Entropy thermodynamic definition

To calculate the entropy of a substance, we use Boltzmann s formula, but the calculations are sometimes very difficult and require a lot of manipulation of Eq. 6. To measure the entropy of a substance, we use the thermodynamic definition, Eq. 1, in combination with the third law of thermodynamics. Because the third law tells us that S(0) = 0 and Eq. 2 can be used to calculate the change in entropy as a substance is heated to the temperature of interest, we can write... 

GENERAL EQUATIONS FOR THE ENTROPY OF GASES 143 conclude from Equation (6.111) that a natural thermodynamic definition of T is... 

The second law of thermodynamics can be expressed in terms of another state function, the entropy (S). The thermodynamics definition considers the change in entropy dS that occurs as a result of a physical or chemical change, and is based on the expression... 

Thermodynamic definitions show that the first term of Eq. (1) is the enthalpy of mixing, AHu, while the second term is the negative of the excess entropy of mixing, ASm, multiplied by T. When all four parameters are zero, the liquid is ideal with a zero enthalpy and excess entropy of mixing. What has been called the quasiregular model, a = b = 0, has been used by Panish and Ilegems (1972) to fit the liquidus lines of a number of III—V binary compounds. The particular extension of this special case of Eq. (1) to a ternary liquid given by... 

In practice, then, it is fairly straightforward to convert the potential energy determined from an electronic structure calculation into a wealth of thennodynamic data - all that is required is an optimized structure with its associated vibrational frequencies. Given the many levels of electronic structure theory for which analytic second derivatives are available, it is usually worth the effort required to compute the frequencies and then the thermodynamic variables, especially since experimental data are typically measured in this form. For one such quantity, the absolute entropy 5°, which is computed as the sum of Eqs. (10.13), (10.18), (10.24) (for non-linear molecules), and (10.30), theory and experiment are directly comparable. Hout, Levi, and Hehre (1982) computed absolute entropies at 300 K for a large number of small molecules at the MP2/6-31G(d) level and obtained agreement with experiment within 0.1 e.u. for many cases. Absolute heat capacities at constant volume can also be computed using the thermodynamic definition... 

The material covered in this chapter is self-contained, and is derived from well-known relationships such as Newton s second law and the ideal gas law. Some quantum mechanical results and the statistical thermodynamics definition of entropy are given without rigorous derivation. The end result will be a number of practical formulas that can be used to calculate thermodynamic properties of interest. 

The next important thermodynamic function that we must obtain is the entropy S. The statistical thermodynamic definition of entropy is... 

The thermodynamic definition of entropy says that the change in entropy dS in a process carried out reversibly is the heat absorbed in the process d Qrev divided by the temperature... 

Entropy is a measure of the degree of randomness in a system. The change in entropy occurring with a phase transition is defined as the change in the system s enthalpy divided by its temperature. This thermodynamic definition, however, does not correlate entropy with molecular structure. For an interpretation of entropy at the molecular level, a statistical definition is useful. Boltzmann (1896) defined entropy in terms of the number of mechanical states that the atoms (or molecules) in a system can achieve. He combined the thermodynamic expression for a change in entropy with the expression for the distribution of energies in a system (i.e., the Boltzman distribution function). The result for one mole is ... 

This is a crude model, but hopefully you now see how the calculus of probabilities, as Maxwell put it, explains why heat flows downhill (from hot to cold), why a gas expands to occupy its container and why the world is. . . getting more disordered and generally going to hell in a hand-basket We also hope that you now have a feel for entropy that cannot be obtained from the purely thermodynamic definition of heat divided by temperature, hi principle, calculating the entropy of a system would now seem to be easy. Just count the num-... 

This very important relationship is the macroscopic (thermodynamic) definition of AS. In our treatment we started with the definition of entropy based on probability, because that definition better emphasizes the fundamental character of entropy. However, it is also very important to know how entropy changes relate to changes in macroscopic properties, such as volume and heat, because these changes are relatively easy to measure. 

The successful development of the thermodynamics of irreversible phenomena depends on the possibility of an explicit evaluation of the production of entropy, and for this it is necessary to assume that the thermodynamic definition of entropy can be applied equally to systems which are not in equilibrium, that is to states whose mean lifetime is limited. We are thus confronted immediately with the problem of the domain of validity of the thermodynamic treatment of irreversible phenomena, which can be determined only by a comparison of the results of the thermodynamic treatment with those obtained by the use of statistical mechanics. This problem wall be dealt with in more detail in the third volume of this work meanwhile the main conclusions can be summarized as follows. 

A thermodynamic definition of microemulsions can be obtained from a consideration of the energy and entropy terms for formation of microemulsions. The process of formation of microemuision from a bulk oil phase (for a O/W microemuision) or from a bulk water phase (for a W/O microemuision) is shown schematically in Figure 15.2. 

From the scientific definition point of view, there is a slight difference between our continuum thermodynamics definition of the Second Law and its statistical mechanical version so that the continuum thermodynamics definition of the Second Law states that an observation of decreased universal entropy is impossible in isolated systems however the statistical mechanical definition says that an observation of universal increased entropy is not probable. 

Thus we have observed that the expected increase in disorder with increasing internal energy expressed by equation (6.27) is directly related to the thermodynamic definition (6.18) of the entropy. It now should make intuitive sense that adding heat to a system will increase its disorder, hence its entropy. 

In all these cases, a further condition must be fulfilled in order to assign a definite entropy increase to the irreversible changes. It must be possible not only to release the constraints and thereby induce the change to state 2, but also, after the release of the constraints and after completion of the process, to bring about a reversible restoration of the change by means of only external thermodynamic effects. After all, only in such case is a thermodynamic definition and determination of entropy possible. 

The Entropy Change for a Process Can Be Calculated Using the Thermodynamic Definition of Entropy 432... 

In general, the thermodynamic definition of entropy (Equations 8.6 to 8.8) yields the same value for the entropy change of a process as Boltzmann s statistical definition (Equation 8.3) for the same process. Consider, for example, the entropy change in the reversible and isothermal (constant teinperature) mmqn n i n i les of an ideal gas from an initial volume Vi to a inB39B9tft0-6K ile heat... 

We learned at the end of Section 8.1 [recall Figure 8.5(d)] that the entropy of a system increases when the temperature of the system is raised from Ti to T2. Using the thermodynamic definition of entropy, we can calculate the change in entropy for a system upon heating (or cooling). Beginning with Equation 8.6,... 

The thermodynamic definition of entropy can be used to calculate changes in entropy on expansion or contraction, heating or cooling, or as a result of a phase change. 

Use the thermodynamic definition of entropy and the second law to explain why heat cannot flow spontaneously from a region of low temperature to one of higher temperature. 

Equation (7.45) is sometimes called the thermodynamic definition of entropy. It shows how to obtain the entropy change (w hich you cannot otherwise measure directly) from experimentally observable quantities (heat transfer and temperature) by using a quasi-static process. 

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