Entropy of an ideal gas

Having found the absolute activity of a M B gas, it is a simple matter to obtain its entropy. The chemical potential was shovm to be given by 

This is the Sakur-Tetrode equation for the entropy of an ideal gas. 

As a last example, we obtain the entropy of an ideal gas as a function of volume, temperature and mole number. For a closed system in which the changes of entropy are only due to flow of heat, if we assume that the changes in volume V and temperature T take place so as to make d S = 0, then we have seen from (3.4.16) that dU = TdS + dW. If dW = —pdV, and if we express dU as a 

---

These equations clearly show that for an ideal gas U, H, Cp, and Cy are functions of temperature only and are independent of f and V The entropy of an ideal gas, however, is a func tion of both T and f or of both T and V... 

Examples.—(1) Prove that the entropy of an ideal gas mixture is the sum of the entropies of the components at the same temperature, each occupying the whole volume of the mixture. 

We have shown thar the change in entropy of an ideal gas when it expands isothermally from a volume V] to a volume V2 is... 

Self-Test 7.3A Calculate the change in molar entropy of an ideal gas when it is compressed isothermally to one-third its initial volume. 

The expressions in Eq. 1 and Eq. 6 are two different definitions of entropy. The first was established by considerations of the behavior of bulk matter and the second by statistical analysis of molecular behavior. To verify that the two definitions are essentially the same we need to show that the entropy changes predicted by Eq. 6 are the same as those deduced from Eq. 1. To do so, we will show that the Boltzmann formula predicts the correct form of the volume dependence of the entropy of an ideal gas (Eq. 3a). More detailed calculations show that the two definitions are consistent with each other in every respect. In the process of developing these ideas, we shall also deepen our understanding of what we mean by disorder. ... 

We can show that the thermodynamic and statistical entropies are equivalent by examining the isothermal expansion of an ideal gas. We have seen that the thermodynamic entropy of an ideal gas increases when it expands isothermally (Eq. 3). If we suppose that the number of microstates available to a single molecule is proportional to the volume available to it, we can write W = constant X V. For N molecules, the number of microstates is proportional to the Nth power of the volume ... 

Partial molar availability, 24 692 Partial molar entropy, of an ideal gas mixture, 24 673—674 Partial molar Gibbs energy, 24 672, 678 Partial molar properties, of mixtures, 24 667-668... 

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. 

The first term on the right-hand side is the entropy of an ideal gas, while the second term gives the entropy of mixing, The prefactor p reflects the fact that both are expressed per unit volume rather than per particle. 

Let us now discuss the entropy of gas expansion in a closed system. Equation 3.42 gives us the molar entropy of an ideal gas at constant temperature T as shown in Eq. 3.47 ... 

We have thus obtained three equivalent formulations for the differential entropy of an ideal gas. We may now carry out an integration between two specific limits selecting Eq. (2.4.6) we obtain... 

Determine the energy and entropy of an ideal gas distributed at constant T in a tall vertical cylinder of height h. 

Equation (16.20) for the molar entropy of an ideal gas allows calculation of absolute entropies for tile ideal-gas state. The data required for evaluation of the last two terms on tlie right are tlie bond distances and bond angles in the molecules, and the vibration frequencies associated witli tlie various bonds, as determined from spectroscopic data. The procedure lias been very successful in the evaluation of ideal-gas entropies for molecules whose atomic stractures are known. 

The free energy of a phase is thus expressed in terms of the amounts of its various components and, through the [xs, in terms of its composition. Pa . .. are called the chemical potentials of A, B,. .. in the phase, at the given temperature and pressure. They generally depend on composition, but sometimes this dependence is relatively simple. Thus for a mixture of ideal gases, U, S, F and G are all sums of terms, one for each gas regarded as though it alone were present. Thus P = — [dFjdV) is a sum of partial pressures (P, Pj, . ..), in accord with Dalton s law, where P F = n RT etc. Now the entropy of an ideal gas has already been found in the form [constant —R log P + Cp log 7] and, since G = H — TS, then, for one mole at partial pressure P the dependence on pressure can be written... 

Another useful expression is that for the entropy of an ideal gas, which can be obtained from (1.77) ... 

The entropy of an ideal-gas mixture is obtained by combining the internal energy of Equation (4.261) and the Helmholtz energy of Equation (4.265) ... 

All types of motion (i.e., translation, rotation, and vibration) contribute to the entropy of an ideal gas. The following equations that have been discussed in this... 

---