Entropy of the Ideal Gas

9 GENERAL EQUATIONS FOR THE ENTROPY OF GASES Entropy of the Ideal Gas 

We can obtain an explicit equation for the entropy of an ideal gas from the mathematical statements of the two laws of thermodynamics. It is convenient to derive this equation for reversible changes in the gas. However, the final result will be perfectly general because entropy is a state function. 

In a system in which only reversible expansion work is possible, the first law can be stated as 

Substituting for DQ from Equation (6.109) into Equation (6.49), we obtain 

Equation (6.111) is sometimes called the combined first and second laws of thermodynamics, and this equation suggests that S and V are natural independent variables for U. Conversely, we can say that U and V are natural variables for S. One can also 

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For most substances, the Third Law and statistical calculations of the entropy of the ideal gas are in agreement, but there are exceptions, some of which are summarized in Table 4.2. The difference results from residual entropy, So, left in the solid at 0 Kelvin because of disorder so that St - So calculated from Cp/TdT is less than the St calculated from statistical methods. In carbon monoxide the residual disorder results from a random arrangement of the CO molecules in the solid. Complete order in the solid can be represented schematically (in two-dimensions) by... 

Li, J.C.M., Pitzer, K.S. (1956) The thermodynamic properties of 1,1-dichloroethane heat capacities from 14 to 294 K, heats of fusion and vaporization, vapor pressure and entropy of the ideal gas. The barrier to internal rotation. J. Am. Chem. Soc. 78, 1077-1080. 

We deal in an analogous fashion with the equivalent forms for the entropy of the ideal gas ... 

Pace and Bobka ( 11.) used low temperature calorimetric data to derive the entropy of the ideal gas at the normal boiling point of 205.47 K. The experimental value of 65.01 0.2 cal K mol" is in good agreement with 65.03 cal K" mol" obtained... 

The adopted molecular structure was derived from electron-diffraction data by Swick and Karle (8). The vibrational frequencies are from the assignment of Carney et al. (9), who reviewed the extensive spectral data. Shlmanouchl ( ) has selected the same frequencies. Low-temperature calorimetric data were used by Pace and Aston (1 ) to derive a value for the entropy of the ideal gas of 69.88 0.20 cal mol at 194.87 K. This value is used to fix the barrier to internal rotation... 

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 entropy of the ideal gas at the same temperature and volume is the observed entropy less the term nR In (F - nb)jV, It may be noted... 

This derivative is always positive in an isothermal transformation, the entropy of the ideal gas increases with increase in volume. The rate of increase is less at large volumes, since V appears in the denominator. 

Vapour pressure measurements have led to heats of sublimation of octafluoro-naphthalene and decafluorobiphenyl of 79.3 2.4 and 85.2 2.3 kJ mol , respectively, and the enthalpy of fusion and heat capacity of octafluorotoluene have been obtained from adiabatic calorimetry at 10—365 K the last study yields a triple-point temperature of 207.69 K, and the standard entropy of the ideal gas is 462.6 J K mol i at 298.15 K. Adiabatic calorimetry has also been used to obtain the heat capacities of the three tetrafluorobenzenes at 11— 353 K, and the entropies of the liquids at 298.15 K are 256.1,257.3, and 250.4 J mol- for the 1,2,3,4-, 1,2,3,5-, and 1,2,4,5-isomers, respectively 1,2,3-trichlorotrifluorobenzene fcry-stals) has an entropy of 243.3 J mol at 298.15 K. . Liquid densities from 293 to 490 K have been determined for eight polyfluoroarenes (CeFe, CeFsH, 1,2,3,4-F4C H2, CsFsa. l,3.5-F3C6Cl3, CsFs CFa, CeFsMe, and CeFsOH) and densities up to the critical temperatures have been estimated the effects of dissolved... 

An ancillary equation is used to evaluate the ideal gas specific heat, C j. The reference entropy of the ideal gas at r" " and p" " is taken from a suitable source for the fluid under investigation. The enthalpy of any state may be calculated from... 

Equation (2.4.10a) also shows that S represents the molar entropy of the ideal gas when the final and initial temperatures, as well as the final and initial pressures in the experiment are identical. However, the theoretical development fails to specify what such an initial condition should be. This indeterminacy allows one to justify the convention that is cited above for the standard state for the ideal gas at whatever temperature prevails in a given experiment, the standard state of an ideal gas is set at a pressure P = 1 bar. 

An interesting application of the above results is the calculation of the entropy of mixing of two idei gases of the same temperature and density. We first rewrite the entropy of the ideal gas in terms of the density n, the energy per particle u and the constant entropy per particle So, defined through... 

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