Mixing, of ideal gases

The conceptual mystery of the Gibbs free energy is largely associated with the entropic contribution. Sidebars 5.10-5.13 describe some alternative ways to think about entropy. Sidebar 5.10 evaluates the entropy change ASmix for the prototypical mixing of ideal gases A, B under isothermal conditions,... 

SIDEBAR 5.10 ISOTHERMAL ENTROPY OF MIXING OF IDEAL GASES... 

Let us illustrate the application of Boltzmann s formula for an elementary example isothermal mixing of ideal gases (Sidebar 5.10). For this purpose, consider a system of Na = Nb = 4 particles. For a particular partition n 4 — n of the four A-type particles between the VA and VB containers, the number of possible ways H of choosing n A-type particles and 4 — n B-type particles for the first container is given by the product of binomial coefficients... 

Solutions of nonpolar solutes in nonpolar solvents represent the simplest type. The forces involved in solute-solvent and solvent-solvent interactions are all London dispersion forces and relatively weak. The presence of these forces resulting in a condensed phase is the only difference from the mixing of ideal gases. As in the latter case, the only driving force is the entropy (randomness) of mixing. In an ideal solution (AW, = 0) at constant temperature the free energy change will be composed solely of the entropy term ... 

The changes of the thermodynamic functions on mixing of ideal gases... 

We will consider further the mixing of ideal gases in Frame 37, section 37.3. 

Show that for mixing of ideal gases at constant temperature and pressure to form an ideal gas mixture. 

Tataiin, V, Borodiouk, O. Entropy calculation of reversible mixing of ideal gases shows absence of Gibbs paradox. Entropy 1(2), 25-36 (1999). [electronic] www.mdpi.org/entropy/... 

Note that since X < 1, the last term on the rhs of (4.1.35) is necessarily positive and likewise those on the rhs of (4.1.36) and (4.1.37) are necessarily negative. This means that, for isothermal-isobaric mixing of ideal gases, the entropy increases, while the Gibbs and Helmholtz energies decrease. But note that this behavior differs from that for isothermal-isometric mixing. 

The change of entropy, AS, is calculated from the Gibbs equation for mixing of ideal gases. The calculated values are always positive. 

In a gas system composed of a mix of ideal gases, it is possible to derive the Stefan-Maxwell equation (4,5), which solves for the mole fraction of a component in terms of the diffusivities, concentration, and diffusion velocities [velocity of a given species U in equation (10-3)] a one-dimensional form of the equation is... 

In particular, we note that in the process of combining the two systems (Fig. 2.1), the species A and B do mix in the process, yet no entropy change is observed. We can therefore conclude that the process of mixing (of ideal gases) by itself has no effect on the entropy of the system. 

Consider the situation depicted in Figure 13-3, which describes the spontaneous mixing of ideal gases. We can represent this process symbolically as... 

So, for both the spontaneous expansion of an ideal gas and the spontaneous mixing of ideal gases, there is no change in internal energy (or enthalpy) but an increase in entropy It seems possible that increases in entropy underlie spontaneous processes. We will soon see that the characteristic feature of a spontaneous process is that it causes the entropy of the universe to increase. 

For the mixing of ideal gases (see Figure 13-3), explain whether a positive, negative, or zero value is expected for AH, AS, and AG. 

Prove 10.22, showing that it implies isobaric, isothermal mixing of ideal gases. 

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