The Calculation of Entropy Changes

No violations of the second law of thermodynamics have ever been observed in a properly done experiment, so there is no reason to doubt its applicability. If it is universally applicable, the ultimate fate of the universe will be to approach a state of thermodynamic equilibrium in which every object in the universe will be at the same temperature. There will be no energy flow from stars to planets, and no life or any other macroscopic processes will be possible. This heat death of the universe will of course not occur for a very long time, but is unavoidable if the second law is universally valid. 

Some people have speculated that the second law might not be universally valid, but might just be a statement of what nearly always occurs. If so, perhaps under some circumstances violations of the second law could be observed (possibly if the universe begins to contract instead of expand). This idea is unsupported speculation, and we have every reason to apply the second law of thermodynamics to any process in any macroscopic system.  

12 Calculate the entropy change for each of the four steps in the Carnot cycle of Problem 3.2, and show that these 

For a process in a closed system that begins at an equilibrium or metastable state and ends at an equilibrium state, the entropy change of the process is given by the line integral on a reversible path from the initial state to the final state. 

Since entropy is a state function, we can calculate AS for a process that is not reversible so long as it has equilibrium or metastable initial and final states by calculating on a reversible path with the same initial and final states. 

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Although derived for a reversible process, this equation relates properties only, and is independent of the process causing the change of state. It is therefore a general equation for the calculation of entropy changes of an ideal gas. 

The thermodynamic temperature is identical to the ideal-gas temperature. Both will be referred to as the absolute temperature. From here on we will use the absolute temperature for the calculation of entropy change according to dS = bqlT. 

What property of entropy allows Hess s law to be used in the calculation of entropy changes ... 

Figure 4-3 Path for the calculation of entropy changes in the ideal-gas state.

Equations 131-1 AO allow the calculation of entropy changes for a pure substance in any state of aggregation [3]. 

Thus, the steric factor may be explained with the help of entropy change. When two molecules come together to produce the activated complex, the total translational degrees of freedom are reduced (from 6 to 3) and rotational degrees of freedom also diminish. This is compensated by an increase in vibrational degrees of freedom. But the definite orientation in forming the activated complex necessarily reduced the entropy, i.e. AS is negative. This decrease in entropy is small when reaction takes place between simple atoms. The calculated value of kbT/h corresponds to collision frequency... 

Now that we have considered the calculation of entropy from thermal data, we can obtain values of the change in the Gibbs function for chemical reactions from thermal data alone as well as from equilibrium data. From this function, we can calculate equilibrium constants, as in Equations (10.22) and (10.90.). We shall also consider the results of statistical thermodynamic calculations, although the theory is beyond the scope of this work. We restrict our discussion to the Gibbs function since most chemical reactions are carried out at constant temperature and pressure. 

The entropy, Spontaneous vs non-spontaneous, Reversible and irreversible processes, Calculation of entropy changes (Isothermal, isobaric, isochoric, adiabatic), Phase changes at equilibrium, Trouton s rule, Calculation for irreversible processes... 

Considerable attention has been given here to heats (enthalpies) of formation, because there are extensive tabulations of these, e.g. [205] and papers on their calculation appear often in the literature, e.g. [201]. However, we should remember that equilibria [147] are dependent not just on enthalpy differences, but also on the often-ignored entropy changes, as reflected in free energy differences, and so the calculation of entropies is also important [206]. 

In equilibrium thermodynamics model A and in model B not far from equilibrium (and with no memory to temperature) the entropy may be calculated up to a constant. Namely, in both cases S = S(V, T) (2.6)2, (2.25) and we can use the equilibrium processes (2.28) in B or arbitrary processes in A for classical calculation of entropy change by integration of dS/dT or dS/dV expressible by Gibbs equations (2.18), (2.19), (2.38) through measurable heat capacity dU/dT or state Eqs.(2.6>, (2.33) (with equilibrium pressure P° in model B). This seems to accord with such a property as in (1.11), (1.40) in Sects. 1.3, 1.4. As we noted above, here the Gibbs equations used were proved to be valid not only in classical equilibrium thermodynamics (2.18), (2.19) but also in the nonequilibrium model B (2.38) and this expresses the local equilibrium hypothesis in model B (it will be proved also in nonuniform models in Chaps.3 (Sect. 3.6), 4, while in classical theories of irreversible processes [12, 16] it must be taken as a postulate). 

Just because this is the simple mixture, the partial entropy s may be interpreted as specific entropy of pure (ideal) gas at a density equal to those in the mixture (see (4.426) and below), and the mixing entropy may be calculated as the sum of entropy changes at the expansion of pure (ideal) gases a (with masses Wa) from starting density (before mixing) to final density (as in the mixture). 

Strategy We are asked to predict, not calculate, the sign of entropy change in the reactions. The factors that lead to an increase in entropy are (1) a transition from a condensed phase to the vapor phase and (2) a reaction that produces more product molecules than reactant molecules in the same phase. It is also important to compare the relative complexity of the product and reactant molecules. In general, the more complex the molecular structure, the greater the entropy of the compound. 

Factors for changes in AF of formation for various structural changes are also available. Using the example given above, one calculates —52,780 cal. and —56,640 cal. from alanine and tyrosine, respectively, for AF of formation of phenylalanine. There is more uncertainty in such a calculation than in the calculation of entropy therefore, it is useful to have a value for AH and to estimate AS in order to calculate AF. 

EXAMPLES. Calculation of Entropy Change for an Irreversible, Isothermal Compression A piston-cylinder device initially contains 0.50 of an ideal gas at 150 kPa and 20°C. The gas is subjected to a constant external pressure of 400 kPa and compressed in an isothermal process. Assume the surroundings are at 20 C. Take Cp = 25R and assume the ideal gas model holds. (a) Determine the heat transfer (in kj) during the process. (b) What is the entropy change of the system, surroundings, and universe (c) Is the process reversible, irreversible, or impossible ... 

Equation refers to molar quantities. To obtain the total entropy of a sample, we must multiply its molar entropy by n, the number of moles. Example illustrates the calculation of A S for a change in concentration. 

Entropy changes are important in every process, but chemists are particularly interested in the effects of entropy on chemical reactions. If a reaction occurs under standard conditions, its entropy change can be calculated from absolute entropies using the same reasoning used to calculate reaction enthalpies from standard enthalpies of formation. The products of the reaction have molar entropies, and so do the reactants. The total entropy of the products is the sum of the molar entropies of the products multiplied by their stoichiometric coefficients in the balanced chemical equation. The total entropy of the reactants is a similar sum for the reactants. Equation... 

The definition of entropy requires that information about a reversible path be available to calculate an entropy change. To obtain the change of entropy in an irreversible process, it is necessary to discover a reversible path between the same initial and final states. As S is a state function, AS is the same for the irreversible as for the reversible process. 

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