Real gas entropy

Wlien H has reached its minimum value this is the well known Maxwell-Boltzmaim distribution for a gas in themial equilibrium with a unifomi motion u. So, argues Boltzmaim, solutions of his equation for an isolated system approach an equilibrium state, just as real gases seem to do. Up to a negative factor (-/fg, in fact), differences in H are the same as differences in the themiodynamic entropy between initial and final equilibrium states. Boltzmaim thought that his //-tiieorem gave a foundation of the increase in entropy as a result of the collision integral, whose derivation was based on the Stosszahlansatz. 

An ideal gas consists of a large number of molecules that occupy the energy levels characteristic of a particle in a box. For simplicity, we consider a one-dimensional box (Fig. 7.9a), but the same considerations apply to a real three-dimensional container of any shape. At T = 0, only the lowest energy level is occupied so W = 1 and the entropy is zero. There is no disorder, because we know which state each molecule occupies. 

In deriving an equation for the entropy of a real gas we can start with Equation (6.110), which is general and not restricted to ideal gases. A suitable substitution for dU in Equation (6.110) can be obtained from the total differential of t/ as a function of y and T [Equation (4.59)] ... 

MPa). Also, a pressure, usually near 1 bar, will exist at which the real gas has a fugacity of unity. bar also real gas at zero pressure. (See Exercise 1, this chapter.) (0.1 MPa). Also, a pressure of the real gas will exist, not zero and not that of unit fugacity, with an entropy equal to that in the standard state. (0.1 Mpa). V =(RT/P°). 

These are, without doubt, somewhat artifieial systems. For real systems, one could not tell if anything happened, unless isotopie labeling could be arranged. An even greater problem would oeeur in the gas phase, sinee the entropy penalty required for these peeuliar geometries would be expected to make them very improbable. 

If one now chooses x, = S and recalls that the xi (k < n) are fieely adjustable, the Second Law would be violated if S were also adjustable at will (by means of non-static adiabatic transitions). Taking continuity requirements into account, it follows that S can either never decrease or never increase. The single example of the sudden expansion of a real gas shows that it can never decrease. One has the Principle of Increase of Entropy The entropy of an adiabatically isolated system can never decrease. 

JK mol-1 the value V°L = 0.91 cm3mol is obtained. An interpretation of the Hildebrand/Trouton Rule is that this free volume, V°L, allows for the freedom of movement of molecules (particles) necessary for the liquid state at the temperature Th. The explanation of the constant entropy of evaporation is that it takes into account only the translational entropy of the vapor and the liquid. It has to be pointed out that V°L does not represent the real molar volume of a liquid, but designates only a fraction of the corresponding molar volume of an ideal gas Vy derived from the entropy of evaporation. The real molar volume VL of the liquid contains in addition the molar volume occupied by the molecules V0. As a result the following relations are valid VL -V°L + V0 and Vc=Vq + V0. However, while V] < V0 and VL is practically independent of the pressure, V0 VaG in the gaseous phase. Only in the critical phase does VCIVL = 1 and the entropy difference between the two phases vanishes. 

M. Planck, Acht Vorleaungen Uber theoret. Phyaik. (Leipzig, 1909), 3rd lecture. The "special physical hypothesis introduced by Planck to exclude the spontaneous occurrence of observable decreases in entropy (he calls it the hypothesis of "elementary disorder") consists of the following statement The number of collisions which take place in a real gas never deviates appreciably from the Stoaazahlanaatz (cf. Section 18). The hypothesis denoted in Section 18c as the "hypothesis of molecular chaos" would, on the other hand, permit such deviations. 

The terms on the right-hand sides of Eqs. (6.62) and (6.63) are readily associated with steps in a calculation path leading from an initial to a final state of a system. Thus, in Fig. 6.14, the actual path from state 1 to state 2 (dashed line) is replaced by a three-step calculational path. Step 1 - 1 represents a hypothetical process that transforms a real gas into an ideal gas at T, and Pi. The enthalpy and entropy changes for this process are... 

Example 1.11 Entropy of a real gas Determine the entropy of a real gas. Solution ... 

For an irreversible expansion of a real gas at constant temperature due to a heat reservoir, the change of entropy flow is d,.S = 8q/T, where 8q is the heat flow between the gas and the reservoir to maintain the constant temperature. The increase of entropy during the expansion is... 

Step 1 —> 1 A hypothetical process that transforms a real gas into an ideal gas at Ti aird Pi. The entlralpy and entropy changes for tins process are ... 

The process will take place in the direction which involves an increase in the entropy of the system. It must therefore be one of the objects of science to determine the entropy of any given system as a function of its variables of condition. On p. 143 we have shown how this may be done for a perfect gas. In other cases the problem is not so simple, but the calculation is always possible if we know the equation of condition, e.g. van der Waals equation for real gases. Yet even when it is not possible to obtain an exphcit expression for the entropy, the entropy law can lead us to important conclusions, just as the law of the conservation of energy is important in many cases in which we are unable to give a numerical or analytical value for the energy of the system. 

The adopted value of S (298.15 K) = 16.718 0.019 cal K mol is taken from the CODATA recommended value (JL). This was calculated by CODATA from the entropy of the ideal gas with appropriate corrections for real gas behavior and vaporization. 

The adopted heat capacity data in the liquid region are taken from the very accurate calorimetric measurements of Osborne et al. 2), The adopted heat capacity data for the real gas at one bar pressure are taken from the recent equation of state formulation of Haar et al. ( ). See the JANAF Table for H OCt. p-1 bar) ( ) for details concerning the entropy. 

Table 8.1 contains values of the standard entropies of a number of important chemical compounds. These are the molar entropies of the real substances, corrected in the case of gases for gas imperfections, at a pressure of 1 atm. and temperature of 25 °C. 

The standard state of a substance is a reference state that allows us to obtain relative values of such thermodynamic quantities as free energy, activity, enthalpy, and entropy. All substances are assigned unit activity in their standard state. For gases, the standard state has the properties of an ideal gas, but at one atmosphere pressure. It is thus said to be a hypothetical state. For pure liquids and solvents, the standard states are real states and are the pure substances at a specified temperature and pressure. For solutes In dilute solution, the standard state is a hypothetical state that has the properties of an infinitely dilute solute, but at unit concentration (molarity, molality, or mole fraction). The standard state of a solid is a real state and is the pure solid in its most stable crystalline form. 

Real compression processes operate between adiabatic and isothermal compression. Actual compression processes are polytropic processes. This is because the gas being compressed is not at constant entropy as in the adiabatic process, or at constant temperature as in the isothermal processes. Generally, compressors have performance characteristics that are analogous to those of pumps. Their performance curves relate flow capacity to head. The head developed by a fluid between states 1 and 2 can be derived from the general thermodynamic equation. 

Plutonium. The famous multi-phase behavior of this troublesome metal is a result of many f-states combined with rapid bandnarrowing at the higher temperatures. Since the thermodynamic contribution of the monoxide gas is now quite small (23), even the earliest studies were little different than later ones with much higher-purity metal. Troublesome second/third law problems were partly due to incomplete or inaccurate thermodynamic functions and questions about the real crystal entropy. The presently accepted value of 13.42 shows the effect of an abnormally large electronic specific heat, which is a measure of the many complexities in energy relationships near the Fermi level for this metal. 

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