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- Volume 2. Statistical Physics and Irreversible Processes
- The kind of motion we call heat: Statistical physics and ..., Book 2
Contents The wave theory of heat.
Non-equilibrium thermodynamics - Wikipedia
Foundations of statistical mechanics Brush Snippet view - Common terms and phrases 19th century assumed assumption atoms body Boltzmann Burbury calculation caloric theory Carnot Chapman Clausius collisions convection density discussion distribution Dover Pubs Dulong Dulong-Petit Dulong-Petit law Edinburgh English trans Enskog entropy equation equilibrium equipartition theorem ergodic ergodic hypothesis experimental experiments finite fluid formula Fourier function H-theorem heat conduction heat conduction coefficient heat transfer hypothesis irreversibility J.
Maxwell kinetic energy kinetic theory later London mathematical Maxwell's mean free path molecular motion molecules Nature Newton's P. Tait paper Paris particles Phil Phys physical physicists Planck pressure probability problem Proc properties published radiant heat radiation random reprinted repulsive force Science scientists Second Law specific heats spheres statistical mechanics Stefan Tait temperature theorem theory of gases theory of heat thermodynamics Thomson tion University Press velocity vibrations vis viva viscosity Waals wave theory Wien Wiss York.
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Maxwell, "The Bakerian lecture- on the viscosity or internal friction of air and other gases", Philos. Meyer, "Ueber die innere Reibung der Case", Ann.
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Leipzig , , , , Leipzig , , 1; Philos. Stokes, "On the friction of fluid In motion and the equilibrium and motion of elastic solids", Philos. Cambridge Soc. Graham, "On the motion of gases", Philos. Maxwell, "Address to the mathematical and physical sections of the British Association", , also in Ref. Maxwell, "Introductory lecture on experimental physics", also in Ref.
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Then the total number of different configurations of micro-systems in m micro-states is defined as. Gibbs introduced a form of entropy as. As the digital representation of integers is binary, the base b is often set as two. 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 outcomes. Their Gibbs entropy values i. The entropy ratio, 7. Diffusion refers to a net flow of matter from a region of high concentration to a region of low concentration, which is an entropy-increasing process, from a more ordered to a less ordered state of molecular locations. For example, when a lump of sugar is added to a cup of coffee for a sweeter taste, the solid cube of sugar dissolves, and the molecules spread out until evenly distributed.
This change from a localized to a more even distribution exists as a spontaneous and, more importantly, irreversible process. In other words, diffusion occurs by itself without external driving forces. In addition, once diffusion occurs, it is not possible for the molecular distribution to return to its original undiffused state. If diffusion does not occur spontaneously, then there is no natural mixing, and one may have a bitter coffee taste and sweet sugar taste in an unmixed liquid phase. In general, diffusion is closely related to mixing and demixing separation within a plethora of engineering applications.
Why does diffusion occur, and how do we understand the spontaneous phenomena? A key stands as the entropy-changing rate from one static equilibrium to the other. Before discussing diffusion as an irreversible phenomenon, however, the following section includes several pictures so as to create a better understanding of diffusion phenomenon as one of the irreversible thermodynamic processes. As temperature increases, molecules gain kinetic energy and diffuse more actively in order to position evenly within the volume. 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. 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 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.
Diffusion in a rectangular container consisting of two equal-sized chambers A and B a before and b after the mid-wall is removed. 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. Stokes derived hydrodynamic force that a stationary sphere experiences when it is positioned in an ambient flow [ 5 ]:.
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.
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Einstein and Langevin independently derived the same equation as 16 of which the general form can be rewritten as. 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. Since the sole molecule exists, there is no concentration gradient. However, the following question arises. Why does a single molecule diffuse without experiencing any collisions in the absence of other molecules? 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 diffusion called self-diffusion. 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 chance. If there are a number of dizzy individuals and we can determine an average for their seemingly random movements, then.
Now let us calculate a mean of x 2 :. Therefore, a sum of them is zero.
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Then, the mean of squared distance at time t is calculated as. Then, the diffusivity for 1D is explicitly. The probability distribution satisfies. In this case, we approximate the integrand of Eq. We finally derive the so-called diffusion equation:. Conditions required for Eq. For simplicity, we shall treat the problem of diffusion in one dimension.
It would be hopeless to deterministically trace all the collisions of this particle with a number of solvent molecules in series. In Eq. We multiply x on both sides of Eq. Because the random fluctuating force A t is independent of the particle position x t , we calculate.
The kind of motion we call heat: Statistical physics and ..., Book 2
Note that the origin of the particle motion exists as the number of its collisions with solvent molecules at temperature T. One more integration with respect to time yields. The root-mean-square distance is. On one hand, if a particle is translating with a constant velocity, its distance from the initial location is linearly proportional to the elapsed time; on the other hand, if particle is diffusing, its root-mean-square distance is proportional to t.
See the next section for the Brownian diffusivity D B. As such, we assume that. The relationship between x , v , and t can be obtained as follows [ 11 ]:. Note that Eq. Let C x be the concentration of particles near the position x of a specific particle. Note that x is not a fixed point in Eulerian space but a moving coordinate of a particle being tracked.