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may also be made in form of large disks, so that they can average the temperature over a larger object.

The principle of resistance measurement involves a dc Wheatstone bridge, as in the sketch of Fig. 4.6. The calculations show how the lead resistances, RC and RT, can be eliminated in precision thermometry by performing two measurements (a and b) with reversed leads connected to the bridge circuit. The measured resistances are represented by RD, the unknown resistance by RX.

Fig. 4.6

A third type of frequently-used thermometer is the thermocouple. The thermocouple is based on the Seebeck effect (see also Fig. 4.36). At the contact points of two dissimilar metals a potential difference is created because some of the electrons in the material 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 with the higher work function as holding the electrons more tightly. When a circuit of two different materials is set up, as is shown in the top drawing in Fig. 4.7, and a voltmeter is inserted in one of the branches, one observes no voltage as long as the two junction points are at equal temperature. The potential difference created in one junction by the drift of electrons from the low-work-function to the high-work-function metal is exactly opposite to the potential difference created in the other junction. If, however, the two junctions are kept at different temperatures, one observes a voltage or electromotive force (emf), EAB. This electromotive force can be used to measure the temperature difference between the junction points of the thermocouple.

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

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

The main difficulty in high-precision temperature measurements with thermocouples is the introduction of spurious voltages by metal junctions outside of the measuring and reference junctions. In Fig. 4.8 two typical measuring circuits are sketched. The right circuit overcomes most of the difficulties by using only thermocouple metals for the circuit (including any switches and connectors!). One only has to watch that the two connections to the measuring instrument are at the same temperature and that no additional thermal emf is introduced inside the meter. The second circuit in Fig. 4.8 eliminates the need to carry the thermocouple wires to the measuring instrument by changing from thermocouple material to normal conductance copper in the reference ice bath. In this arrangement all subsequent Cu/Cu junction potentials cancel.

The measuring of the emf must naturally be carried out in such a way that practically 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 zero current. At the bottom of Fig. 4.8 the circuit diagram is given for such a potentiometer. Today electronic voltmeters draw practically no current and potentiometers are becoming old-fashioned. A modern, high-impedance voltmeter can, in addition, be digital and be already calibrated for a given thermocouple. Also, it is possible to eliminate the reference junction at to by providing an appropriate counter-emf. But note 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 is concluded with a listing of some additional instruments which offer promise for special applications:

Quartz thermometer (measurement of the frequency of an oscillator, controlled by a quartz crystal with linear temperature dependence—resolution 0.0001 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 the maintenance of ITS 90 as given in Appendix 8. The calibration needs to be done at one temperature only).

Bimetallic thermometer (bimetallic strip which shows a deflection due to differential expansivity that is proportional to temperature—frequently used for temperature control by coupling the bimetallic strip to a mechanical switch to control the chosen device).

Vapor pressure thermometer (pressure measurement above a liquid in contact with the unknown system—particularly useful at low temperatures where other types 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 thermal agitation of electrons in conductors, detected by high amplification of the signal).

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4.1.4 Application of Thermometry

Several typical thermometry applications are treated in connection with the molar mass determination of macromolecules in Sect. 1.4.3 and in a study of melting with optical and atomic force microscopy in Figs. 3.95–97. The evaluation of phase diagrams 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 is shown 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

Fig. 4.9

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

where Cp is 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-flow rate, dQ/dt:

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

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where L is the latent heat coupled to the change in n. This equation describes the heating curve at the eutectic temperature. Above the eutectic temperature, both heating and melting occur. In this case, both partial differentials must be properly combined, as suggested in Fig. 2.10 (dH = CpdT + Ldn). The changing amount of melting with temperature accounts for the changing slope between the eutectic and the liquidus temperature in Fig. 4.9. A special complication arises with the crystallization illustrated in Fig. 1.67, where a certain amount of supercooling is necessary to nucleate the crystallization of the sample (see Sect. 3.5). The equilibrium melting/crystallization is passed to lower temperature, followed by self-heating to equilibrium, 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 phase diagram (see Chap. 7). In addition, the heating curve indicates a heating apparatus that delivers a constant heat input as a function of time. In the first part of the experiment the temperature-increase is linear, indicating a constant heat capacity.

Whenever the eutectic temperature, Teutectic is reached, the temperature is constant, it increases only after all B-crystals are melted, along with enough A crystals to give the

eutectic concentration ce. Beyond the eutectic temperature, the slope of the temperature-versus-time curve is less than before since melting of A crystals continues, increasing the concentration of A in the melt from the eutectic concentration to the overall concentration, cs, given by the liquidus line. At the liquidus

temperature, Tliquidus, melting of A is completed and the original slope of the temperature increase is resumed. A simple thermometry experiment, thus, permits the

measurements of two temperatures, Teutectic and Tliquidus, and fixes two points in the phase diagram. Starting with different concentrations, the complete phase diagram

can be mapped. When calibrating K, it may even be possible to evaluate heat capacity and 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 conductivities and heat capacities, progress in calorimetry by scanning experiments was only made after 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 radiation wavelength described in Appendix 8 and listed in Fig. 4.10. From the present best value of the pole-to-pole circumference of the earth of 40.009160×106 m one can see that originally the meter was chosen to make this circumference come out to be 4×107 m. Such a definition changes, however, with time since measurements of the circumference of the earth are improving steadily and natural changes may occur.

The unit for volume is the cubic meter. For density measurements this makes an unhandy, 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).

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Fig. 4.11

how accurately historical standards could be reproduced. The Roman foot varied over several centuries by as little as 1/400 about the mean.

The establishment of volume standards is even more difficult. An interesting example is the ladder of units in use in Elizabethan England. It started with the smallest unit, the mouthful. Two mouthfuls gave the jigger, two of which, in turn, made a jack, and two of those were the jill. Two jills gave a cup, two cups a pint, two pints a quart, two quarts a pottle. Then follows the gallon, next the peck, the double peck, and the bushel. Doubling further, one gets, in sequence, to the cask, the barrel, the hogshead, the pipe, and finally, the tun. Many of these units had their own, fluctuating history. The mouthful, for example, was already in use as a unit in ancient Egypt. It is mentioned in the Papyrus Ebers as the basic unit for mixing medicines (the ro, equal to 15 cm3). It may be difficult, however, to calibrate a barrel with 8192 mouthfuls. The volume tun has the same root as the mass unit ton. When referring to water, their sizes are similar. The present-day metric ton represents 106 cm3 of water and leads to a size of the mouthful of 15.3 cm3. Even mouths seem not to have changed much over the centuries. The connection between mass and volume was the most difficult branch of metrology. Hundreds of units have been described, each pointing 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 a standard meter as shown at the top of Fig. 4.12. Help in reading the divisions of the standard scale can be given by optical magnification or by designing a micrometer screw that allows one turn of the screw to be divided into 360 degrees, readable perhaps to a fraction of one degree. The length shown can be read as 22.45 units.

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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 0.9 units, 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 the length is read at the exactly matching divisions. The calculations in the figure show the validity of the method. Similarly, one can construct angular verniers which, in addition, can be coupled with a micrometer screw for an even more precise length measurement.

For higher precision, the scale is magnified with an optical microscope. Accuracies of 0.2 m are possible in this way. Precision techniques for subdivision of scales and special instruments for comparisons have been developed. The highest precision can be reached by observing differences in interference fringes, set up by monochromatic light between the ends of the objects to be compared. For the maintenance of the standard meter a precision of 1 in 108 is possible as mentioned in Appendix 8.

For the thermomechanical analyses described in Sect. 4.5, which require measurement of small changes in length, and similar applications, an electrical measurement of length is chosen. It involves a linearly variable differential transformer, LVDT. A change in the position of the core of the LVDT, which floats without 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 of Fig. 4.13. Variations in the length due to changes in temperature, force, or structure can then be registered. To eliminate the changes in length of the rods connected to the

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Fig. 4.13

LVDT due to temperature changes, a differential setup may be used, as indicated in the bottom sketch. A reference sample is connected to the coil, so that only the differential expansion of the sample is registered. Quartz is used frequently for the construction of the connecting rod and also as reference material. It has a rather small expansivity when compared to other solids as can be seen from Table 4.1. Another material 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 by finding the appropriate lengths. Almost always, the volume measurement will be based on a mass determination, as described in Sect. 4.6. For routine liquid volume measurements, common in chemistry laboratories, one uses the Type 1 volumetric equipment 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 only and either for delivery of the measured liquid, as in case of burettes and pipettes, or the volume contained, as in case of volumetric flasks and pycnometers. Calibrations are done by weighing a calibration liquid delivered when emptying, or weighing the instrument filled with the liquid and correcting for the container weight.

Table 4.1. Typical Linear Expansivities for Solids

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Fig. 4.14

To determine the bulk volume of a solid, one uses a calibrated pycnometer as shown in Fig. 4.14. After adding the weighed sample, the pycnometer is filled with mercury 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 measuring fluid and the sample weight, the density of the sample is computed. Today one hesitates to work with mercury without cumbersome safety precautions, but other liquids or gas pycnometers do not quite reach the Hg precision.

Figure 4.15 illustrates on the left as instruments of Type 3 a dilatometer usable over a wider temperature range at atmospheric pressure. It consists of a precision-bore capillary, 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 the dilatometer by the glass blower. The dilatometer is then evacuated through the top ground-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 reference temperature and the temperature of measurement is calculated using the indicated equation where Vg is the volume change due to the glass or quartz of the dilatometer and 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 to evaluate the absolute volume as a function of temperature.

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