Gas, chemical potential

The chemical potential pa on the left is the full chemical potential including ideal and excess parts. In this chapter we will scale the chemical potentials by (3 and often refer to this unitless quantity as the chemical potential. 3/ia yields the absolute activity. The first term on the right is the ideal-gas chemical potential, where pa is the number density, Aa is the de Broglie wavelength, and q 1 is the internal (neglecting translations) partition function for a single molecule without interactions with any other molecules. 

To interpret the phase diagram in Fig. 7.1 quantitatively, we must return to Eq. (7.3) and more fully define the chemical potential. For ideal gases, the chemical potential can be rigorously derived from statistical mechanics. A useful definition of the ideal-gas chemical potential for O2 is... 

A Statistical Thermodynamic Approach to Hydrate Phase Equilibria and the ideal gas chemical potential fx is calculated by... 

The Ising spin model does not consider the coexistence of the ordered (ferromagnetic) and non-ordered (paramagnetic) phases at subcritical temperatures. As a result, there is no latent heat r/T) and disorder parameter associated with the ferromagnetic transition. The condition dh/dT = 0 must be added to h=0. The known CXC-dependence of the lattice-gas chemical potential ... 

Deviations from ideal gas (IG) behavior. Taking the limit P — 0, or p-r —> 0, we have the ideal-gas chemical potential... 

We are interested in how liquid and gas chemical potentials depend on xb and r, at fixed pressure p. For the liquid solution, we have peiliquid, T, xg) = Pg (liquid, T) + fcTlnygXg, where the superscript ° for the liquid state in the solvent convention means that the liquid B is pure and contains no solute. 

Using the ideal gas chemical potential Mg = fcBTln(P/fcgj ) and the... 

Using the temperature dependence of 7 from Eq. III-l 1 with n - and the chemical potential difference Afi from Eq. K-2, sketch how you expect a curve like that in Fig. IX-1 to vary with temperature (assume ideal-gas behavior). 

The chemical potential now includes any such effects, and one refers to the gmvochemicalpotential, the electrochemical potential, etc. For example, if the system consists of a gas extending over a substantial difference in height, it is the gravochemical potential (which includes a tenn m.gh) that is the same at all levels, not the pressure. The electrochemical potential will be considered later. 

Note that a constant of integration p has come mto the equation this is the chemical potential of the hypothetical ideal gas at a reference pressure p, usually taken to be one ahnosphere. In principle this involves a process of taking the real gas down to zero pressure and bringing it back to the reference pressure as an ideal gas. Thus, since dp = V n) dp, one may write... 

Given this experimental result, it is plausible to assume (and is easily shown by statistical mechanics) that the chemical potential of a substance with partial pressure p. in an ideal-gas mixture is equal to that in the one-component ideal gas at pressure p = p. 

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. 

Note that this has resulted in the separation of pressure and composition contributions to chemical potentials in the ideal-gas mixture. Moreover, the themiodynamic fiinctions for ideal-gas mixing at constant pressure can now be obtained ... 

It follows that, because phase equilibrium requires that the chemical potential p. be the same in the solution as in the gas phase, one may write for the chemical potential in the solution ... 

This is the same as that in the canonical ensemble. All the thennodynamic results for a classical ideal gas tlien follow, as in section A2.2.4.4. In particular, since from equation (A2.2.158) the chemical potential is related to which was obtained m equation (A2.2.88). one obtains... 

In an ideal Bose gas, at a certain transition temperature a remarkable effect occurs a macroscopic fraction of the total number of particles condenses into the lowest-energy single-particle state. This effect, which occurs when the Bose particles have non-zero mass, is called Bose-Einstein condensation, and the key to its understanding is the chemical potential. For an ideal gas of photons or phonons, which have zero mass, this effect does not occur. This is because their total number is arbitrary and the chemical potential is effectively zero for tire photon or phonon gas. 

The chemical potential for an ideal Bose gas has to be lower than the ground-state energy. Otherwise the occupancy (n.p of some state j would become negative. 

At equilibrium, in order to achieve equality of chemical potentials, not only tire colloid but also tire polymer concentrations in tire different phases are different. We focus here on a theory tliat allows for tliis polymer partitioning [99]. Predictions for two polymer/colloid size ratios are shown in figure C2.6.10. A liquid phase is predicted to occur only when tire range of attractions is not too small compared to tire particle size, 5/a > 0.3. Under tliese conditions a phase behaviour is obtained tliat is similar to tliat of simple liquids, such as argon. Because of tire polymer partitioning, however, tliere is a tliree-phase triangle (ratlier tlian a triple point). For smaller polymer (narrower attractions), tire gas-liquid transition becomes metastable witli respect to tire fluid-crystal transition. These predictions were confinned experimentally [100]. The phase boundaries were predicted semi-quantitatively. 

The excess chemical potential, that is the difference between the actual value and that of equivalent ideal gas system, is given by ... 

Here d/l is the additional wall area exposed when the uptake diminishes by dn moles through evaporation from the capillary p." is the chemical potential of the capillary condensate and p° that of the bulk liquid adsorptive. The negative sign is necessary because the area A exposed increases as the uptake diminishes. If the adsorptive vapour behaves as a perfect gas,... 

The chemical potential in the dyebath solution, is defined in equation 1 where is the standard chemical potential in the solution, R is the gas constant, Tis temperature in K, and a is activity. 

Because only differences in chemical potential can be measured, the chemical or electrochemical potential of each species is broken down as in equation 3. An arbitrary secondary reference state is defined for each compound. For instance, the chemical potential of chlorine gas is expressed as... 

Chemical Potential. Equilibrium calculations are based on the equaHty of individual chemical potentials (and fiigacities) between phases in contact (10). In gas—soHd adsorption, the equiHbrium state can be defined in terms of an adsorption potential, which is an extension of the chemical potential concept to pore-filling (physisorption) onto microporous soHds (11—16). 

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... 

Driving Force Gas moves across a membrane in response to a difference in chemical potential. Partial pressure is sufficiently proportional to be used as the variable for design calculations for most gases of interest, but fugacity must be used for CO9 and usually for Hg... 

Figure 3.2 Chemical potential diagrams for the transport of silicon carbide by chlorine, showing that the much greater stability of SiCU than CCI4 makes this process very inefficient, while the use of HCl as the transporting gas can be operated under optimum conditions...

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