State entropy and

The two thermodynamic quantities AS° and A H° are the net difference in standard-state entropy and enthalpy of the energized molecule C and the stabilized reactant molecule C. 

We have already shown, in Figs. XI-4, XI-5, XI-6, and XI-7, the equation of state, entropy, and Gibbs free energy of a substance in all of its three phases. Examination of the parts of those figures dealing with solids will show the similarity of those curves to the ones found in the present section in a more explicit and detailed way. 

The standard-state entropy and enthalpy changes for the overall reaction are approximately independent of temperature and are given by A77° = 8,400 cal/g mole and AiS° = —2.31 cal/(g mole)(°K). Derive expressions for the forward- and reverse-rate constants as functions of temperature. 

In the previous examples we only considered electronic energy changes and approximated the entropy as all or nothing. In essence, we assumed that gas-phase species have 100% of their standard state entropy and surface species possess no entropy at all. These assumptions can certainly be improved and in order to construct thermodynamically consistent microkinetic models this is not just optional, but absolutely necessary. Entropy and enthalpy corrections for surface species can be calculated using statistical thermodynamics from knowledge of the vibrational frequencies, and the translational and rotational degrees of freedom (DOF). In contrast to gas-phase molecules, adsorbates cannot freely rotate and move across the surface, but the translational and rotational DOF are frustrated within the potential energy well imposed by the surface. In the harmonic limit the frustrated translational and rotational DOF can conveniently be described as vibrational modes, which in turn means that any surface adsorbate will have iN vibrational DOFs that are all treated equally. 

A quantitative theory of rate processes has been developed on the assumption that the activated state has a characteristic enthalpy, entropy and free energy the concentration of activated molecules may thus be calculated using statistical mechanical methods. Whilst the theory gives a very plausible treatment of very many rate processes, it suffers from the difficulty of calculating the thermodynamic properties of the transition state. 

Vibrational energy states are too well separated to contribute much to the entropy or the energy of small molecules at ordinary temperatures, but for higher temperatures this may not be so, and both internal entropy and energy changes may occur due to changes in vibrational levels on adsoiption. From a somewhat different point of view, it is clear that even in physical adsorption, adsorbate molecules should be polarized on the surface (see Section VI-8), and in chemisorption more drastic perturbations should occur. Thus internal bond energies of adsorbed molecules may be affected. 

Figure A2.1.10. The impossibility of reaching absolute zero, a) Both states a and p in complete internal equilibrium. Reversible and irreversible paths (dashed) are shown, b) State P not m internal equilibrium and with residual entropy . The true equilibrium situation for p is shown dotted.

It is also instructive to start from the expression for entropy S = log(g(A( m)) for a specific energy partition between the two-state system and the reservoir. Using the result for g N, m) in section A2.2.2. and noting that E = one gets (using the Stirling approximation A (2kN)2N e ). 

A simple cooling cycle serves to illustrate the concepts. Figure 1 shows a temperature—entropy plot for an actual refrigeration cycle. Gas at state 1 enters the compressor and its pressure and temperature are increased to state 2. There is a decrease in efficiency represented by the increase in entropy from state 1 to state 2 caused by friction, heat transfer, and other losses in the compressor. From state 2 to states 3 and 4 the gas is cooled and condensed by contact with a heat sink. Losses occur here because the refrigerant temperature must always be above the heat sink temperature for heat transfer to take... 

TABLE 2-389 Standard-State Entropy of Elements at 298.15 K and 1 Atmosphere... 

The thermodynamic stability of a protein in its native state is small and depends on the differences in entropy and enthalpy between the native state and the unfolded state. From the biological point of view it is important that this free energy difference is small because cells must be able to degrade proteins as well as synthesize them, and the functions of many proteins require structural flexibility. 

Also of importance is the effect of temperature on the gas solubility. From this information it is possible to determine the enthalpy and entropy change experienced by the gas when it changes from the ideal gas state (/z and ) to the mixed liquid state ( andT,). 

A = 3/4 appears to represent a state of maximal disorder (as deduced from computing certain entropy and correlation measures see next section). 

Pagels show that thermodynamic depth is proportional to the difference between the state s thermodynamic entropy (i.e. its coarse grained entropy) and its finegrained entropy, given by fcex volume of points in phase space corresponding to the system s trajectory, where k], is Boltzman s constant. 

You will also notice that gases, as a group, have higher entropies than liquids or solids. Moreover, among substances of similar structure and physical state, entropy usually increases with molar mass. Compare, for example, the hydrocarbons... 

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