Energy-temperature relationship

The empirically fitted free-energy-temperature relationship (in joules) is... 

Earlier reports of complex energy-temperature relationships (e.g. dapsone and ethambutol chloride (Kuhnert-Brandstatter and Moser 1979)) or unusual crystal chemistry (e.g. oxyclozanide (Pearson and Varney 1973)) also warrant serious consideration for reinvestigation. The formulation implications of chemical reactivity in solid-state pharmaceuticals have recently been reviewed by Byrn et al. (2001). 

Although amorphous pharmaceutical materials can be readily isolated and may persist for many thousands of years,they are in fact a thermodynamically metastable state and will eventually revert to the more stable crystalline form. Fig. 4 shows a snapshot in time of the free energy-temperature relationship for a material that can be isolated as both an amorphous form and a crystalline form. This quasi-equilibrium thermodynamic view of the amorphous state shows that the amorphous form has a significantly higher free energy than the crystalline form, and illustrates why it is expected to have a much higher aqueous solubility and significantly different physical properties (e.g., density). 

It is important to remember that the energy-temperature relationships discussed above are thermodynamic only. Metastable forms may readily convert to a more stable form in the solid state (Fig. 39) or may never convert on the human... 

Figure 3. Schematic energy-temperature relationships for (a) an enantiotropic systoi, (b) a monotropic system, (c) a hypothetical trimorphic system.

The transformation of triglycerides from unstable or metastable forms to stable polymorphs occurs spontaneously and irreversibly at the expense of the unstable forms. Figure 19 shows the Gibbs Ifee energy-temperature relationships... 

Figure 19 Gibbs free energy-temperature relationship of monoacid triglyceride polymorphs. 

The temperature dependence of a rate is often described by the temperature dependence of the rate constant, k. This dependence is often represented by the Arrhenius equation, /c = Aexp(- a/i T). For some reactions, the temperature relationship is instead written fc = AT" exp(- a/RT). The A term is the frequency factor for the reaction, which reflects the number of effective collisions producing a reaction. a is known as the activation energy for the reaction, and is a measure of the amount of energy input required to start a reaction (see also Benson, 1960 Moore and Pearson, 1981). 

The number of defects is maximal in the amorphous and liquid states. The phase diagram in Figure 5 shows the volume-temperature relationships of the liquid, the crystalline form, and the glass (vitreous state or amorphous form) [14], The energy-temperature and enthalpy-temperature relationships are qualitatively similar. 

Figure 8.5 Pressure—temperature relationship for gases. As the temperature increases, the gas particles have greater kinetic energy (longer arrows) and collisions...

The Celsius scale is a relative scale. It was designed so that water s boiling point is at lOO C and water s melting point is at 0 C. The Kelvin scale, on the other hand, is an absolute scale. It was designed so that 0 K is the temperature at which a substance possesses no kinetic energy. The relationship between the Kelvin and Celsius scales is shown in Figure 5.2, and by the following equation. 

If one is able to collect the combustion products after a combustion experiment, the combustion temperature can be determined from the energy conservation relationship for the reactants and products. For example, when iron and potassium perchlorate react to produce heat, the reaction products and heat of reaction, Q(r), can be determined by reference to thermochemical tables (NASA SP-273). In this case, the reaction of iron (0.84 mass fraction = 0.929 moles) and potassium perchlorate (0.16 mass fraction = 0.071 moles) is represented by... 

Therefore phase changes can be predicted from thermodynamic information on the free energy-composition-temperature relationship. 

A certain amount of energy will be required to raise the molecules to a level of kinetic energy where they will escape. In the special case of a liquid passing to the vapor state, the energy put into the system to cause volatilization is the latent heat of vaporization. The property is characteristic for a given chemical and may vary with temperature. The temperature relationship of the latent heat of vaporization may be calculated by the Clausius-Clapyeron equation. 

Write the material- and energy-balance expressions for the reactor. This problem must be solved by simultaneous solution of the material- and energy-balance relationships that describe the reacting system. Since the reactor is well insulated and an exothermic reaction is taking place, the fluid in the reactor will heat up, causing the reaction to take place at some temperature other than where the reaction rate constant and heat of reaction are known. 

Electromotive force measurements of the cell Pt, H2 HBr(m), X% alcohol, Y% water AgBr-Ag were made at 25°, 35°, and 45°C in the following solvent systems (1) water, (2) water-ethanol (30%, 60%, 90%, 99% ethanol), (3) anhydrous ethanol, (4) water-tert-butanol (30%, 60%, 91% and 99% tert-butanol), and (5) anhydrous tert-butanol. Calculations of standard cell potential were made using the Debye-Huckel theory as extended by Gronwall, LaMer, and Sandved. Gibbs free energy, enthalpy, entropy changes, and mean ionic activity coefficients were calculated for each solvent mixture and temperature. Relationships of the stand-ard potentials and thermodynamic functons with respect to solvent compositions in the two mixed-solvent systems and the pure solvents were discussed. 

Fig. 4.15 Characteristic free-energy temperature diagram (a) and DSC traces (b) for the enantiotropic relationship between polymorphs. The Gi and Gu curves cross at the transition temperature 7[ n below their melting points mpi, and mpn all indicated on the temperature axis. DSC trace A at the transition temperature modification I undergoes an endothermic transition to modification II, and the heat absorbed is A/fi n for that transition. Modification II then melts at mpn, with the accompanying AHfu. DSC trace B Modification I melts at mpi with A//n followed by crystaUization of II with A//ni at the intermediate temperature. Modification II then melts with details as above. DSC trace C modification II, metastable at room temperature, transforms exothermically to modification I with A/fn i at that transition temperature. Continued heating leads to the events in trace A. DSC trace D modification II exists at room temperature and no transition takes place prior to melting at mpn, with the appropriate A//ni- (After Giron 1995, with permission.)...

This result implies that the energy equipartition relationship of Eq. (2.S) applies as well as the general definitions of Chapter I. Note that for Af m the variable turns out to be coupled weakly to the thermal bath. This condition generates that time-scale separation which is indispensable for recovering an exponential time decay. To recover the standard Brownian motion we have therefore to assiune that the Brownian particle be given a macroscopic size. In the linear case, when M = w we have no chance of recovering the properties of the standard Brownian motion. In the next two sections we shall show that microscopic nonlinearity, on the contrary, may allow that the Markov characters of the standard Brownian motion be recovered with increasing temperature. 

According to the adsorption-site theory, the model constants should follow an Arrhenius-type temperature relationship. An Arrhenius-type plot of the adsorption model constants is shown in Figure 3. The rate constant, ko, increases with an increase in temperature, and the adsorption constants decrease with an increase in temperature. These opposing effects are in agreement with a physically realistic model. The activation energies found from these data are 29.3 kcal/mole for reaction, —28.9 kcal/mole for hydrocarbon adsorption, and —35.4 kcal/mole for hydrogen adsorption. 

The liquid solvent added to a pharmaceutical material generally exists in a variety of states. Some will condense or be pulled by capillary forces into macroscopic pores and fissures or into the interstitial spaces between particles. A state of local equilibrium can be assumed to exist at the interface between the liquid and vapor phases of solvent so situated. As a result, the temperature and vapor pressure exerted by the condensed solvent will not be independent of one another. Fig. 4 shows the equilibrium vapor pressure vs. temperature relationship for a number of common solvents. Heats of vaporization are shown parenthetically. Among common solvents, acetone has the highest vapor pressure and water the lowest. Water requires three-five times the energy of the common organic solvents to vaporize. 

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