Thermal Analysis of Polymeric Materials Part 6 doc

In the first part of theexperiment the temperature-increase is linear, indicating a constant heat capacity.Whenever the eutectic temperature, Teutecticis reached, the temperature is cons

Fig 4.5

increasing atomic vibration amplitudes at higher temperatures The resistanceincreases almost linearly with temperature The best-known resistance thermometersare made of platinum They are also used for the maintenance of the internationaltemperature scale, ITS 90, as shown in Appendix 8 Over a wide temperature range,the change of its resistance is 0.4% per degree In order to make a platinum resistancethermometer, a wire is wound noninductively Most conveniently, the total resistance

is made to be 25.5 6 at 273.15 K Under this condition, the resistance of thethermometer will change by about 0.1 6 K1 The best precision that has beenachieved with platinum resistance thermometers is ±0.04 K at 530 K and ±0.0001 K

at 273.15 K It decreases to ±0.1 K at 1700 K

Semiconductor properties are also summarized in Fig 4.5 The conductionmechanism of semiconductors is more complicated At low temperatures, semicon-ductors have a very high resistance because their conductance band is empty ofelectrons As the temperature increases, electrons are promoted out of the relativelylow-lying valence band into the conductance band, or they may also be promoted fromimpurity levels into the conductance band It is also possible that positive holes,created by the electrons promoted out of the valence band, carry part or all of thecurrent All these effects increase the conductance with increasing temperature bycreating mobile charge carriers that more than compensate the decrease in mobilitywith increasing temperature Semiconductors, thus, have over a wide temperaturerange the opposite dependence of resistance of metals on temperature Typically, onemay have as many at 1017charge carriers per cubic centimeter at room temperature,and the specific resistance may vary from 102 to 10+9 6 cm The temperaturecoefficient of the resistivity may be ten times greater than that of a metal resistancethermometer Semiconductor thermometers can be built in many shapes Frequentlythey are very small beads, so that their heat capacity and thermal lag are small They

4.1 Thermometry and Dilatometry _287

A third type of frequently-used thermometer is the thermocouple The ple is based on the Seebeck effect (see also Fig 4.36) At the contact points of twodissimilar metals a potential difference is created because some of the electrons in thematerial of the lower work function drift into the metal with the higher work function.The work function is the energy needed to remove one electron from a metal surface,i.e., from its Fermi level—usually measured in eV One can think of the metal withthe higher work function as holding the electrons more tightly When a circuit of twodifferent materials is set up, as is shown in the top drawing in Fig 4.7, and a voltmeter

thermocou-is inserted in one of the branches, one observes no voltage as long as the two junctionpoints are at equal temperature The potential difference created in one junction bythe drift of electrons from the low-work-function to the high-work-function metal isexactly opposite to the potential difference created in the other junction If, however,the two junctions are kept at different temperatures, one observes a voltage orelectromotive force (emf), EAB This electromotive force can be used to measure thetemperature difference between the junction points of the thermocouple

The table in Fig 4.7 lists the change in the emf per kelvin of temperaturedifference for a number of well-known thermocouples Copper–constantanthermocouples, which have been given the letter T by the Instrument Society of

Fig 4.7

Fig 4.8

America, are used frequently because of their reproducible temperature-to-emfrelation The chromel–alumel thermocouple (K) can be used all the way up to 1600 K.Its main advantage lies in the fact that the emf per kelvin of temperature change isrelatively constant between 300 and 1250 K A reading of the emf can in this wayeasily be converted into temperature A voltmeter can be supplied with a linear scale,reading in K (or °C) Figure 4.8 illustrates how the emf of the thermocouples can be

4.1 Thermometry and Dilatometry _289

expressed mathematically The three constants a, b, and c must be fitted at fixedpoints of the ITS 90 which are listed in Fig A.8.1

The main difficulty in high-precision temperature measurements with couples is the introduction of spurious voltages by metal junctions outside of themeasuring and reference junctions In Fig 4.8 two typical measuring circuits aresketched The right circuit overcomes most of the difficulties by using onlythermocouple metals for the circuit (including any switches and connectors!) Oneonly has to watch that the two connections to the measuring instrument are at the sametemperature and that no additional thermal emf is introduced inside the meter Thesecond circuit in Fig 4.8 eliminates the need to carry the thermocouple wires to themeasuring instrument by changing from thermocouple material to normal conductancecopper in the reference ice bath In this arrangement all subsequent Cu/Cu junctionpotentials cancel

thermo-The measuring of the emf must naturally be carried out in such a way thatpractically no current flows This can be done by bucking the thermocouple emf with

an identical potential of a calibrated potentiometer and reading the position of zerocurrent At the bottom of Fig 4.8 the circuit diagram is given for such a potentiome-ter Today electronic voltmeters draw practically no current and potentiometers arebecoming old-fashioned A modern, high-impedance voltmeter can, in addition, bedigital and be already calibrated for a given thermocouple Also, it is possible toeliminate the reference junction at toby providing an appropriate counter-emf Butnote that the condition of temperature constancy on all dissimilar metal junctions, up

to and including the voltmeter, remains

The discussion of the experimental aspects of temperature measurement isconcluded with a listing of some additional instruments which offer promise forspecial applications:

Quartz thermometer (measurement of the frequency of an oscillator,controlled by a quartz crystal with linear temperature dependence—resolution0.0001 K, range 200 to 500 K)

Pyrometer (measurement of the total light intensity or the intensity of a given,

narrow frequency range to obtain the temperature—absolute thermometer, used in themaintenance of ITS 90 as given in Appendix 8 The calibration needs to be done atone temperature only)

Bimetallic thermometer (bimetallic strip which shows a deflection due to

differential expansivity that is proportional to temperature—frequently used fortemperature control by coupling the bimetallic strip to a mechanical switch to controlthe chosen device)

Vapor pressure thermometer (pressure measurement above a liquid in

contact with the unknown system—particularly useful at low temperatures where othertypes of thermometers may not be applicable, see also Appendix 8 for the application

of vapor pressure thermometers in the maintenance of the ITS 90)

Gas thermometer (measurement of p and V, followed by calculation of T

through the gas laws—see Figs 2.8, 2.99, and 4.2)

Noise thermometer (measurement of random noise caused by thermalagitation of electrons in conductors, detected by high amplification of the signal)

Fig 4.9

4.1.4 Application of Thermometry

Several typical thermometry applications are treated in connection with the molarmass determination of macromolecules in Sect 1.4.3 and in a study of melting withoptical and atomic force microscopy in Figs 3.95–97 The evaluation of phasediagrams involves often the recording of time-temperature curves A cooling curve

to find the crystallization temperature is illustrated in Fig 1.67 A heating curve isshown in Fig 4.9, together with the corresponding phase diagram and Newton’s law

of temperature change on heating The sample with its thermometer, initially at a

temperature T1, is inserted with a reproducible thermal resistance into a constanttemperature bath at To, as shown in Fig 1.67 For small changes in temperaturedifference, K of Newton’s law remains constant The heat exchanged, dQ, during thechange in temperature, dT, during the time interval, dt, can be expressed as given inFig 2.10 Without any change in composition, n, one can write:

where Cpis the heat capacity of the system at temperature T Equating dT with the dT

of the Newton’s law expression in Fig 4.9, one can get an expression for the heat-flowrate, dQ/dt:

If a pure sample undergoes a first-order phase transition, as described in Sect 2.5, itstemperature remains constant until the heat of transition is absorbed or evolved, butone can assume that the heat-flow rate is the same as without transition At constanttemperature, T, one can then write:

4.1 Thermometry and Dilatometry _291

where L is the latent heat coupled to the change in n This equation describes theheating curve at the eutectic temperature Above the eutectic temperature, bothheating and melting occur In this case, both partial differentials must be properlycombined, as suggested in Fig 2.10 (dH = CpdT + Ldn) The changing amount ofmelting with temperature accounts for the changing slope between the eutectic and theliquidus temperature in Fig 4.9 A special complication arises with the crystallizationillustrated in Fig 1.67, where a certain amount of supercooling is necessary tonucleate the crystallization of the sample (see Sect 3.5) The equilibriummelting/crystallization is passed to lower temperature, followed by self-heating toequilibrium, once nucleated, as indicated by the dashed line in Fig 1.67

The heating curve of the two-component sample in Fig 4.9 has a eutectic phasediagram (see Chap 7) In addition, the heating curve indicates a heating apparatusthat delivers a constant heat input as a function of time In the first part of theexperiment the temperature-increase is linear, indicating a constant heat capacity.Whenever the eutectic temperature, Teutecticis reached, the temperature is constant, itincreases only after all B-crystals are melted, along with enough A crystals to give theeutectic concentration ce Beyond the eutectic temperature, the slope of thetemperature-versus-time curve is less than before since melting of A crystalscontinues, increasing the concentration of A in the melt from the eutectic concentra-tion to the overall concentration, cs, given by the liquidus line At the liquidustemperature, Tliquidus, melting of A is completed and the original slope of thetemperature increase is resumed A simple thermometry experiment, thus, permits themeasurements of two temperatures, Teutecticand Tliquidus, and fixes two points in thephase diagram Starting with different concentrations, the complete phase diagramcan be mapped When calibrating K, it may even be possible to evaluate heat capacityand latent heat, i.e., perform calorimetric experiments Because of the different parts

in the heating-curve apparatus in Fig 1.67, all with different thermal conductivitiesand heat capacities, progress in calorimetry by scanning experiments was only madeafter developing differential techniques, described in Sects 4.3 and 4.4

4.1.5 Principle and History of Dilatometry

A dilatometer is an instrument to measure volume or length of a substance as a

function of temperature (from L dilatare, to extend, and metrum, a measure) The SI

unit of length is the meter, which is maintained as a multiple of a krypton-86 radiationwavelength described in Appendix 8 and listed in Fig 4.10 From the present bestvalue of the pole-to-pole circumference of the earth of 40.009160×106

m one can seethat originally the meter was chosen to make this circumference come out to be4×107

m Such a definition changes, however, with time since measurements of thecircumference of the earth are improving steadily and natural changes may occur.The unit for volume is the cubic meter For density measurements this makes anunhandy, large sample, but one may remember that the g/cm3

is numerically identical

to the SI unit Mg/m3

For the present discussion, the experimental pressure is assumed

to be constant and, if not indicated otherwise, is the atmospheric pressure (0.1 MPa)

The change of length and volume with respect to temperature is described by theexpansivities L and, respectively, as written in Fig 4.10 Both are frequentlypositive, but macromolecular crystals have often a close to zero or negativeLin thechain direction The best known, exceptional substance with a negative volumeexpansivity is water from the melting temperature 273 up to 277 K The correlationbetween the linear and volume expansivities is shown at the bottom of the figure forthe case thatLis the same in all directions of space The volume of the cube as afunction of temperature can then be expressed in terms of the linear and volumeexpansivities, as shown at the bottom of Fig 4.10 The quantity5ois the length of thecube at the temperature To Recognizing that terms higher than the first power inLare small and can be neglected, one can see that  3L.

The obvious length measurement is a side-by-side comparison with a standardmeter, the obvious volume measurement, the evaluation of the content of a standardvessel Early length measurements of this type were based on anatomical lengths.Naturally, the variation in human size was a basic problem that was solved either byaveraging, or by arbitrary choice The sixteenth century woodcut of how to produce

a “right and lawful” rood is illustrated in Fig 4.11 It shows that one should line upsixteen men, tall and short, “as they happen to come out of the church” after theservice One sixteenth of this rood was “the right and lawful” foot.1

It is surprising

4.1 Thermometry and Dilatometry _293

to water, their sizes are similar The present-day metric ton represents 106

cm3ofwater and leads to a size of the mouthful of 15.3 cm3

Even mouths seem not to havechanged much over the centuries The connection between mass and volume was themost difficult branch of metrology Hundreds of units have been described, eachpointing to a different method of dilatometry

4.1.6 Length, Volume, and Density Measurement

The easiest length measurement involves direct placement of the sample against astandard meter as shown at the top of Fig 4.12 Help in reading the divisions of thestandard scale can be given by optical magnification or by designing a micrometerscrew that allows one turn of the screw to be divided into 360 degrees, readableperhaps to a fraction of one degree The length shown can be read as 22.45 units

Fig 4.12

A further refinement is given by the vernier which was invented by Pierre Vernier

in 1631 Figure 4.12 shows examples of the use and construction of an advanced and

a retarded vernier The example lengths in the figure are of 22.16 and 24.67 units.One can see, that the advanced, linear vernier has 10 divisions for an interval of 0.9units, and the retarded vernier, similarly, has 10 divisions for an interval of 1.1 units.The object to be measured is lined up with the zero position of the vernier and then thelength is read at the exactly matching divisions The calculations in the figure showthe validity of the method Similarly, one can construct angular verniers which, inaddition, can be coupled with a micrometer screw for an even more precise lengthmeasurement

For higher precision, the scale is magnified with an optical microscope.Accuracies of 0.2m are possible in this way Precision techniques for subdivision

of scales and special instruments for comparisons have been developed The highestprecision can be reached by observing differences in interference fringes, set up bymonochromatic light between the ends of the objects to be compared For themaintenance of the standard meter a precision of 1 in 108

is possible as mentioned inAppendix 8

For the thermomechanical analyses described in Sect 4.5, which requiremeasurement of small changes in length, and similar applications, an electricalmeasurement of length is chosen It involves a linearly variable differentialtransformer, LVDT A change in the position of the core of the LVDT, which floatswithout friction in the transformer coil, results in a linear change in output voltage.For length measurement, the sample is placed as indicated in the top sketch ofFig 4.13 Variations in the length due to changes in temperature, force, or structurecan then be registered To eliminate the changes in length of the rods connected to the

4.1 Thermometry and Dilatometry _295

Supra Invar (63% Fe,

LVDT due to temperature changes, a differential setup may be used, as indicated inthe bottom sketch A reference sample is connected to the coil, so that only thedifferential expansion of the sample is registered Quartz is used frequently for theconstruction of the connecting rod and also as reference material It has a rather smallexpansivity when compared to other solids as can be seen from Table 4.1 Anothermaterial of interest is Invar, an alloy, developed to have a low expansivity

Four types of experimental setups for volume and density measurements are given

in Figs 4.14 and 4.15 Rarely is it possible to make a volume determination byfinding the appropriate lengths Almost always, the volume measurement will bebased on a mass determination, as described in Sect 4.6 For routine liquid volumemeasurements, common in chemistry laboratories, one uses the Type 1 volumetricequipment in the form of calibrated cylinders and flasks, pipettes, and burettes, as well

as the Type 2 pycnometers These instruments are calibrated at one temperature onlyand either for delivery of the measured liquid, as in case of burettes and pipettes, orthe volume contained, as in case of volumetric flasks and pycnometers Calibrationsare done by weighing a calibration liquid delivered when emptying, or weighing theinstrument filled with the liquid and correcting for the container weight

Fig 4.14

To determine the bulk volume of a solid, one uses a calibrated pycnometer asshown in Fig 4.14 After adding the weighed sample, the pycnometer is filled withmercury or other measuring fluid and brought up to the temperature of measurement.The excess measuring fluid is brushed off and the exact weight of the pycnometer,sample and measuring fluid is determined From the known density of the measuringfluid and the sample weight, the density of the sample is computed Today onehesitates to work with mercury without cumbersome safety precautions, but otherliquids or gas pycnometers do not quite reach the Hg precision

Figure 4.15 illustrates on the left as instruments of Type 3 a dilatometer usableover a wider temperature range at atmospheric pressure It consists of a precision-borecapillary, fused to a bulb containing the sample, indicated as black, irregular shapes.The spacers are made out of glass to act as thermal insulators during the sealing of thedilatometer by the glass blower The dilatometer is then evacuated through the topground-glass joint, and filled with mercury The whole dilatometer is, next, immersed

in a constant-temperature bath, and the mercury position in the 30-cm-long capillary

is read with a cathetometer The change in sample volume between a referencetemperature and the temperature of measurement is calculated using the indicatedequation where Vgis the volume change due to the glass or quartz of the dilatometerand spacers Routine accuracies of ±0.001 m3

/Mg can be accomplished The equation

in Fig 4.15, however, gives only changes from a fixed reference temperature One,thus, must start with a sample of known density at the reference temperature toevaluate the absolute volume as a function of temperature

To the right, Fig 4.15 illustrates Type 4 instrumentation, a particularly easymethod of density determination at a fixed reference temperature, the density gradientmethod Figure 4.16 depicts the analysis method The sample, checked for uniformityand freedom from attached air bubbles, is placed in a density gradient column, and its

4.1 Thermometry and Dilatometry _297

Fig 4.16

Fig 4.15

floatation height is measured by a cathetometer The two Erlenmeyer flasks A and Bcontain a heavy and a light, miscible liquid, respectively The cylinder is slowly filledthrough the capillary from the bottom up Stirring in flask B is sufficiently rapid tofully mix the liquid in flask B The first liquid delivered to the cylinder is thus thelight liquid As the meniscus in B drops, heavy liquid out of A is mixed into B andthe liquid delivered at the bottom of the density gradient column gets denser After

Fig 4.17

filling, one waits for a few hours to establish a stable density gradient Onceestablished, the gradient slowly flattens out by diffusion, but usually it is usable forseveral weeks The calibration of the density gradient is shown in the left graph ofFig 4.16 The points in the graph are established by finding the height of glass floats

of calibrated density The floats can be precise to ±0.0001 Mg m3 The height of afloating piece of sample can easily be measured to ±0.5 mm, which means that four-digit accuracy in density is possible

4.1.7 Application of Dilatometry

A typical length measurement by thermomechanical analysis, TMA (see Sect 4.5), isshown in Fig 4.17 on the change in dimensions of a printed circuit board made of anepoxy-laminated paper Measurements of this type are important for matching theexpansivities of the electronic components to be fused to the board, so that strain and

eventual fracture of the printed metal can be avoided The measurement is made inthis case under zero load, so that the bottom curve directly gives the change in lengthrelative to a reference length The derivative, simultaneously recorded, yields theexpansivity after changing from time to temperature The glass transition at 401 K iseasily established, and quantitative expansivities are derived, as is shown

Figure 4.18 illustrates volume dilatometry of an extended-chain, high-crystallinitypolyethylene sample (see also Chap 6) A close to equilibrium melting is observed

by such slow dilatometry with equipment illustrated on the left of Fig 4.15.The dilatometry at different pressures leads to a full p-V-T phase diagram Linearmacromolecules in the liquid state can reach equilibrium and have then beensuccessfully described by a single p-V-T diagram The semicrystalline and glassy

4.1 Thermometry and Dilatometry _299

1V = 3b; p = a/(3V ) = a/(27b); T = 8pV/(3R) = 8a/(27Rb); volume V is for one mole.

Fig 4.20

A schematic, three-dimensional, one-component, p-V-T diagram is reproduced inFig 4.20 Its surface represents all possible equilibrium states of the system The gasarea, especially at high temperature and volume, is well described by the ideal gas law,

at lower temperatures, the van der Waals equation is applicable as seen in Fig 2.99

The critical point can be derived mathematically from the van der Waals equation byidentifying the temperature for the p-V curve with a single horizontal tangent.1

At thecritical point, all gases are in corresponding states, i.e., they behave similarly Abovethe critical temperature there is a continuous change from the liquid to the gaseousstate The liquid-to-gas transition can thus occur either below Tcvia a first-ordertransition with an abrupt change in volume, enthalpy, and entropy, or above Tcwithcontinuous changes in the thermodynamic functions

Figure 4.21 displays the projections of the three-dimensional diagram of Fig 4.20into the pressure-volume and pressure-temperature planes Such curves are used tocharacterize the equilibrium phase behavior of a one-component system The diagram

on the right is simple, it collapses the two-phase areas into lines The left diagramshows the details of the two-phase areas and the critical point At T4a van der Waalscurve is seen, as in Fig 2.99 More detailed descriptions of the phase equilibria aregiven in Sect 2.5 Actual data for the system of ice and water are reproduced inFig 4.22 The most fascinating is perhaps the behavior of ice-I, the common ice atatmospheric pressure It is larger in volume than water and, thus, has according to theClausius-Clapeyron equation a decreasing melting temperature with pressure (seeSect 5.6) This trend is reversed with ice III, V, VI, and VII Many of the geologicaland biological developments on Earth are based on this abnormal behavior of ice

4.1 Thermometry and Dilatometry _301

Fig 4.22

Fig 4.21

To allow a better understanding of the condensed phase, the volume of a samplecan be divided into two parts: the van der Waals volume, Vw, which represents theactual volume of the molecules or ions in the hard-sphere approximation, taken fromFig 4.23, and the total, experimental volume V It must be remembered that the vander Waals radius depends somewhat on the forces that determine the approach of the

Fig 4.24

Fig 4.23

atoms considered For similar types of crystals and liquids, however, the hard-sphereapproximation of the atoms is useful The ratio of these two volumes gives thepacking fraction, k, as listed in in Fig 4.24 A large packing fraction k means that themolecules are well packed; a low k indicates a large amount of empty space.Restricting the discussion for the moment to identical spherical motifs, one can easilycompute that the highest packing fraction is 0.74 Such close packing of spheres leads

4.1 Thermometry and Dilatometry _303

to a coordination number, CN, of 12 nearest neighbors The resulting crystal structure

is a cubic, close-packed crystal or one of the various trigonal or hexagonal closepacks By placing the spheres randomly, but still packed as closely as possible, thepacking fraction drops to 0.64 An irregular pack with a coordination number of three,the lowest possible coordination number without building a structure which wouldcollapse, yields a very open structure with k = 0.22 Packing of spherical molecules

in the condensed phase could thus vary between 0.22 and 0.74

The packing fraction of rods is another easily calculated case It could serve as amodel for extended-chain, linear macromolecules Motifs of other, more irregularshapes are more difficult to assess The closest packing of rods with circular crosssection reaches a k of 0.91 with a coordination number of six Packing withcoordination number four reduces k to 0.79 A random heap of rods which do notremain parallel can result in quite low values for k which should also depend on thelengths of the rods

Making further use of packing fractions, one may investigate the suggestion that

at the critical point the packing fraction is at its minimum for a condensed phase, andthat at the glass transition temperature, packing for the random close pack is perhapsapproached, while on crystallization closest packing is achieved via the cubic orhexagonal close pack Unfortunately such a description is much too simplistic Anadditional accounting for differences in interaction energies and the more complicatedgeometry of actual molecules is necessary for an understanding of the various phases

of matter Volume considerations alone can only give a preliminary picture At best,molecules with similar interaction energies can be compared Looking at the packingfractions of liquid macromolecules at room temperature, discussed in Sect 5.4, sometrends can be observed A packing fraction of 0.6 is typical for hydrocarbon polymers.Adding >NH, O, >C=O, CF2, >S or >Se to the molecule can substantiallyincrease the packing fraction

It is also of interest to compare the expansivity, the derivative of the extensivequantity volume, to specific heat capacity, the derivative of the extensive quantityenthalpy A detailed discussion of heat capacity is possible by considering harmonicoscillations of the atoms (see Sect 2.3) A harmonic oscillator does not, however,change its average position with temperature Only the amplitudes of the vibrationsincrease To account for the expansivity of solids, one thus, must look at models thatinclude the anharmonicity of the vibrations Only recently has it been possible tosimulate the dynamics of crystals with force fields that lead to anharmonic vibrations(see Figs 1.44–48) Despite this difference, the expansivity and heat capacity forliquids and glasses behave similarly (see Sect 2.3) The reason for this is the largerinfluence of the change in potential energy (cohesive energy) with volume

This concludes the discussion of thermometry and dilatometry The tools tomeasure temperature, length, and volume have now been analyzed The tools formeasurement of heat, the central theme of this book, will take the next three sectionsand deal with calorimetry, differential scanning calorimetry, and temperature-modulated calorimetry The mechanical properties which involve dilatometry ofsystems exposed to different and changing forces, are summarized in Sect 4.5 Themeasurement of the final basic variable of state, mass, is treated in Sect 4.6 whichdeals with thermogravimetry

4.2.1 Principle and History

The SI unit of heat, as well as of work and energy is the joule, J as summarized inFig 2.3 Its dimension is expressed in [kg m2

s2] Heat and work describe the energyexchanged between thermodynamic systems, as discussed in Sect 2.1.5 with theequation

specific heat capacity of water (1 calthermochemical= 4.184 J) Since the early 20th

century,however, energy, heat, and work are more precisely determined in joules, making thecalorie a superfluous unit The calorie is not part of the SI units and should beabandoned All modern calorimetry is ultimately based on a comparison with heatgenerated by electrical work

Almost all calorimetry is carried out at constant pressure, so that the measured heat

is the change in enthalpy, as discussed in Chap 2 The thermodynamic functions thatdescribe a system at constant pressure are listed in Fig 4.25 These functions containthe small correction caused by the work term

called Gibbs function or Gibbs energy replaces the Helmholtz free energy and is used

as measure of thermodynamic stability (see Sect 2.2.3) The added pV term in thesefunctions represents the needed volume-work for creation of the space for the system

at the given, fixed pressure p As indicated in Fig 4.25, the pV-term is small Adifference of 0.1 J raises the temperature of one cm3

of water by a little more than0.02 K In addition, many processes have only a small change in volume ( V) Theforce-times-length term, fl, provides a similar correction for work exchanged bytensile force during a calorimetric experiment as occurs on changing the length, orgeometry in general, of a sample, which is rubber-elastic (see Sect 5.6.5)

A calorimeter does not allow one to find the total heat content (H) of a system in

a single measurement, such as one can for other extensive quantities like volume bydilatometry in Sect 4.1 or mass by thermogravimetry in Sect 4.6 A calorimeter isthus not a total-heat meter Heat must always be determined in steps as H, and thensummed from a chosen reference temperature The two common referencetemperatures are 0 K and 298.15 K (25°)

Three common ways of measuring heat are listed at the top of Fig 4.26 First, thechange of temperature in a known system can be observed and related to the flow ofheat into the system It is also possible, using the second method, to follow a change

of state, such as the melting of a known system, and determine the accompanying flow

of heat from the amount of material transformed in the known system Finally, inmethod three, the conversion to heat of known amounts of chemical, electrical, ormechanical energy can be used to duplicate or compensate a flow of heat

4.2 Calorimetry _305

Fig 4.25

Fig 4.26

The prime difficulty of all calorimetric measurements is the fact that heat cannot

be perfectly contained There is no ideal insulator for heat During the time oneperforms the measurement, there are continuous losses or gains from the surroundings.Even when a perfect vacuum surrounds the system under investigation, heat is lost andgained by radiation Because of these heat-loss difficulties, experimental calorimetryhas not received as much development as one would expect from its importance

Joseph Black, 1728–1799 British chemist and physicist who discovered carbon dioxide,which he also found in air, and the concept of latent heat He noticed that when ice melts, ittakes up heat without change of temperature He held a position as Professor of Chemistry andAnatomy at the University of Glasgow, Scotland and also practiced Medicine

Fig 4.27

The earliest reasonably accurate calorimetry seems to have been carried out in the

18th

century In 1760 Joseph Black1

described calorimetry with help of two pieces ofice, as sketched in Fig 4.26 The sample is placed into the hollow of the bottompiece A second slab is put on top After the sample has acquired temperature-equilibrium at 273.15 K (0°C), the amount of water produced is mopped out of thecavity and weighed (Wwater) Equations (1) and (2) show the computation of theaverage specific heat capacity, c, from the latent heat of water, L One expects theresults are not of highest accuracy, although with care, and perhaps working in a coldroom at about To, an accuracy of perhaps ±5% might be possible This is a respectableaccuracy compared to the much more sophisticated calorimeters used today whichoften does not exceed ±1%

In 1781 de la Place published the description of a much improved calorimeter Apicture of it can be found in the writings of Lavoisier [4] and is shown in Fig 4.27

The outer cavity, a, and the lid, F, are filled with ice to insulate the interior of the calorimeter from the surroundings Inside this first layer of ice, in space b, a second

layer of ice is placed, the measuring layer Before the experiment is started, this

measuring ice is drained dry through the stopcock, y Then, the unknown sample in

basket LM, closed with the lid HG and kept at a known temperature T1, is quickly

dropped into the calorimeter, f, and the lid, F, is closed After 812 hours for equilibration of the temperature, the stopcock, y, is opened and the drained water,

4.2 Calorimetry _307

1Robert Wilhelm Bunsen, 1911–1899 Professor of Chemistry at the University of Heidelberg,Germany He observed in 1859 that each element emits light of a characteristic wavelength.These studies of spectral analysis led Bunsen to his discovery of cesium and rubidium

weighed The heat that flowed from the sample to the measuring ice is thus measured

as in the experiment of Black At the bottom of Fig 4.27, some of the experimentaldata are listed that were obtained by Lavoisier This calorimeter was also used todetermine the thermal effects of living animals such as guinea pigs

4.2.2 Isothermal and Isoperibol Calorimeters

The name calorimeter is used for the combination of sample and measuring system,kept in well-defined surroundings, the thermostat or furnace To describe the nextlayer of equipment, which may be the housing, or even the laboratory room, one usesthe term environment For precision calorimetry the environment should always bekept from fluctuating The temperature should be controlled to ±0.5 K and the roomshould be free of drafts and sources of radiating heat

Calorimeters can be of two types, 1 isothermal and isoperibol, or 2 adiabatic.Isothermal calorimeters have both calorimeter and surroundings at constant To If onlythe surroundings are isothermal, the mode of operation is isoperibol (Gk.­)#+, equal,

%J'?-7, surround) In isoperibol calorimeters the temperature changes withtime, governed by the thermal resistance between the calorimeter and surroundings

In adiabatic calorimeters, the exchange of heat between a calorimeter and ings is kept close to zero by making the temperature difference small and the thermalresistance large

surround-To better assess heat losses, twin calorimeters have been developed that permitmeasurement in a differential mode A continuous, usually linear, temperature change

of calorimeter or surroundings is used in the scanning mode The calorimetry,described in Sect 4.3 is scanning, isoperibol twin-calorimetry, usually less preciselycalled differential scanning calorimetry (DSC)

Perhaps the best-known isothermal phase-change calorimeter is the Bunsen1

calorimeter, invented in 1870 [5] This calorimeter is strictly isothermal and has, thus,practically no heat-loss problem The drawing in Fig 4.28 shows the schematics Themeasuring principle is that ice, when melting, contracts, and this volume contraction

ice-is measured by weighing the corresponding amount of mercury drawn into thecalorimeter The unknown sample is dropped into the calorimeter, and any heatexchange is equated to the heat exchanged with the measuring ice Heat-loss or -gain

of the calorimeter is eliminated by insulation with a jacket of crushed ice and water,contained in a Dewar vessel An ice-calorimeter is particularly well suited to measurevery slow reactions because of its stability over long periods of times The obviousdisadvantage is that all measurements must be carried out at 273.15 K, the meltingtemperature of ice For a more modern version see [6]

Figure 4.29 represents an isoperibol drop calorimeter [7] The surroundings are

at almost constant temperature and are linked to the sample via a heat leak Therecipient is a solid block of metal making it an aneroid calorimeter (Fr anérọde, notusing a liquid, derived from the Gk.) The solid block eliminates losses due to

Fig 4.28

Fig 4.29

evaporation and stirring that occur in a liquid calorimeter, as shown in Fig 4.30 Butits drawbacks are less uniformity of temperature relative to the liquid calorimeter andthe need of a longer time to reach steady state

For measurement, the sample is heated to a constant temperature in a thermostatabove the calorimeter (not shown), and then it is dropped into the calorimeter, whereheat is exchanged with the block The small change in temperature of the block is

4.2 Calorimetry _309

Fig 4.30

used to calculate the average heat capacity Heats of reaction or mixing can also bemeasured by dropping one of the components into the calorimeter or breaking anampule of one of the reactants to initiate the process More advanced versions of thistype of calorimeter involve compensation of heat flow, as shown in Sect 4.2.5.Figure 4.30 illustrates the liquid calorimeter It also operates in an isoperibolmanner The cross-section represents a simple bomb or reaction calorimeter, as isordinarily used for the determination of heats of combustion [8] The reaction iscarried out in a steel bomb, filled with oxygen and the unknown sample The reaction

is started by electrically burning the calibrated ignition wire The heat evolved during

the ensuing combustion of the sample is then dissipated in the known amount of waterthat surrounds the bomb, contained in the calorimeter pail From the rise in

temperature and the known water value, W, of the whole setup, the heat of reaction

can be determined:

H = W Twhere H is the heat of reaction and T, the measured increase in temperature Thewater value is equivalent to the heat capacity of a quantity of water which equals that

of the whole measuring system

Much of the accuracy in bomb calorimetry depends upon the care taken in theconstruction of the auxiliary equipment of the calorimeter, also called the addenda

It must be designed such that the heat flux into or out of the measuring water is at aminimum, and the remaining flux must be amenable to a calibration In particular, theloss due to evaporation of water must be kept to a minimum, and the energy inputfrom the stirrer must be constant throughout the experiment With an apparatus such

as shown in Fig 4.30 anyone can reach, with some care, a precision of ±1%, but it ispossible by most careful bomb calorimetry to reach an accuracy of ±0.01%

Fig 4.31

In both the aneroid and liquid calorimeters, a compromise in the block and pailconstruction has to be taken The metal or liquid must be sufficient to surround theunknown, but it must not be too much, so that its temperature-rise permits sufficientaccuracy in T measurement The calibration of isothermal calorimeters is best donewith an electric heater in place of the sample, matching the measured effect as closely

as possible To improve the simple calculation of the output of the aneroid and liquidcalorimeters, a loss calculation must be carried out as described in the next section

4.2.3 Loss Calculation

To improve the measured T, in isoperibol calorimetry, the heat losses must naturally

be corrected for Loss calculations are carried out using Newton's law of cooling,written as Eq (1) in Fig 4.31 (see also Fig 4.9) The change in temperature with

time, dT/dt, is equal to some constant, K, multiplied by To, the thermal head (i.e., theconstant temperature of the surroundings), minus T, the measured temperature of thecalorimeter In addition, the effect of the stirrer, which has a constant heat input withtime, w, must be considered in the liquid calorimeter in Eq (2) The same term w alsocorrects any heat loss due to evaporation

The graph in Fig 4.32 and the data in Table 4.2 show a typical example ofcalorimetry with a liquid calorimeter The experiment is started at time t1 andtemperature T1 The initial rate of heat loss is determined in the drift measurement

If the thermal head of the calorimeter, To, is not far from the calorimeter temperature,

a small, linear drift should be experienced The measurement is started at t2, T2 Thisprocess may be combustion, mixing of two liquids, or just dropping a hot or coldsample into the calorimeter A strong temperature change is noted between t and t

4.2 Calorimetry _311

t (min) T (K) (index)

This is followed by the period of final

drift, between t3 and t4 The experi

ment is completed at time t4and

tem-perature T4

The detailed analysis of the curve

is continued in Fig 4.32 The

tem-peratures and times, T2, T3and t2, t3,

respectively, are established as the

points where the linear drifts of the

initial and final periods are lost or

gained The equations for the initial

and final drifts, Ri and Rf, are given

by Eq (3) and Eq (4) They are used

to evaluate Newton’s law constant K,

as shown by Eq (5), and the stirrer

correction w, as shown by Eq (6)

With these characteristic constants

evaluated, the actual jump in

tempera-ture ... Peltier effect of hundreds of thermocouples A short summary of the threethermoelectric effects, of which one is the Peltier effect, is given in Fig 4. 36 (see alsoFig 4.7 for the application of the Seebeck... accuracy of ±0.1% and a temperature range of 170

to 60 0 K A sample of 100–300 g is placed in two sets of silver trays, one outside andone inside a cylindrical heater In the middle of the... 1911–1899 Professor of Chemistry at the University of Heidelberg,Germany He observed in 1859 that each element emits light of a characteristic wavelength.These studies of spectral analysis led