Entropy of a Real Gas

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

The difference between real and ideal systems of the Helmholtz energies yields 

Example 1.12 Chemical potential of a real gas Similar to Eq. (1.142), the Gibbs free energy for a real gas is 

Using the compressibility factor Z, the volume of a real gas is Ereal = ZRT/P. Therefore, the chemical potential in terms of Z in Eq. (1.147) is 

The chemical potential can also be expressed in terms of fugacity / 

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)]  

The entropy S also can be considered to be a function of V and T thus, a second equation for the total differential dS is 

A comparison of the coefficients of the dT terms in Equations (6.115) and (6.116) leads to the following equality  

It can be shown also, by a procedure to be outlined in Chapter 7, that the following relationship is valid  

Substituting from Equations (6.117) and (6.118) into Equation (6.116), we obtain 

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Using Eqn (b) in Areal = Ureal TSreal, we can calcnlate the entropy of a real gas. For example, using the van der Waals equation, we get... 

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. 

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... 

Real-gas thermodynamic properties may be expressed as functions of the variables of state, according to relations which may be developed from first principles. A comprehensive list of such relations has been given by Beattie and Stockmayer. For example, the molar enthalpy H, the molar entropy S, and the molar Gibbs energy G of a real gas can be written in terms of p, T, and p (the amount density) as follows ... 

For a solid or liquid, the standard state is the actual substance at pressure P° (exactly 1 bar). For a gas, the standard state is a hypothetical ideal gas state at the standard pressure P° (1 bar). That is, a correction must be made for the difference between the entropy of the real gas at pressure P° and the corresponding ideal gas at pressure F . We will discuss how to make this correction in Chapter 4, but the correction is small for ordinary pressures, and we can usually neglect it. These standard states are the same as for the enthalpy. 

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°). 

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... 

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. 

Be able to calculate the entropy change between two states of an ideal gas Be able to calculate the entropy change of a real fluid using thermodynamic properties charts and tables... 

You may wonder why such 0-conditions are possible in the first place. Is it a mere coincidence that at a certain point repulsion and attraction are so perfectly balanced For instance, such balancing, or compensation, never quite happens in a real gas. Historically, Boyle found that his law pV = const for a gas at fixed temperature) is followed at some temperatures more accurately than at others, but never quite perfectly in modern language, we can say that the gas should be close to ideal at the temperature (called Boyle s point) when B = 0, but it is not quite ideal because C = 0. By contrast, compensation between attraction and repulsion is indeed nearly perfect for a polymer coil. Why The answer is that the cancelation only works because three-body interactions (and all the higher ones) are not important. Their contribution to U is always very small. As for the binary collision term (8.8), it is proportional to B, so it falls to zero at the 0-point. Hence, all that really remains of the free energy F at T = 0 is the entropy term (see (7.19)). This is why the coil s behavior becomes ideal. 

Values of Cp, Cy, internal energy E, enthalpy H, and entropy S of the real gas are also available from calculations using reference values for the ideal gas and a modified Benedict-Webb-Rubin equation of state, the melting curve, vapor and liquid density curves, and the vapor pressure curve mentioned above. They are parameterized in the same way (along coexistence lines and isochores) as the pgT data, see p. 202 [2]. In a similar manner, H and S were calculated earlier from a Martin-Hou equation of state, see p. 202 [3]. 

Like the enthalpy departure function, the entropy departure function can be used to find the entropy change of a real fluid. It is defined as the difference in that property between the real, physical state and that of a hypothetical ideal gas at the same T and P ... 

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. 

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. 

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. 

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