Functions at Constant Temperature

Storage moduli in shear and extension, and loss tangents, of poly(methyl methacrylate) near 2S°C, plotted logarithmically against frequency. The numbers between E and G denote values of Poisson s ratio. (After Koppelmann. ) 

In principle, the frequency dependence of the complex bulk modulus could be obtained from the data for E, E , G, and G , In Fig. 15-6, the shear and extension data have been combined to calculate Poisson s ratio /it as a function of frequency. The values indicated show that /it passes through a minimum between the inflections in the dispersion of G and of E.  

Although the data of Fig. 15-6 have not been used to calculate relaxation and retardation spectra, it is clear from the very gradual frequency dependence of the moduli and tan 6 that the spectra are relatively flat over a wide range of frequency scale, as already illustrated as curves V in Figs. 3-3 and 3-4 for this same polymer in the glassy state. The latter were derived from shear creep data of Iwayanagi, confirmed by comparison with similar data of Lethersich. 2 

The loss tangent for the /(3 mechanism in poly(methyl methacrylate) can now be compared with those of other homologs, the ethyl, tertiary butyl, and cyclohexyl methacrylates, in Fig. 15-7, cited by Heijboer. The maximum for ethyl occurs at about the same frequency, but with greater magnitude. From other (isochronal) measurements by Heijboer, it is evident that the magnitude of tan S ax increases in the order methyl, ethyl, n-propyl, and n-butyl, while the frequency remains about the same. This is also evident in the maximum in the retardation spectrum as derived from shear creep measurements by Hideshima when the methyl, -propyl, and 

Plots of loss tangent against log v for four poly(methaciylate)s at 40°C. (Et) Ethyl (Me) methyl (t-Bu) t-butyl (CH) cyclohexyl. (Heijboer. ) Reproduced by permission from The International Journal of Polymeric Materials. 

---

Since in addition to the chemical potentials also the electrical potential 99, affects the charged species, electrochemical potentials //, must be used. We use the symbol 99 instead of -ip because this is the Galvani potential (see Section 5.5). The Gibbs-Duhem equation for changes of state functions at constant temperature is... 

Structure and orientation of a Me deposit on S in the initial stage of 3D Me bulk phase formation can be either independent of or influenced by the surface structure of S, which can be modified by 2D Meads overlayer formation and/or 2D Me-S surface alloy phase formation in the UPD range. Epitaxial behavior of 2D and 3D Me phases exists if some or all of their lattice parameters coincide with those of the top layer of S. The epitaxy is determined by a minimum of the Gibbs function at constant temperature and pressure. 

Living organisms, however, function at constant temperature. As would be necessary for warm-blooded animals and as is apparent in... 

J(A.,T, ) has known the meaning of the creep function at constant temperature as a function of the effective time A.. All these results may be transferred immediately to the problem of viscoelastic behavior under the influence of aging at constant temperature [9,15]. The temperature history has to be replaced by the degree of aging. A, of the sample and the time-temperature shift function, a(T,T ), by the time-age shift function, b (A,A ). The degree of aging, also called the age of the sample, defines the time elapsed from the last quench from the equilibrium state down to the aging temperature so far A, can then be expressed ... 

Besides the Thomson statement of the second law in Fig. 2.2, there exists a statement by Clausius "It is impossible to devise an engine which, working in a cycle, shall produce no effect other than the transfer of heat from a colder to a hotter bodjy." Prove in a manner similar to the Thomson argument that this also a statement which forbids an increase in entropy [see also Eqs. (6) to (8) in Fig. 2.6]. The equation (Q2 Q )IQ2 ( 2 l)/ 2 expression for the theoretical efficiency of a heat engine which takes, for example, high temperature and pressure steam at 700 K (T2), drives a turbine, and releases the relaxed steam at about 400 K (Tj) and at atmospheric pressure. Calculate the theoretical effiden(y and compare with actual data. Biological systems function at constant temperature. How can they produce work ... 

An alternative method is to consider the A-V surface (Rowlinson Swinton 1982), where the saturation pressure is given by the slope of the common tangent to the curve at the saturated volumes. The objective function, at constant temperature, is... 

A thermodynamic function for systems at constant temperature and pressure that indicates whether or not a reaction is favorable (AG < 0), unfavorable (AG > 0), or at equilibrium (AG = 0). 

The basic measurement of adsorption is the amount adsorbed v, which usually is given in units of cm of gas adsorbed per gram of adsorbent. Usually this quantity is measured at constant temperature as a function of pressure p (in mm Hg), and hence is termed an isotherm. Isobars and isosteres also can be measured, but have little practical utility. It has been found that isotherms of many types exist, but the five basic isotherm shapes are shown in Figure 1, where />ois the vapor pressure. 

AA is sometimes referred to as the change in work function. This equation simply states that energy will be available to do work only when the heat absorbed exceeds the increase in internal energy. For proeesses at constant temperature and pressure there will be a rise in the heat content (enthalpy) due both to a rise in the internal energy and to work done on expansion. This can be expressed as... 

The computation of quantum many-body effects requires additional effort compared to classical cases. This holds in particular if strong collective phenomena such as phase transitions are considered. The path integral approach to critical phenomena allows the computation of collective phenomena at constant temperature — a condition which is preferred experimentally. Due to the link of path integrals to the partition function in statistical physics, methods from the latter — such as Monte Carlo simulation techniques — can be used for efficient computation of quantum effects. 

In order to reinforce such conclusion for this part of the phase boundary we also studied the cumulants as functions of the rotational constant at constant temperature in the range T < 0.2424. The fourth-order cumulant plots show that the intersection occurs in this low temperature range at a... 

The rate of reaction nearly always depends upon reactant concentrations and (for reversible reactions) product concentrations. The functional relationship between rate of reaction and system concentrations (usually at constant temperature, pressure, and other environmental conditions) is called the rate equation. 

In general, gas solubilities are measured at constant temperature as a function of pressure. Permanent gases (gases with critical temperatures below room temperature) will not condense to form an additional liquid phase no matter how high the applied pressure. However, condensable gases (those with critical temperatures above room temperature) will condense to form a liquid phase when the vapor pressure is reached. The solubilities of many gases in normal liquids are quite low and can be adequately described at ambient pressure or below by Henry s law. The Henry s law constant is defined as... 

For the chain (homogenous) consisting of one con-former, osmotic forces are similar to the ones stretching the molecule by the ends. Then, labor of the distance being estimated at constant temperature T , one can estimate 5ch value from the condition = F AR = T ASch)- If a more accurate estimation of the distance change valRe between the ends is required, one may calculate the R value, taking into account the distribution function of the distances between the ends R. The value of the mean-square distances between the ends of the chain, being stretched by forces, applied to the ends equals [14] ... 

This expression shows that the maximum possible useful work (i.e., reversible work) that can be obtained from any process occurring at constant temperature and pressure is a function of the initial and final states only and is independent of the path. The combination of properties U + PV - TS or H - TS occurs so frequently in thermodynamic analysis that it is given a special name and symbol, F, the free energy (sometimes called the Gibbs Free Energy). Using this definition, Equation 2-143 is written... 

If the total pressure of the vapour at constant temperature is represented as a function of the compositions of the two phases, the p-liquid and p-vapour curves are obtained. The p-liquid curves—that is, the curves representing the total vapour pressures of liquid binary mixtures as functions of the composition of the liquid phase—are most important they are usually referred to simply as the vapour-pressure curves of the mixture. Each curve is an isotherm. 

At constant temperature, p is a function of s orV alone. In, all cases we have the inequalities ... 

In the previous sections, we emphasized that at constant temperature, the liquid-phase activity coefficient is a function of both pressure and composition. Therefore, any thermodynamic treatment of gas solubility in liquids must consider the question of how the activity coefficient of the gaseous solute in the liquid phase varies with pressure and with composition under isothermal conditions. 

For those dilute mixtures where the solute and the solvent are chemically very different, the activity coefficient of the solute soon becomes a function of solute mole fraction even when that mole fraction is small. That is, if solute and solvent are strongly dissimilar, the relations valid for an infinitely dilute solution rapidly become poor approximations as the concentration of solute rises. In such cases, it is necessary to relax the assumption (made by Krichevsky and Kasarnovsky) that at constant temperature the activity coefficient of the solute is a function of pressure but not of solute mole fraction. For those moderately dilute mixtures where the solute-solute interactions are very much different from the solute-solvent interactions, we can write the constant-pressure activity coefficients as Margules expansions in the mole fractions for the solvent (component 1), we write at constant temperature and at reference pressure Pr ... 

In earlier days, A was called the work function because it equals the work performed on or by a system in a reversible process conducted at constant temperature. In the next chapter we will quantitatively define work, describe the reversible process and prove this equality. The name free energy for A results from this equality. That is, A A is the energy free or available to do work. Work is not a state function and depends upon the path and hence, is often not easy to calculate. Under the conditions of reversibility and constant temperature, however, calculation of A A provides a useful procedure for calculating u ... 

The combination of fundamental variables in equation (l.23) that leads to the variable we call G turns out to be very useful. We will see later that AG for a reversible constant temperature and pressure process is equal to any work other than pressure-volume work that occurs in the process. When only pressure-volume work occurs in a reversible process at constant temperature and pressure, AG = 0. Thus AG provides a criterion for determining if a process is reversible. Again, since G is a combination of extensive state functions... 

Equation (5.16) can be integrated. We expect the partial molar properties to be functions of composition, and of temperature and pressure. For a system at constant temperature and constant pressure, the partial molar properties would be functions only of composition. We will start with an infinitesimal quantity of material, with the composition fixed by the initial amounts of each component present, and then increase the amounts of each component but always in that same fixed ratio so that the composition stays constant. When we do this. Z, stays constant, and the integration of equation... 

The liquid line and vapor line together constitute a binary (vapor + liquid) phase diagram, in which the equilibrium (vapor) pressure is expressed as a function of mole fraction at constant temperature. At pressures less than the vapor (lower) curve, the mixture is all vapor. Two degrees of freedom are present in that region so that p and y2 can be varied independently. At pressures above the liquid (upper) curve, the mixture is all liquid. Again, two degrees of freedom are present so that p and. v can be varied independently/... 

The rate of a chemical reaction is always taken as a positive quantity, and the rate constant k is always positive as well. A negative rate constant is thus without meaning. An equation such as Eq. (1-4), which gives the reaction rate as a function of concentration, usually at constant temperature, is referred to as a rate law. The determination of the form in which the different concentrations enter into the rate law is one of the initial goals of a kinetic study, since it allows one to infer certain features of the mechanism. 

Because entropy is a state function, the change in entropy of a system is independent of the path between its initial and final states. This independence means that, if we want to calculate the entropy difference between a pair of states joined by an irreversible path, we can look for a reversible path between the same two states and then use Eq. 1 for that path. For example, suppose an ideal gas undergoes free (irreversible) expansion at constant temperature. To calculate the change in entropy, we allow the gas to undergo reversible, isothermal expansion between the same initial and final volumes, calculate the heat absorbed in this process, and use it in Eq.l. Because entropy is a state function, the change in entropy calculated for this reversible path is also the change in entropy for the free expansion between the same two states. 

---