The Chemical Potential of an Ideal Gas

In section 1.6 we derived the thermodynamic quantities of an ideal gas. Of particular importance is the chemical potential, which is now written as 

q is the internal partition function of a single molecule, V the volume of the system, N the number of particles, and the momentum partition function. 

It is convenient to assign meaning to the two terms in (2.2.1) as follows the chemical potential is defined, in the T, F, N ensemble, as 

Suppose that instead of just adding one particle to the system, we introduce the particle at a fixed position in the system, say, at Rq. The corresponding change in the Helmholtz energy is defined by 

We see that this quantity is equal to the first term on the rhs of (2.2.1). We shall refer to /i as the pseudo chemical potential 

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The chemical potential of an ideal gas at temperature T depends on pressure according to the following relation ... 

Here /i j3 is the chemical potential of the ideal gas at the standard pressure. It will be seen subsequently that qi for an ideal gas depends linearly on the volume V, so fif is a function only of the temperature. It does of course depend on the distribution of energy levels of the ideal gas molecules. The form of Equation 4.59 for the chemical potential of an ideal gas component is the same as that previously derived from thermodynamics (Equation 4.47). The present approach shows how to calculate m through the evaluation of the molecular partition function. Furthermore, the... 

In the preceding chapters we considered Raoult s law and Henry s law, which are laws that describe the thermodynamic behavior of dilute solutions of nonelectrolytes these laws are strictly valid only in the limit of infinite dilution. They led to a simple linear dependence of the chemical potential on the logarithm of the mole fraction of solvent and solute, as in Equations (14.6) (Raoult s law) and (15.5) (Heiuy s law) or on the logarithm of the molality of the solute, as in Equation (15.11) (Hemy s law). These equations are of the same form as the equation derived for the dependence of the chemical potential of an ideal gas on the pressure [Equation (10.15)]. 

The ideal gas data in these tables were calculated directly from a statistical mechanical description of the isolated molecules. In terms of the quantities defined in these tables, jL02(T,p°)=[H°(T) H°(Tr) TS(T) [H°(0) H°(Tr). This expression is derived by noting that the chemical potential of an ideal gas is equal to the ideal gas free energy, G = H TS. 

Equation 5.17 may be simplified somewhat by finding expressions for the absolute activity A./ and the individual particle partition function qjj in terms of experimentally measured or fitted parameters. To achieve such a simplification, we first consider the chemical potential of an ideal gas and its relation to the particle partition function. 

The procedure used to derive Eq. 3.5 is the same as for the derivation of the Gibbs-Duhem equation. Substituting the expression for the chemical potential of an ideal gas in Eq. 3.5 provides the classical Gibbs isotherm ... 

The residual chemical potential is equal to the difference between the chemical potential of the system and the chemical potential of an ideal gas at the same temperature, molar volume, and composition. The difference given in Eq. (11.20) is between the system and an ideal gas at the same temperature, pressure, and composition. This is not the same as the residual chemical potential. However, the two quantities are related. If we consider an ideal gas with the same molar volume as the system, its pressure will be equal to p = RT/V, which will, in general, not be equal to the actual pressure p of... 

For the chemical potential of an ideal gas we obtained the following expression (5.3.6) Section 5.3... 

If we understand that equation 4.57 gives the chemical potential of an ideal gas yUideai in terms of pressure and equation 4.59 gives the chemical potential of our real gas pL in terms of fugacity, we can use them to evaluate pi — /UideaE... 

Equation (2.4.15) relates the chemical potential of an ideal gas to / Tln P,- in accordance with (i), this suggests that for a real gas should be specified by an analogous expression, namely RT In i, where f is termed the fugacity of the ith constituent of the gas. To satisfy (ii), this quantity must converge on the pressure P, at ideality. Since /x,- is specified only to within an arbitrary constant, only the difference in chemical potential of the nonideal gas in two states, 1 and 2, may be uniquely specified as... 

Solutes are treated differently from convention I. The standard state for a solute in convention II is the same as that for a dilute solute the hypothetical pure substance that obeys Henry s law. The chemical potential of the solute in its standard state is equal to the chemical potential of an ideal gas at pressure equal to kj, which is the same as the Henry s law standard state ... 

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