Temperature differences

When there are sufficient data at different temperatures, the temperature dependence of the parameters is reflected in the confidence ellipses (Bryson and Ho, 1969 Draper and Smith,... 

Figure 15.1a shows a single-stage evaporator represented on both actual and shifted temperature scales. Note that in shifted temperature scale, the evaporation and condensjftion duties are shown at different temperatures even though they are at the same actual temperature. Figure 15.16 shows a similar plot for a three-stage evaporator. 

The measurement of a crude oil s viscosity at different temperatures is particularly important for the calculation of pressure drop in pipelines and refinery piping systems, as well as for the specification of pumps and exchangers. 

The corrosion rate of steel in carbonic acid is faster than in hydrochloric acid Correlations are available to predict the rate of steel corrosion for different partial pressures of CO2 and different temperatures. At high temperatures the iron carbonate forms a film of protective scale on the steel s surface, but this is easily washed away at lower temperatures (again a corrosion nomogram is available to predict the impact of the scale on the corrosion rate at various CO2 partial pressures and temperatures). 

The experiment could be repeated at a number of different temperatures and initial pressures to determine the shape of the two-phase envelope defined by the bubble point line and the dew point line. These two lines meet at the critical point, where it is no longer possible to distinguish between a compressed gas and a liquid. 

The curve of Fig. XVII-15 is essentially a characteristic curve of the Polanyi theory, but in the form plotted in might better be called a characteristic isotherm. Furthermore, as would be expected from the Polanyi theory, if the data for a given adsorbate are plotted with RTln P/f ) as the abscissa instead of just ln(P/P ), then a nearly invariant shape is obtained for different temperatures. The plot might then be called the characteristic adsorption curve. 

It suffices to carry out one such experiment, such as the expansion or compression of a gas, to establish that there are states inaccessible by adiabatic reversible paths, indeed even by any adiabatic irreversible path. For example, if one takes one mole of N2 gas in a volume of 24 litres at a pressure of 1.00 atm (i.e. at 25 °C), there is no combination of adiabatic reversible paths that can bring the system to a final state with the same volume and a different temperature. A higher temperature (on the ideal-gas scale Oj ) can be reached by an adiabatic irreversible path, e.g. by doing electrical work on the system, but a state with the same volume and a lower temperature Oj is inaccessible by any adiabatic path. 

For an ideal gas and a diathemiic piston, the condition of constant energy means constant temperature. The reverse change can then be carried out simply by relaxing the adiabatic constraint on the external walls and innnersing the system in a themiostatic bath. More generally tlie initial state and the final state may be at different temperatures so that one may have to have a series of temperature baths to ensure that the entire series of steps is reversible. 

Figure A2.3.22 (a) The free energy Gand (b) tire magnetizationas a fiinetion of the magnetie field// at different temperatures, (e) The magnetization m(H,T) and (d) the suseeptibility x as a fiinetion of temperature.

Figure A2.3.24 Plot of tanh[p qJm 0,T) versus m(0,T) at different temperatures.

Figure A2.3.25 The magnetic field versus the magnetization at different temperatures.

Measuring tire pressure dependence of k at different temperatures shows that the apparent activation energy at constant viscosity decreases with increasing viscosity [46, ( figure A3,6,8). From a detailed analysis one... 

There are two main applications for such real-time analysis. The first is the detemiination of the chemical reaction kinetics. Wlien the sample temperature is ramped linearly with time, the data of thickness of fomied phase together with ramped temperature allows calculation of the complete reaction kinetics (that is, both the activation energy and tlie pre-exponential factor) from a single sample [6], instead of having to perfomi many different temperature ramps as is the usual case in differential themial analysis [7, 8, 9, 10 and H]. The second application is in detemiining the... 

Figure C2.1.5. Reduced osmotic pressure FT / (RTc as a function of the weight concentration c of polystyrene (M = 130 000 g mor ) in cyclohexane at different temperatures. At 7"= 35 °C and ambient pressure, tire solution is at tire 0 conditions. (Figure from 1741, reprinted by pennission of EDP Sciences.)...

As an example of steam distillation, let us consider bromobenzene which has a normal boiling point of 155°. The vapour pressures of water and bromobenzene at different temperature.s are given in the following table. 

System in which the solid phases consist of the pure components and the components are completely miscible in the liquid phase. We may now conveniently consider the general case of a system in which the two components A and B are completely miscible in the liquid state and the solid phases consist of the pure components. The equilibrium diagram is shown in Fig. 1,12, 1. Here the points A and B are the melting points of the pure components A and B respectively. If the freezing points of a series of liquid mixtures, varying in composition from pure A to pure B, are determined, the two curves represented by AC and BC will be obtained. The curve AC expresses the compositions of solutions which are in equilibrium, at different temperatures, with the solid component A, and, likewise, the curve BC denotes the compositions... 

FIGURE 2.1 Fraction of molecules that will be found at various energies above the ground-state energy for two different temperatures. 

Conversion of Specific Gravity at 25°/25°C to Density at any Temperature from 0° to 40°C. Liquids change volume with change in temperature, but the amount of this change, /3 (coefficient of cubical expansion), varies widely with different liquids, and to some extent for the same liquid at different temperatures. 

The values in the table below are to be subtracted from the observed readings to correct for the difference in the expansion of the mercury and the glass scale at different temperatures. 

When the weight of the water is determined at a temperature of t°C, and that of the liquid at a different temperature t, the equations above are modified as follows ... 

The electrons have a range of kinetic energies and are therefore at different temperatures. Depending on the strength of the applied electric field, some electrons in the swarm will have... 

Eor two different esters of cellulose, 1q values are listed here in two different temperature regions, 30 and 130-140°C ... 

Viscosity is considerably more sensitive to temperature than elasticity. By varying the temperature, the relaxation time of the polymer will be changed. Hence different mechanical response might be expected on a fixed laboratory time scale for samples examined at different temperatures. 

Equivalent mechanical behavior can be achieved by either time (or frequency) or temperature manipulation. As noted in Sec. 3.2, results measured at different temperatures can be reduced to a common temperature to describe response over a wide range of times. We shall consider data reduced to a common temperature in this chapter and discuss the reduction process in Chap. 4. 

Figure 3.16 Some experimental dynamic components, (a) Storage and loss compliance of crystalline polytetrafluoroethylene measured at different frequencies. [Data from E. R. Fitzgerald, J. Chem. Phys. 27 1 180 (1957).] (b) Storage modulus and loss tangent of poly(methyl acrylate) and poly(methyl methacrylate) measured at different temperatures. (Reprinted with permission from J. Heijboer in D. J. Meier (Ed.), Molecular Basis of Transitions and Relaxations, Gordon and Breach, New York, 1978.)...

As noted above, not all techniques which provide information regarding crystallinity are useful to follow the rate of crystallization. In addition to sufficient sensitivity to monitor small changes, the method must be rapid and suitable for isothermal regulation, quite possibly over a range of different temperatures. Specific volume measurements are especially convenient for this purpose. We shall continue our discussion using specific volume as the experimental method. 

Figure 4.8a shows how this quantity varies with time for polyethylene crystallized at a series of different temperatures. Several aspects of these curves are typical of all polymer crystallizations and deserve comment ... 

Next let us examine an experimental test of the Avrami equation and the assortment of predictions from its various forms as summarized in Table 4.3. Figure 4.9 is a plot of ln[ln(l - 0)" ] versus In t for poly (ethylene terephtha-late) at three different temperatures. According to Eq. (4.35), this type of... 

Figure 4.9 Log-log plot of ln(l - 6) versus time for poly(ethylene tereph-thalate) at three different temperatures. [Reprinted from L. B. Morgan, Philos. Trans. R. Soc. London 247A 13 (1954).]...

This equation is the basis for viscosity determination by measuring flow times through a capillary. It can also be used to describe a single liquid at two different temperatures, as required for Eq. (4.63). Combining Eqs. (4.63) and (4.64) yields... 

In describing the various mechanical properties of polymers in the last chapter, we took the attitude that we could make measurements on any time scale we chose, however long or short, and that such measurements were made in isothermal experiments. Most of the experimental results presented in Chap. 3 are representations of this sort. In that chapter we remarked several times that these figures were actually the result of reductions of data collected at different temperatures. Now let us discuss this technique our perspective, however, will be from the opposite direction taking an isothermal plot apart. 

The crystallization of poly(ethylene terephthalate) at different temperatures after prior fusion at 294 C has been observed to follow the Avrami equation with the following parameters applying at the indicated temperatures ... 

The time-temperature superpositioning principle was applied f to the maximum in dielectric loss factors measured on poly(vinyl acetate). Data collected at different temperatures were shifted to match at Tg = 28 C. The shift factors for the frequency (in hertz) at the maximum were found to obey the WLF equation in the following form log co + 6.9 = [ 19.6(T -28)]/[42 (T - 28)]. Estimate the fractional free volume at Tg and a. for the free volume from these data. Recalling from Chap. 3 that the loss factor for the mechanical properties occurs at cor = 1, estimate the relaxation time for poly(vinyl acetate) at 40 and 28.5 C. 

Fox and Schneckof carried out the free-radical polymerization of methyl methacrylate between -40 and 250 C. By analysis of the a-methyl peaks in the NMR spectra of the products, they determined the following values of a, the probability of an isotactic placement in the products prepared at the different temperatures ... 

Figure 8.13 shows the reduced osmotic pressure for solutions of polyisobutylene in benzene plotted against C2 at several different temperatures. The... 

The osmotic pressure of solutions of polystyrene in cyclohexane was measuredf at several different temperatures, and the following results were obtained ... 

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