Chemical potential pure ideal gases

The right-hand equality in Equation (10.10), which gives the molar free energy of a pure ideal gas, is of the same form as Equation (10.15), which gives the chemical potential of a component of an ideal gas mixture, except that for the latter, partial pressure is substituted for total pressure. If the standard state of a component of the mixture is defined as one in which the partial pressure of that component is 0.1 MPa, then... 

The chemical potential of a pure ideal gas is given explicitly by Eq. (10.47) ... 

It is generally more convenient in aqueous solution thermodynamics to describe the chemical potential of a species i, in terms of its activity, a. The basic relationship between activity and chemical potential was developed by G.N. Lewis who first established a relationship for the chemical potential for a pure ideal gas, and then generalized his results to all systems to define the chemical potential of species i in terms of its activity aj as... 

Because the system shown in Fig. 9.4 is in an equilibrium state, gas A must have the same chemical potential in both phases. This is true even though the phases have different pressures (see Sec. 9.2.7). Since the chemical potential of the pure ideal gas is given by IX = fx°(g) + RT n(p/p°), and we assume that pa in the mixture is equal to p in the pure gas, the chemical potential of A in the mixture is given by... 

Here yi is the mole fraction of i in the gas. The chemical potential of the pure ideal gas... 

The chemical potential of any species in an ideal gas mixture, is then related to the pure ideal gas component chemical potential,, according to the following relation ... 

For precise measurements, diere is a slight correction for the effect of the slightly different pressure on the chemical potentials of the solid or of the components of the solution. More important, corrections must be made for the non-ideality of the pure gas and of the gaseous mixture. With these corrections, equation (A2.1.60) can be verified within experimental error. 

However, the chemical potential is given by Eq. (4-341) for gas-phase reactions and standard states as the pure ideal gases at T°, this equation becomes... 

For any single-component system such as a pure gas the molar Gibbs energy is identical to the chemical potential, and the chemical potential for an ideal gas is thus expressed as... 

The difference between the chemical potential of a pure and diluted ideal gas is simply given in terms of the logarithm of the mole fraction of the gas component. As we will see in the following sections this relationship between the chemical potential and composition is also valid for ideal solid and liquid solutions. 

The activity of the solvent is related directly to the vapor pressure when the vapor is an ideal gas. As the pure solvent is the standard state, the chemical potential p,i of the solvent in any solution is given by the expression [Equation (16.1)]... 

Consider, for example, a pure gas, under conditions in which it behaves as an ideal gas. The chemical potential is given by d l = dGm = -SmdT + V flP, where the subscript m indicates molar quantities. Thus, at constant temperature, d l = V jdP = (RT/P)dP. Integration of this equation from P to P gives... 

Now is becomes necessary to evaluate changes in chemical potential with changes in compn and the relative values of the chemical potentials of the pure components. This will be done below From the definition of p. for the ideal gas ... 

Since the vast majority of organic molecules behave almost ideally in the gas phase for temperatures below 200°C, Eq. (7.6) allows us to calculate the vapor pressure of such compounds from the chemical potential difference of the molecule in the ig phase and in the respective liquid state. As a special case, we can express the vapor pressure of the pure compound X as... 

The quantity gk T) in Equation (7.67) is again a molar quantity, characteristic of the individual gas, and a function of the temperature. It can be related to the molar Gibbs energy of the fcth substance by the use of Equation (7.67). The first two terms on the right-hand side of this equation are zero when the gas is pure and ideal and the pressure is 1 bar. Then gk(T) is the chemical potential or molar Gibbs energy for the pure fcth substance in the ideal gas state at 1 bar pressure. We define this state to be the standard state of the fcth substance and use the symbol 1 bar, yk = 1] for the... 

Real gases are usually non-ideal. Thermodynamics describes both ideal and non-ideal gases with the same type of formulas, except that for non-ideal gas mixtures the fugacity f is substituted in place of the pressure pi and that the activity at is substituted in place of the molar fraction xi or concentration c, of constituent substance i. We have already seen that in the ideal gas of a pure substance the chemical potential is expressed by Eq. 7.5. By analogy, we write Eq. 7.9 for the non-ideal gas of a pure substance i ... 

An ideal gas mixture conforms to the gas equation and PV = nRTbut there is an additional aspect to the definition, namely that the chemical potential of a component, i, of the mixture at a partial pressure, pi is equal to the chemical potential of the pure material, i, at the same total pressure. 

For a gas phase. The standard state for a gaseous substance, whether pure or in a gaseous mixture, is the (hypothetical) state of the pure substance B in the gaseous phase at the standard pressure p = p and exhibiting ideal gas behaviour. The standard chemical potential is defined as... 

To avoid some possible difficulties in determining chemical potentials, Lewis proposed a new property called the fugacity /. At low pressure and concentration, the fugacity is a well-behaved function. The fugacity function can define phase equilibrium and chemical equilibrium. For an ideal gas, the fugacity of a species in an ideal gas mixture is equal to its partial pressure. As the pressure decreases to zero, pure substances or mixtures of species approach an ideal state, and we have... 

Equation (1) is the central equation of LAST, specifying the equality of chemical potential in the bulk gas and the adsorbed phase (which is assumed to be ideal in the sense of Raoult s law). Equation (2) calculates the spreading pressure from the pure-component isotherm. The total amount adsorbed and the selectivity are given by equations (3) and (4), respectively. 

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