Constant-temperature work

EXAMPLE 7.8 Constant-temperature work. A gas does work when it expands quasi-statically in a piston at constant temperature. Now both the pressure and volume can change during the expansion, so you need to know the functional form ofp V) for this process. A quasi-static process is slow enough for the gas pressure inside to equilibrate with the external pressure, pint = Pext-To compute the work of an ideal gas at constant T, integrate ... 

The definition of polymer thermal stabiUty is not simple owing to the number of measurement techniques, desired properties, and factors that affect each (time, heating rate, atmosphere, etc). The easiest evaluation of thermal stabiUty is by the temperature at which a certain weight loss occurs as observed by thermogravimetric analysis (tga). Early work assigned a 7% loss as the point of stabiUty more recentiy a 10% value or the extrapolated break in the tga curve has been used. A more reaUstic view is to compare weight loss vs time at constant temperature, and better yet is to evaluate property retention time at temperature one set of criteria has been 177°C for 30,000 h, or 240°C for 1000 h, or 538°C for 1 h, or 816°C for 5 min (1). 

Another instance in which the constant-temperature method is used involves the direc t application of experimental KcO values obtained at the desired conditions of inlet temperatures, operating pressure, flow rates, and feed-stream compositions. The assumption here is that, regardless of any temperature profiles that may exist within the actu tower, the procedure of working the problem in reverse will yield a correct result. One should be cautious about extrapolating such data veiy far from the original basis and be carebil to use compatible equilibrium data. 

We assume (Fig. 5.5) that all parts of the system and of the environment are at the same constant temperature T and pressure p. Let s start with a mixture of ice and water at the melting point T, (if p = 1 atm then T, = 273 K of course). At the melting point, the ice-water system is in a state of neutral equilibrium no free work can be extracted if some of the remaining water is frozen to ice, or if some of the ice is melted... 

This expression states that there will be energy free to do work when Q exceeds AE. Expressed in another way work ean be done, that is an action can proceed, if AE - 0 is negative. If the difference between AE and Q is given the symbol AA, then it can be said that a reaction will proceed if the value of AA is negative. Since the heat term is the product of temperature T and change of entropy AS, for reactions at constant temperature then... 

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... 

For points 2-3, there is constant entropy (S) compression for a one pound of air from Pg to P3. From points 3-5 the air cools at constant pressure, and gives up heat, Q, to the intercooler. From points 5-6 the air is compressed at constant S to the final pressure Pg. Note that point Tj = point Tg for constant temperature. For minimum work Tg = T3. Then the heat, Q, equals the Work, Wl of Figure 12-36B. Figure 12-38 is convenient for estimating the moisture condensed from an airstream, as well as establishing the remaining water vapor in the gas-air. 

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... 

We shall now define what is to be understood by equal intervals of temperature. Let us imagine that we have a system of reversible engines [1,2], [2,8], [8,4],. . . , working between constant temperature reservoirs (1), (2), (8), (4),. . . , so that the refrigerator of any engine (except the last) forms the source of the next engine. Let each perform a cycle so that... 

The suffix x indicates that besides T, all the variables xu a, . . . during the change of which external work is done, are maintained constant (adynamic condition). Thus, if the only external force is a normal and uniform pressure p, then x — v, the volume of the system, and (11) is the condition of equilibrium at constant temperature and volume. 

We shall now prove that P, for fixed values of 7r and the temperature, is definite for a given solution. For this purpose we have first of all to show that the dilution or concentration of the solution can be effected isothermally and reversibly. If the above apparatus is constructed of some good conductor of heat, placed in a large constant-temperature reservoir, and if all processes are carried out very slowly, the isothermal condition is satisfied. Further, suppose the end pistons fixed, and then apply to the septum an additional small pressure SP towards the solution. There will be a slight motion of the septum, through a small volume SV, and work... 

Corollary.—The work done in the isothermal and reversible dilution of a dilute solution is equal to the heat absorbed from the constant temperature reservoir. 

If a chemical reaction occurs spontaneously, the available energy of the system necessarily diminishes by an amount equal to the work which could be done by the system if the given change were executed reversibly. If the reaction occurs at constant temperature, this is equal to the diminution of free energy of the system, this being the energy available at constant temperature. It is usual to refer to the work available at constant... 

If J, are the free energies of a system before and after dissociation at a constant temperature, the maximum external work obtainable is — 4/ This may be calculated directly. Let us take the case of nitrogen peroxide ... 

The general theory was worked out by Roozeboom (Zeitschr. physik. Chem., 1899) from the standpoint of the theory of thermodynamic potential. The equations (2a, h), (3a, h) of the preceding section apply equally well to the present case, and details need not be given here. The liquid solidifies at a constant temperature when it has the same composition as the solid deposited— the so-called eutectic point. 

Chueh s method for calculating partial molar volumes is readily generalized to liquid mixtures containing more than two components. Required parameters are and flb (see Table II), the acentric factor, the critical temperature and critical pressure for each component, and a characteristic binary constant ktj (see Table I) for each possible unlike pair in the mixture. At present, this method is restricted to saturated liquid solutions for very precise work in high-pressure thermodynamics, it is also necessary to know how partial molar volumes vary with pressure at constant temperature and composition. An extension of Chueh s treatment may eventually provide estimates of partial compressibilities, but in view of the many uncertainties in our present knowledge of high-pressure phase equilibria, such an extension is not likely to be of major importance for some time. 

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... 

The Isothermal Process In an isothermal (constant temperature) expansion, heat is added to balance the work removed, so that the temperature of the system does not change. The amount of work can be calculated from the line... 

Thus, in a reversible process that is both isothermal and isobaric, dG equals the work other than pressure-volume work that occurs in the process." Equation (3.96) is important in chemistry, since chemical processes such as chemical reactions or phase changes, occur at constant temperature and constant pressure. Equation (3.96) enables one to calculate work, other than pressure-volume work, for these processes. Conversely, it provides a method for incorporating the variables used to calculate these forms of work into the thermodynamic equations. 

FIGURE 6.16 (a) On the reversible path, the work done in Example 6.5 is relatively large (w = -2.12 kj) because the change in internal energy is zero, heal flows in to maintain constant temperature and constant internal energy. Therefore, q =... 

To find the relation between the Gibbs free energy and the maximum nonexpansion work, we start with Eq. 15 for an infinitesimal change (denoted d) in G at constant temperature ... 

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