdifferent rates of heat exchange. Therefore, Eqs. (12)(14)must be multiplied by as shown in Eqs. (21)(23) .

ididt

= ikii + i i (21)

rdrdt

= rkrr + rr + r (22)

f df dt

= f kf f + f f (23)

i and f must be determined with heater calibration runsmade respectively before and after the reaction is run. Datataken during the heater calibrations also supply values of k i, i, k f and f . Integration of Eq. (22) gives Eq. (24)

r dt = r dr + rkrr dt rr dt (24)where the term on the left side of the equation is equal to theheat of the reaction.The integrals are taken from b to e or t bto t e where the b subscript indicatesa data point taken shortlybefore the reaction is initiated and the e subscript indicatesa data point taken some time after the reaction has reachedequilibrium. An exact solution of Eq. (24) is only possibleif r , k r and r can each be expressed as both a function of r and of t . Careful design of the experimental conditionsand of the calorimeter make this possible for most reactions.Note that the values of the beginning and endpoints of thefunctions describing r , k r and r are always known fromthe heater calibration data.

If the objective of the experiment is to determine the totalheat effect of the reaction, the integrals on the right side areevaluated between t b and t e and the corresponding values of where the time interval t et b includes all heat effects of the chemical reaction(s). If the objective is to determine thetotal heat liberated to any point in time during the course

of the reaction, as in a continuous titration, the integrals areevaluated from t b to t r , where t r is a time between t b and t e.If the objective is to obtain the kinetics of the reaction, thedifferential form of Eq. (24) is simpler to use since rr isthe heat rate at any point during the course of the reactionand dr /dt is readily obtained by numerical methods. The

latter procedure, followed by integration of rr over time orsummation of rr t , can also be used to obtain total heatto any point or the overall total heat of the reaction.

The many reports in the literature on heat exchangecorrections for isoperibol, temperature-change calorimeters(e.g. [711] ) are descriptions of various means to evalu-ate the integrals in Eq. (24) under various circumstances.Although these reports appear to arrive at quite different so-lutions, the differences are due to different approximationsforced by the conditions of the experiment and not to differ-ences in the fundamental approach. The possible situationsthat may usefully arise are too numerous to delineate here,but two examples will illustrate some of the more commonmethods of approximating the integrals required in Eq. (24) .

First, consider a simple laboratory dewar partially lledwith a solution of one reactant. The solution is stirred witha magnetic stir bar and the relative temperature is recordedwith a recording thermometer. The reaction is to be initiatedby lifting the stopper and adding the second reactant. Sucha simple system can be used to do fairly accurate calorime-try if certain conditions are established. The rate of energyinput from stirring must be kept constant, the amount of ma-terial added at reaction initiation must be kept small enoughthat does not change signicantly, and the calorimeter iscalibrated with a chemical reaction with a known enthalpy

change under very similar conditions. Under these condi-tions, Eqs. (16)(20) apply. Any errors resulting from smallchanges in or will tend to cancel between the calibra-tion reaction and the test reaction if the temperature changesfrom the two reactions are very close in magnitude and doneat the same temperature. Note that under these conditions,the thermometer can be read in any arbitrary units and thatthe temperature of added reactant need not be known as longas it is close to the temperature of the experiment.

Some of the problems of using magnetic stirrer bars alsobecome apparent from the requirement that the rate of en-ergy input from stirring be kept constant. The energy in-put from any stirrer depends on the viscosity of the stirredliquid, but in addition a stirring bar creates heat from thefriction of rubbing against the vessel or submerged bearingwhich is dependent on the lubricity of the liquid. Thus, mostcalorimeters are stirred with synchronous-motor driven pro-pellers with shaft bearings above the liquid.

If a large amount of reactant is added, will change andEqs. (16)(19) do not apply. Assuming the response time of the system is short compared to the data interval, r = i, but

r = f and electrical calibration will allow determinationof these parameters. Under these conditions, it is also likelythat k and will change to their nal values immediatelyon addition of the second reactant. Application of Eqs. (14),


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(16) and (20) to data from an electrical calibration run onthe nal system provides values of f , k f and f for use inEq. (24) . The applicable equation then becomes

Q = f (e b) + f kf dt + f f (t e t b) (25)where the integral is readily evaluated by a numericalmethodsuch as a summation (e.g. Eq. (17) ), the trapezoidal rule,or Simpsons rule. Note that data in the initial period arenot used except to determine b so no electrical calibrationon the initial system is needed. The specic heat capacityC , mass m, and relative temperature addition of the addedreactant material must be known for this case. The heat effectfor addition of the reactant is given by Eq. (26) ,

Q addition = C( addition b)m (26)

and must be subtracted from the total measured heat Q . Acommon error is to use some temperature other than b in

Eq. (26) , but the thermodynamics of the situation requirethat the reactants be mixed at the beginning temperature. If the thermal equivalent of the nal system is then applied incalculating Q, the H value calculated applies at b [12].

If the kinetics of a moderately slow reaction are to bestudied, the problem becomes only slightly different unlessthe reaction changes the physical properties of the solutionsignicantly. To determine the kinetics, the rate of heat dueto the reaction at each data point during the reaction mustbe determined. The data from an electrical calibration onthe nal system, i.e. f , k f and f and a rearrangement of Eq. (22) provides the means for this calculation.

r = f dr

dt + f kf r + f f (27)

In Eq. (27) , is the heat rate at each data point r in thereaction region. The derivative d r /dt must be evaluated fromthe t data by a numerical method. Usually / t is asufciently accurate approximation. Again the results mustbe corrected for the heat effect caused by addition of materialat a different temperature than b.

For a second example, consider the case of a continuoustitration which might be done in the same calorimeter asdescribed above if the time constant is very short [7,8] . Forsuch an experiment to be more useful than an experimentdone with a single injection, the heat produced to each datapoint in the titration must be calculated. Eq. (24) can beused to do this calculation if values of r , k r and r can becalculated at each data point in the titration. Values of i,k i and i and f , k f and f can be obtained from electricalcalibrations made on the initial and nal systems. Thesevalues x the beginning andendpoint values for the functionsdescribing r , k r and r , but the function must be determinedin other experiments or assumed from auxiliary data. If thephysical properties of the material in the reaction vessel donot change signicantly with the extent of titration, then

r , k r and r are functions only of the reaction vessel and

the amount of contents and linear functions as shown inEqs. (28)(30) may be safely assumed.

r = i + r( f i) (28)

kr = ki + r(k f ki) (29)

r = i + r( f i) (30)

In Eqs. (28)(30) , is the fraction of titration completed topoint r . These functions then provide the values needed tointegrate Eq. (24) to any point in the titration.

The time constant of the calorimeter is unimportant if onlya total heat is to be measured, but a short time constant is acondition for doing continuous titrations and for readily ex-tracting reaction kinetics from temperaturetime data. Con-struction of a calorimeter system with a short time constantrequires that three design constraints be met. The thermome-ter must have a short time constant, the unstirred mass inthe reaction vessel must be very small, and the boundary be-tween the system and surroundings must be well dened [6].Glass dewars with time constants < 1 s can be constructedif careful attention is paid to these constraints [7,8]. Suchcalorimeters have been available commercially since about1965.

When the physical properties of the reaction mixturechange signicantly in a nonlinear fashion with the extentof reaction, or if chemical reactions do not completelycease within a reasonable time, accurate correction for heatexchange with the surroundings may not be possible. Forexample, in polymerization reactions, the viscosity of thereaction mixture often changes exponentially with the de-

gree of polymerization. In such a case, it is not possible toaccurately correct for the heat of stirring. The heat trans-fer characteristics may also change, affecting the outputof self-heated temperature sensors and the heat exchangecharacteristics of the reaction vessel. Situations are oftenencountered where the products of a reaction are not stableand continue to react or a side-reaction continues into thenal period. In this situation, it may be difcult at best andoften impossible to obtain accurate values for f , k f , and f . The best that can be done in such cases is to measure

i, k i, and i and attempt to estimate values for these pa-rameters during the reaction period. Errors resulting fromsuch estimates should be determined by assuming worstcase estimates and calculating the extreme Q value.

5. Constant temperature baths

Operation of any calorimeter always also involves a tem-perature bath that surrounds the calorimeter. Whether simplythe air in the laboratory or a sophisticated, high-precisionbath, effects of uctuations in the temperature of the sur-roundings must always be considered in making calorimet-ric measurements. This section considers constant temper-ature baths for isoperibol calorimeters, but considerations


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for operation of an adiabatic calorimeter in which the bathis controlled to be the same temperature as the reactionvessel are the same except that the response time for thebath must also be taken into consideration. The primarypurpose of this section is to provide the calorimetrist withthe information needed to select a commercially available

bath or bath components. Detailed procedures for designof constant temperature baths such as constant temperaturerooms, uid ow loops, and laboratory water baths havebeen previously published elsewhere [13].

Perhaps the simplest type of constant temperature bathsare those in which temperature control is achieved by hav-ing two or more phases of the bath material present. Thecontrol temperature is thus the melting point, triple point orboiling point of the bath material. No controller or temper-ature sensor is needed for such a bath. External power needonly be supplied to recharge the bath for liquidsolid sys-tems, and only roughly controlled continuous power is re-quired to maintain liquidgas baths at the boiling point. Suchbaths must be well mixed to achieve temperature uniformityif signicant amounts of heat are to be dissipated to thebath.

Liquid baths with temperature control to a few hundredthsof a degree are commonly found in routine laboratory use.These baths usually consist of a large container (comparedto the calorimeter reaction vessel) lled with water, a stirrer,an electrical heater controlled by an electronic controller,and a coil of tubing with coolant owing through it. Thecontainer is usually insulated if the bath is to operate veryfar from room temperature. The insulation should be sealedto prevent condensation if the bath is operated below room

temperature and must be open to allow for expansion if thebath is operated above room temperature. The water must bereplaced with some other uid if the bath is operated aboveabout 70 C or below 0 C. Use of air as the bath uid avoidsthe problems associated with submerging the reaction vesselin a liquid, but because of the low heat capacity, heat transferis slow and control is usually not better than a few tenthsof degree. Table 1 gives a list of suitable uids and theirtemperature ranges. The coolant may simply be tap water,uid from a refrigeration system, or may even be from aheated bath if the bath is operated at high temperature. If

Table 1Fluids for use in constant temperature baths

Fluid Useful temperature range ( C)

Halocarbon 0.8 100 to + 70Ethylene glycol/water (1:1) 30 to + 110Silicone oil type 200.10 30 to + 160Silicone oil type 200.20 + 10 to + 230Silicone oil type 200.50 + 30 to + 275Silicone oil type 550 0 to + 230Silicone oil type 710 + 80 to + 300Mineral oil 0 to + 120KNO3 /NaNO2 /NaNO3

eutectic (52:40:7wt.%)+ 150 to + 550

carefully designed, liquid baths of this type can be controlledto about a millidegree.

Thermostatted metallic blocks into which the calorimeteris inserted are usually used to control the temperature of thesurroundings at high temperature, and have found wide usein heat-conduction and power-compensation calorimeters

at all temperatures. Metallic blocks can be controlled inthe microdegree range, but obtaining such control requiresa carefully designed system which includes the operat-ing environment. Such baths require at least two layersof temperature control, an outer bath actively controlledin the millidegree range and an inner bath passively con-trolled by the outer bath through carefully designed thermalconnections.

There are several characteristics that must be specied todescribe temperature control in a bath. Gradient describesthe long-term average temperature differences that existbetween different parts of the bath. Noise is used to referto short-term, random temperature uctuations. Cyclingrefers to a regular pattern, usually sinusoidal, that mayoccur in the bath temperature. Drift denotes a long-termshift in the mean bath temperature. The dead band istwice the temperature offset from the control temperaturerequired to activate the controller. Settling time is thetime required for the bath to regain control after a smallupset occurs. Slew rate is the maximum rate at which thebath temperature can change and is important in applica-tions where large upsets occur and a rapid return to controlis needed. The temperature sensor used to characterize anyof these parameters must have a time constant shorter thanthe characteristic being determined. Measurements must be

made over a period of days to distinguish between cycling,which may have a frequency of per day, from drift.

A temperature controlled bath is a steady state system, notan equilibrium system. Constant temperature is maintainedby balancing the rates of heat input and output. Becauseuctuations in the temperature of the surroundings will af-fect the rate of heat ow to or from the bath, the bath sur-roundings must also be considered in operation of the bath.The system will undergo a transient response whenever thesteady state changes, and since control is not maintained dur-ing the transient, precise control depends on precise controlof heat ows into and out of the bath and on keeping theseheat ows as constant as possible. Heat ows that must beconsidered are (a) evaporation and condensation of the bathuid, (b) the heat due to stirring, (c) heat exchange by con-duction through the walls of the container, (d) heat loss tothe cooler, and (e) heat provided by the heater. Either theheater or the cooler should be operated at constant powerwhile the other is controlled. The control element should beoperated near the middle of its power range, and the heaterand cooler should be no larger than necessary to providerm control. Stirring is often a major part of the heat input.Heat exchange with the bath surroundings through the wallsof the container should be minimized by insulation since thesurroundings are difcult to control.


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Because heat can only ow if there is a temperature gra-dient, elimination of signicant gradients requires carefulattention to placement of the various heat sources and thetemperature control sensor in the bath. The only point in thebath that is actually controlled is the sensor. The uid owin the bath will determine which parts of the bath are free

from temperature gradients. Optimum control of the tem-perature surrounding the calorimeter reaction vessel will beobtained with the stirrer, heater, and cooler clustered in onelocation so that uid ow is from the stirrer to the constantcontrol element (usually the cooler), to the variable controlelement (usually the heater), to the sensor, and then to thereaction vessel.

The size of a bath also has an important impact on theachievable temperature control. The minimum achievabletemperature noise at a 10s time constant in water lled bathshas been shown to increase linearly with bath size from about0.1 to 100L [13]. This result is counterintuitive because thetotal heat capacity and thus obviously the thermal inertia in-crease with bath volume. Also, the thermal inhomogeneityis inversely proportional to the product of the specic heatcapacity and volume of the bath. However, the thermal in-homogeneity is also directly proportional to the product of the power dissipation through the bath and the eddy currentlifetime in the uid, and eddy current lifetime increases asa power of bath volume. Because of the effects on mixing,the shape of a bath also inuences eddy current lifetime, andhence temperature noise. Baths should not have square cor-ners or other dead spots in which uid is not well mixed.The bath uid should have a high heat capacity and a lowviscosity to minimize noise.

Temperature sensors used to control constant tem-perature baths are of two types. Mercury thermoregula-tors and bimetallic switches are examples of off/on ormake-and-break sensors. These sensors provide onlytoo-hot or too-cold information and can only be used withon-off or two-level control. These sensors are typicallyused in baths with control in the range of 0.1 to 0.02 K.The other type of sensors are resistors, thermistors, thermo-couples or other electronic devices in which the propertieschange continuously with temperature. These sensors alsoprovide information about how far the temperature is fromthe set point and this information can be used with more so-phisticated controllers to reduce noise, cycling and drift inthe bath temperature. The time constant of the sensor mustbe adjusted to match the response time of the other compo-nents of the bath such as the heater, uid mixing time, etc.

In order of increasing sophistication, controllers areon/off, two-level, proportional (or type zero), proportionalwith reset (or type one), and proportional with reset andlook ahead. As implied by the name, an on/off controllersimply turns the power to the control element on or off depending on the signal from the temperature sensor. Atwo-level controller turns a portion of the power on or off. Aproportional controller changes the power to the control ele-ment in proportion to the signal from the sensor. Automatic

reset is used to adjust for long-term changes in the steadystate power needed to control the bath, and look ahead(also known as derivative or rate control) is used to dampinherent instabilities in type one control. Cycling and driftare inherent with on/off and two-level control. Proportionalcontrol eliminates cycling, but not drift which is eliminated

with automatic reset. Control may be done by an analogdevice or may be done with a simple digital computer.

6. Initiating the reaction

All calorimetric measurements on reactions or phasechanges must somehow involve a mechanism for initiat-ing the reaction or process of interest. The four distinctmechanisms are (a) mixing of the reactants, (b) additionof a catalyst, (c) passing electrical current through the sys-tem, and (d) changing the temperature, pressure or volume.Changing the temperature is the process that gives rise tothe temperature scanning methods commonly known asDSC and DTA. Changing the pressure and volume can alsobe used to initiate reactions. The system pressure may bechanged by addition of an inert gas (e.g. see [14]) or bycompression of a liquid linked to the reaction vessel (e.g. see[15]). In parallel with the temperature scanning methods,such methods should properly be called pressure scanningcalorimetry, but the name has been preempted to describemethods in which the pressure of a gaseous reactant isvaried, and thus the method is called pressure-controlledscanning calorimetry or transitiometry [16]. Electrolyticmethods have been used almost entirely to study reactions

during both discharge and charging of batteries (e.g. see[17]), although a few studies have been done for other pur-poses [18,19] . Although electrolytic methods appear to bean elegant means for initiating and controlling reactions,only in rare cases of either electrolysis or cell dischargedoes the reaction go cleanly without side reactions or recy-cling of products. Catalyst addition is most commonly donewith enzymes, but is a method that should be considered instudies of non-biological systems. The major advantages of this method are that only relatively tiny amounts of materialneed be added to the reaction vessel and that the reactionrate can be controlled since it will be directly proportionalto the amount of catalyst added. Direct mixing of reactantsis the most common method for initiating the process to bestudied and is the subject of the remainder of this section.

For very slow reactions, the reactants may simply bepremixed shortly before insertion into the calorimeter. Thismethod is commonly used in studies of degradation andcorrosion reactions, e.g. see [20]. Methods may be furthersubdivided as to whether the reactants are all within thereaction vessel before mixing, or if one is added from out-side the reaction vessel. In the latter case, a correction mustbe made for the temperature difference between the addedreactant and the contents of the reaction vessel at the timeof reaction initiation. In the former case, correction must


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be made for the effects of opening the barrier between thereactants. Addition from outside the reaction vessel is oftenmuch simpler than maintaining and breaking separationinside the reaction vessel.

Both solids and liquids can be added from outside thereaction vessel. Liquids may be added with a syringe and

solids may simply be dropped or pushed out of a tube with asyringe-like device. The latter is often easily constructed bycutting the delivery end from a liquid syringe. Solids mustbe retained in the syringe with a foil or plug in the end of the syringe. The plug or foil also prevents contact betweenvapors in the calorimeter and the solid prior to addition.Difcult-to-wet solids may be contained within a weighted,porous bag to prevent them from oating on the surface.Correction for incomplete transfer of a reactant can oftenbe done by measuring the amount remaining in the deliverysystem at the end of the experiment.

Techniques for mixing of reactants within the reactionvessel usually involve some type of syringe or pipet. Sy-ringes for delivery of liquids can often simply be immersedin the liquid in the reaction vessel. Separation can often bemaintained with a gas bubble in a narrow bore needle at-tached to the syringe. Addition of solids can be done with thesame devices used to inject from outside the reaction ves-sel if the solid is contained between two plugs. Breaking aglass bulb containing the solid is a common method of mix-ing a solid reactant into a system. The glass must be madevery thin to minimize the heat of breaking, and the bulbsmust be kept very uniform to obtain a constant blank value.Fig. 9 shows a simple device that replaces the bulb with mi-croscope cover slips [21]. The advantages of the cover slips

over bulbs are the ready availability and uniformity. Thecover slip device is also much easier to ll with solids thannarrow-necked bulbs. Glass bulbs and the device shown inFig. 9 can also be used for addition of liquids that cannot behandled with a syringe because they are too reactive or tooviscous. Many devices used for addition of solids have in-corporated a bulb smashing or opening device as an integralpart of the stirrer. Combining the stirrer and addition deviceis usually unnecessary and only results in complicating themechanical design of both stirrer and the addition device.

Addition of liquids with syringes is convenient, but caremust be taken that the liquid is not signicantly heated by toorapid passage through a narrow bore tube. Eq. (31) gives thetemperature rise in K in a uid from frictional ow in a tube

T =21 10 10l( v)

Cd 4(31)

where is the uid viscosity in centipoise, l is the lengthof the tube in cm, v is volume ow rate in cm 3 min 1, is the uid density in g cm 3 , C is the uid heat capacityin Jg 1 K 1, and d is the tube diameter in cm. For wa-ter owing through a 0.4mm diameter tube 30 cm long at1 cm3 min 1 , T = 0.002 K.

Another means of mixing reactants within the reactionvessel is to invert the reaction vessel and allow the reactants

Fig. 9. Device for mixing solids or liquids by breaking glass cover slips.

to ow together. Further mixing is done by repeated inver-sion of the reaction vessel. Inversion of the reaction vesselmay be done by inverting the vessel or the entire calorimeter.A similar, two container method can be used with calorime-ters and vessels that cannot be inverted. The device is shownin Fig. 10. A slight difference in pressure between the twocontainers is used to transfer and mix the reactants. If further

Fig. 10. Device for transfer of a liquid reactant.


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mixing is desired, the transfer can be repeated in reverse. Toaccount for incomplete transfer, the two containers can sim-ply be weighed after the experiment. The device is simpleto construct and can be made from disposable materials forstudy of reactions that result in difcult to remove materialssuch as polymers and glues.

7. Continuous titration

Continuous titration is a very efcient means for collec-tion of data as a function of the total concentration of areactant. It has been much used to determine the thermody-namics of reactions in solution, especially for metalligandand proton ionization where several reactions may occursimultaneously. The primary requirement of a buret forcontinuous delivery is that the delivery rate be constant.Constancy of delivery rate should not be confused with ac-curacy of delivery of a total volume. The latter only requiresthat the distance of movement of the piston be accurate, theformer that the piston move at a constant speed. A commonfault of many syringe-type burets is a sinusoidal deliveryrate because of wobble in the lead screw or rack-and-piniondriving mechanism. A buret that is submersible in a constanttemperature bath is convenient, but thermal equilibration of titrant can be done with a coil of tubing placed in a constanttemperature environment.

8. Stirrer design

The stirrer used to mix the reaction vessel contents is animportant component of the calorimeter. Stirring controlsthe rate of mixing of reactants and is often the major heatinput to the calorimeter during baseline periods. Stirringof liquids is often done with a simple rotating propeller.The rate of energy input with a rotating propeller dependssomewhat on the pitch and aspect ratio of the propeller andon the width, height and shape of the reaction vessel withrespect to the propeller, but is a much more sensitive func-tion of the liquid viscosity and paddle length and speed.The equation for power input from a rotating propeller is

P = sf 3d 3 (32)

where P is the power, s is a nondimensional shape factor, is viscosity, f is the frequency of rotation, and d is the diam-eter of the propeller. Because of the sensitivity to frequency,the stirrer must be driven with a constant speed motor.Wobble should also be avoided since its effect is to changed . Reactions that produce a signicant change in viscosity,such as polymerizations, will cause a change in the base-line power input by the stirrer and this must be taken intoaccount in analyzing the data. To prevent settling of densematerial and a vortex, the propeller should be placed nearthe bottom of the reaction vessel and should rotate in thedirection that causes lift along the propeller shaft. The speed

of rotation should be the minimum that still gives the nec-essary mixing. Minimizing power input from the stirrer alsominimizes the magnitude of variations in baseline heat rate.

The efciency of a stirrer may be dened as the ratio of power input to mixing time. A at paddle is often more ef-cient than a pitched propeller. Unless bafes are included, a

propeller provides only circular motion with very little verti-cal mixing. Obtaining efcient vertical mixing is a problemin many reaction vessel designs. A tall, narrow, notched, atpaddle is useful in vessels with large height to width ratios.Another approach is to use a rotating tube with holes in thesides. Liquid is thrown out through the holes and drawn inthe open end of the tube. Such a design may be the most ef-cient type of stirrer when the aspect ratio and hole size arecorrectly adjusted to match viscosity and shape parameters.

Penetration of the stirrer shaft through the reaction vessellid is a problem when the vessel must be sealed because of volatility of the contents or for work above ambient pressure.In either case the rate of heat loss from evaporation is directlyproportional to the product of the heat of vaporization, thevapor pressure, and the effective area through which thevapor must pass.

There are basically three approaches to solving the prob-lem of sealing the stirrer shaft: (a) use of a rotating seal, (b)replacement of the shaft with a magnetic coupling, and (c)use of a non-rotating, mechanical coupling. Three types of rotating seals are commonly available. Although they do nottotally seal the system, lubricated rubber ring seals similar tothose used on automobile axles are often sufcient to controlloss of volatiles. Ferrouid and mercury seals as shown inFig. 11 are useful to a differential pressure of several atmo-

spheres. Ferrouid seals cannot be used with volatile organ-ics because of dissolution of the ferrouid. Mercury sealsare a problem because of the toxicity of Hg vapor. Magneticstirring avoids the problem of sealing a shaft, but the paddlebearing must be placed within the reaction vessel and therotating magnetic elds can cause problems with sensors byinduction of electric currents. Stirring can also be erratic if the magnetic coupling is not strong enough. The bearing fora magnetic stirrer should be placed above the liquid level if possible. The drive shaft can then support the stirring magnetnear the bottom of the reaction vessel. The third approachmakes use of a rubber seal glued to both a shaft and the lidof the vessel. The stirrer can then be oscillated vertically orhorizontally or rotated through a circle with an eccentric.The eccentric arrangement is probably the most efcient.

9. Design of calibration heaters

The accuracy of a calorimetric measurement can be nobetter than the calibration accuracy. Electrical calibrationcan be subject to large systematic error because of poorheater design. The accuracy of electrical calibration shouldbe veried whenever possible with a well-known chemicalsystem.


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Fig. 11. Designs for ferrouid and mercury seals on a rotating shaft.

At least two conditions must be met for electrical cali-brations to be accurate: (a) all of the heat must be generatedin the heater, and (b) all of the heat must be transferred tothe calorimeter. Because an electrical heater must be con-nected to the calorimeter surroundings by wires that haveboth nite thermal conductivity and electrical resistance,neither condition can be exactly met, only approached. The

requirements of reducing both thermal conductivity andelectrical resistance are antithetical, and thus any heaterdesign must be a compromise. The rst condition requireslarge diameter, short lead wires with negligible resistancecompared to the heater. The second condition requiresthin, long lead wires to minimize thermal conductivitybetween the heater and surroundings. Designing a heater


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thus requires nding a compromise that makes both errorsnegligible.

The fraction of the resistance, and thus the fraction of the heat generated in the lead wires, can be reduced by in-creasing heater resistance, but there is a practical upper limitto heater resistance. As heater resistance increases, voltage

must also be increased to achieve a given power. (Note thatpower = V 2/R .) The higher the voltage, the more difcultit becomes to avoid leakage currents bypassing the heater.Increasing electrical insulation on the heater exacerbates theproblem of getting all of the heat generated in the heatertransferred to the calorimeter. Experience shows that electri-cal calibration heaters should have a resistance between 100and 1000 , lead wire resistance should be < 0.1 , and leadwire thermal conductance should be < 0.1% of the thermalconductance between the heater and the calorimeter.

Accurate low power ( < 1 W) heaters are easier to con-struct than heaters for higher powers. A low power heaterfor solution calorimeters can be constructed from a variable,12 V power supply with output stable to better than 0.1%;a 100 , quarter-watt, wire-wound, low thermal-coefcientresistor; small gauge, varnished Cu wire; and a length of thin wall, shrinkable teon tubing. Break the case and endcaps from the resistor by squeezing with pliers, solder theCu wires to the resistor (two lengths on each end, one tocarry the current, one to measure voltage), thread the assem-bled heater through the teon tube, and shrink the tubingby heating. The lead wires and resistor must be carefullysupported during the last operation because the solder willmelt at the temperature required to shrink the tubing. Theteon tubing can be replaced with a carefully applied coat of

epoxy or other varnish in some circumstances. Higher powerheaters (up to about 10W) can be similarly constructed fromseveral resistors connected in series with short lengths of heavy gauge Cu wire so the total resistance is about 100 .The set of resistors replaces the single resistor as describedabove. Using several resistors increases the surface area of the heater and hence the thermal conductance to the solu-tion. Heaters for ow calorimeters are conveniently made bywinding varnished, low thermal-coefcient resistance wiredirectly on the ow tubing. Ceramic and metal clad wiresold for construction of thermocouples is also very useful

for winding heaters on cylindrical tubes and vessels. In allcases heat losses through the lead wires can often be reducedby thermally connecting the leads to the calorimeter.

References

[1] L.D. Hansen, Thermochim. Acta 371 (2001) 1922.[2] H.J. Hoge, Rev. Sci. Instrum. 50 (1979) 316320.[3] L.D. Bowers, P.W. Carr, Thermochim. Acta 10 (1974) 129.[4] L.D. Bowers, P.W. Carr, Thermochim. Acta 11 (1975) 225233.[5] R.L. Berger, W.S. Friauf, H.E. Cascio, Clin. Chem. 20 (1974) 1009

1012.[6] H.A. Skinner, J.M. Sturtevant, S. Sunner, The design and opera-

tion of reaction calorimeters, in: H.A. Skinner (Ed.), ExperimentalThermochemistry, vol. II, Interscience Publishers, New York, 1962,Chapter 9, pp. 157219.

[7] J.J. Christensen, R.M. Izatt, L.D. Hansen, Rev. Sci. Instrum. 36(1965) 779783.

[8] L.D. Hansen, T.E. Jensen, S. Mayne, D.J. Eatough, R.M. Izatt, J.J.Christensen, J. Chem. Thermodyn. 7 (1975) 919926.

[9] J. Coops, R.S. Jessup, K. van Nes, Calibration of calorimeters forreactions in a bomb at constant volume, in: F.D. Rossini (Ed.),Experimental Thermochemistry, Interscience Publishers, New York,1956, Chapter 3, pp. 2758.

[10] C.E. Vanderzee, J. Chem. Thermodyn. 13 (1981) 11391150.[11] L.D. Hansen, E.A. Lewis, D.J. Eatough, Instrumentation and data

reduction, in: K. Grime (Ed.), Analytical Solution Calorimetry, Wiley,New York, 1985, Chapter 3, pp. 5795.

[12] F.D. Rossini, Introduction: general principles of modern thermo-chemistry, in: F.D. Rossini (Ed.), Experimental Thermochemistry,Interscience Publishers, New York, 1956, Chapter 1, pp. 1618.

[13] L.D. Hansen, R.M. Hart, Constant temperature baths, in: P.J. Elving(Ed.), Treatise on Analytical Chemistry, second ed., Part 1, vol. 12,Wiley, New York, 1983, Chapter 4, pp. 135164.

[14] R.S. Criddle, R.W. Breidenbach, A.J. Fontana, L.D. Hansen, Ther-

mochim. Acta 216 (1993) 147155.[15] S.L. Randzio, D.J. Eatough, E.A. Lewis, L.D. Hansen, J. Chem.

Thermodyn. 20 (1988) 937948.[16] S.L. Randzio, Chem. Soc. Rev. 25 (1996) 383.[17] L.D. Hansen, H. Frank, J. Electrochem. Soc. 134 (1987) 17.[18] E.I. Khanaev, Izv. Sib. Otd. Akad. Nauk SSSR, Ser. Khim. Nauk (2)

21-4, CA 71(14): 64918d (1971).[19] J.E. Jones, L.D. Hansen, S.E. Jones, D.S. Shelton, J.M. Thorne, J.

Phys. Chem. 99 (1995) 69736979.[20] L.D. Hansen, E.A. Lewis, D.J. Eatough, R.G. Bergstrom, D.

DeGraft-Johnson, Pharmaceut. Res. 6 (1989) 2027.[21] R.A. Winnike, D.E. Wurster, J.K. Guillory, Thermochim. Acta 124

(1988) 99108.

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