Probability, entropy and

Since the entropy of gases is much larger than the entropy of condensed phases, there is a large decrease in entropy in this reaction a gas, hydrogen, is consumed to form condensed materials. Conversely, in reactions in which a gas is formed at the expense of condensed materials, the entropy will increase markedly. 

From the value of AS° for a reaction at any particular temperature Tq, the value at any other temperature is easily obtained by applying Eq. (9.38)  

Differentiating this equation with respect to temperature at constant pressure, we have (d AS° (5iS°(products) /55°(reactants)  

Writing Eq. (9.61) in differential form and integrating between the reference temperature Tq and any other temperature T, we obtain 

The entropy of a system in a definite state can be related to what is called the probability of that state of the system. To make this relation, or even to define what is meant by the probability of the state, it is necessary to have some structural model of the system. In contrast, the definition of the entropy from the second law does not require a structural model the definition does not depend in the least on whether we imagine that the system is composed of atoms and molecules or that it is built with waste paper and baseball bats. For simplicity we will suppose that the system is composed of a very large number of small particles, or molecules. 

The thermodynamic probability of a system is defined as the ratio of the probability of an actual state to one of the same total energy and volume in which the molecules are completely ordered. This suggests that entropy is a function of probability (P) that is. 

However, the entropy of two systems is equal to the sum of the entropies of the individual ones that is, it is additive S = Sj + S2. On the other hand, the probabilities of two independent individual events (Pj and P ) are multiplied together to obtain the probability of the combined event that is. 

The only relation that satisfies both Equations 9.2 and 9.3 is a logarithmic one  

This relationship has been well confirmed and the value of the constant k (Boltzmann s constant) has been found to be 1.38 x 10 J/degree. 

The value of W then becomes large and the increase in entropy of mixing (and therefore the decrease in free energy) is appreciable. By applying Equation 9.5 to molar quantities and assuming that the molecules of solute and solvent a and b, respectively) are the same size, we can arrive at a relatively simple general expression (not derived here) for the entropy of mixing (S )  

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Traditional thermodynamics gives a clear definition of entropy but unfortunately does not tell us what it is. An idea of the physical nature of entropy can be gained from statistical thermodynamics. Kelvin and Boltzmann recognised diat there was a relationship between entropy and probability (cf., disorder) of a system with the entropy given by... 

Now we will connect entropy and probability quantitatively by defining the entropy function S as follows ... 

Basing our calculation on the relationship between entropy and probability, we have thus been able to deduce the complete equation of condition for a perfect gas from the kinetic theory. Transforming equation (2) by means of (3) and (3a), and remembering that, for a perfect gas, %R=c , we have, for the entropy per mol.,... 

Thus there seems to be a relationship between entropy and probability. In the example mentioned above where Vi/(Vi + V2) = 0.5, (6.2) becomes, for one mole of gas,... 

We should be able to relate S and z because the partition function and Boltzmann distribution were originally derived by combining equation (6.7) for the energy of a system with equation (6.4) for the probability of its equilibrium configuration, W, and we already know that entropy and probability are related by... 

This expresses the increase in entropy and probability which accompanies rise in temperature or expansion of volume. 

Legendre Transformations, Maxwell Relations, Linking Entropy and Probability, and Derivation of dS/dt... 

In an effort to understand the relation between microscopic behavior of matter, which was in the realm of mechanics, and macroscopic laws of thermodynamics, Ludwig Boltzmann (1844-1906) introduced his famous relation that related entropy and probability (Box 3.1) ... 

For two-state systems (e.g. binary spins) there is a simple connection between temperature, entropy, and probability. The difference in probability between the two states is called the polarization bias. Consider a single spin particle in a constant magnetic field. At equilibrium with a thermal heat-bath the probabilities of this spin to be up or down (i.e., parallel or anti-parallel to the magnetic field) are given by and p = We refer to a spin as a bit, so that... 

We said earher that the entropy change for a process is related to our knowledge about the system before and after the process has taken place. That is, the entropy change is related to our information about the state of the before and after it has occurred. Let us now examine the relationship between entropy and probability. Consider a deck of 52 playing cards. The probability of drawing a spade is 1/4, and the probability of drawing an ace is 1/13. The probabihty of drawing an ace of spades is (1/4 X 1/13 = 1/52). 

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