3.1 Mutual diffusion

S kB

Non-Equilibrium Particle Dynamics

Finally, we have

2.2.3 Shannon's entropy

outcomes.

(i.e., entropy per object) are

and their ratio is

82

scoin kB

The system entropies of the coin and the dice are

Sdice Scoin

entropy ratio, 7.754, is higher than the ratio of available states, 3.

¼ � ∑ 2 k¼1

> sdice kB

<sup>¼</sup> <sup>6</sup> � ln 6 <sup>2</sup> � ln 2 <sup>¼</sup> <sup>3</sup> �

pk ln pk ¼ � ∑

¼ � ∑ 6 k¼1 � 1 <sup>6</sup> ln <sup>1</sup>

≃ N ln ð Þ� N=e ∑

namic state k. Gibbs introduced a form of entropy as

m k¼0

Nk ln ð Þ¼� N=e N ∑

m k¼0

where pk ¼ Nk=N exists as the probability of finding the system in thermody-

m k¼0 pk ln pk

pk ln pk

pk ln pk

pi log bpi (13)

¼ ln 2 ¼ 0:6931

m k¼0

S ¼ �kBN ∑

sG ¼ �kB ∑

which is equal to the system entropy per object or particle, denoted as

<sup>N</sup> ¼ �kB <sup>∑</sup>

sG <sup>¼</sup> <sup>S</sup>

In information theory, Shannon's entropy is defined as [2]

SSh ¼ � ∑ i

As the digital representation of integers is binary, the base b is often set as two. Note that Shannon's entropy is identical to Gibbs entropy, if Boltzmann's constant kB is discarded and the natural logarithm ln ¼ log <sup>e</sup> is replaced by log 2. Entropy only takes into account the probability of observing a specific event, so the information it encapsulates is information about the underlying probability distribution, not the meaning of the events themselves. Example 3 deals with tossing a coin or a dice and how the entropy S increases with respect to the number of available

Example 3: Let's consider two conventional examples, i.e., a coin and a dice. Their Gibbs entropy values

<sup>6</sup> <sup>¼</sup> ln 6 <sup>¼</sup> <sup>1</sup>:<sup>791</sup>

ln 2 <sup>¼</sup> <sup>3</sup> � <sup>2</sup>:<sup>5850</sup> <sup>¼</sup> <sup>7</sup>:754><sup>3</sup>

2 k¼1 1 <sup>2</sup> � ln <sup>1</sup> 2 

Scoin=kB ¼ 2 � 0:6931 ¼ 1:386

Sdice=kB ¼ 6 � 1:791 ¼ 10:750

ln 2 � 3

where three indicates the ratio of the number of available cases of a dice (6) to that of a coin (2). The

m k¼0

Nk N  � ln Nk N 

> Diffusion is often driven by the concentration gradient referred to as ∇c, typically in a finite volume, temperature, and pressure. As temperature increases, molecules gain kinetic energy and diffuse more actively in order to position evenly within the volume. A general driving force for isothermal diffusion exists as a gradient of the chemical potential ∇μ between regions of higher and lower concentrations.

> As shown in Figure 1, diffusion of solute molecules after removing the mid-wall is spontaneous. Initially, two equal-sized rectangular chambers A and B are separated by an impermeable wall between them. The thickness of the mid-wall is negligible in comparison to the box length; in each chamber of A and B, the same amount of water is contained. Chamber A contains seawater of salt concentration 35,000 ppm, and chamber B contains fresh water of zero salt concentration. If the separating wall is removed slowly enough not to disturb the stationary solvent medium but fast enough to initialize a sharp concentration boundary between the

Figure 1.

Diffusion in a rectangular container consisting of two equal-sized chambers A and B (a) before and (b) after the mid-wall is removed.

two concentration regions, then the concentration in B increases as much as that in A decreases because mass is neither created nor annihilated inside the container. This spontaneous mixing continues until both concentrations become equal and, hence, reach a thermodynamic equilibrium consisting of a half-seawater/half-fresh water concentration throughout the entire box. Diffusion occurs wherever and whenever the concentration gradient exists, and diffusive solute flux is represented using Fick's law as follows [3, 4]:

$$J\_s = -D\frac{\mathbf{d}c}{\mathbf{d}\mathbf{x}} \quad \text{in} \quad \mathbf{1} - D \tag{14}$$

3.3 Diffusion pictures

diffusion called self-diffusion.

position is

85

3.3.1 Self-diffusion and random walk

their seemingly random movements, then

Now let us calculate a mean of x2:

because Δx has a 50:50 chance of þl and �l:

cule and absolutely zero anywhere else: <sup>c</sup> <sup>¼</sup> <sup>V</sup>�<sup>1</sup>

DOI: http://dx.doi.org/10.5772/intechopen.86607

Fundamentals of Irreversible Thermodynamics for Coupled Transport

Several pictures of diffusion phenomena are discussed in the following section, which give probabilistic and deterministic viewpoints. If one considers an ideal situation where there exists only one salt molecule in a box filled with solvent (e.g., water) of finite T, P, and V. Since the sole molecule exists, there is no concentration gradient. Mathematically, the concentration is infinite at the location of the mole-

position of the solute and r is an arbitrary location within the volume. However, the following question arises. Why does a single molecule diffuse without experiencing any collisions in the absence of other molecules? The answer is that the solvent medium consists of a number of (water) molecules having a size of an order of O 10�<sup>10</sup> m. The salt molecule will suffer a tremendous number of collisions with solvent molecules of a certain kinetic energy at temperature T. Since each of these collisions can be thought of as producing a jump of the molecule, the molecule must be found at a distance from its initial position where the diffusion started. In this case, the molecule undergoes Brownian motion. Note that the single molecule collides only with solvent molecules while diffusing, which exists as a type of

A particle initially located at r<sup>0</sup> has equal probabilities of 1/6 to move in ð Þ �x; �y; �z directions. For mathematical simplicity, we restrict ourselves to 1D random walk of a dizzy individual, who moves to the right or to the left with a 50:50 chance. Initially (at time t ¼ 0), the individual is located at x<sup>0</sup> ¼ 0 and starts moving in a direction represented by Δx ¼ �l where þl and �l indicate the right and left distances that the individual travels with an equal probability, respectively.

where Δx<sup>1</sup> can be þl or �l. At the time of the second step, t<sup>2</sup> ¼ t<sup>1</sup> þ Δt ¼ 2Δt, the

where Δx<sup>2</sup> ¼ �l. At tn ¼ nΔt (n ≫ 1), the position may be expressed as

xn ¼ Δx<sup>1</sup> þ Δx<sup>2</sup> þ ⋯ þ Δxn�<sup>1</sup> þ Δxn ¼ ∑

h i xn ¼ ∑ n i¼1

h i <sup>Δ</sup><sup>x</sup> ¼ þð Þ<sup>l</sup> <sup>1</sup>

If there are a number of dizzy individuals and we can determine an average for

<sup>2</sup> þ �ð Þ<sup>l</sup> <sup>1</sup>

At the next step, t<sup>1</sup> ¼ t<sup>0</sup> þ Δt ¼ Δt, the individual's location is found at

δð Þ r � r<sup>0</sup> where r<sup>0</sup> is an initial

x<sup>1</sup> ¼ x<sup>0</sup> þ Δx<sup>1</sup> ¼ Δx<sup>1</sup> (19)

x<sup>2</sup> ¼ x<sup>1</sup> þ Δx<sup>2</sup> ¼ Δx<sup>1</sup> þ Δx<sup>2</sup> (20)

n i¼1

h i Δxi ¼ nh i Δx ¼ 0 (22)

<sup>2</sup> <sup>¼</sup> <sup>0</sup> (23)

Δxi (21)

or

$$J\_s = -D\nabla c \quad \text{in } \mathfrak{Z} - D \tag{15}$$

where D is diffusion coefficient (also often called diffusivity) of a unit of m<sup>2</sup>=s. A length scale of diffusion can be estimated by ffiffiffiffiffiffiffiffi <sup>D</sup>δ<sup>t</sup> <sup>p</sup> where <sup>δ</sup><sup>t</sup> is a representative time interval. In molecular motion, δt can be interpreted as a time duration required for a molecule to move as much as a mean free path (i.e., a statistical averaged distance between two consecutive collisions).

### 3.2 Stokes-Einstein diffusivity

When the solute concentration is low so that interactions between solutes are negligible, the diffusion coefficient, known as the Stokes-Einstein diffusivity, may be given by

$$D\_0 = \frac{k\_B T}{6\pi\eta a} \tag{16}$$

where kB is the Boltzmann constant, η is the solvent viscosity<sup>1</sup> , and a is the (hydrodynamic) radius of solute particles. Stokes derived hydrodynamic force that a stationary sphere experiences when it is positioned in an ambient flow [5]:

$$F\_H = \mathfrak{G}\pi\eta av$$

where v represents a uniform fluid velocity, which can be interpreted as the velocity of a particle relative to that of an ambient fluid. FH is linearly proportional to v, and its proportionality 6πηa is the denominator of the right-hand side of Eq. (16). Einstein used the transition probability of molecules from one site to the another, and Langevin considered the molecular collisions as random forces acting on a solute (see Section 3.3 for details). Einstein and Langevin independently derived the same equation as (16) of which the general form can be rewritten as

$$D\_0 = \frac{k\_B T}{(2d)\pi\eta} \tag{18}$$

where d is the spatial dimension of the diffusive system (i.e., d ¼ 1, 2, and3 for 1D, 2D, and 3D spaces).

<sup>1</sup> Greek symbol μ is also often used for viscosity in fluid mechanics literature. In this book, chemical potential is denoted as μ.

Fundamentals of Irreversible Thermodynamics for Coupled Transport DOI: http://dx.doi.org/10.5772/intechopen.86607
