Molecular basis of entropy

Thermodynamics rests largely on the consohdation of many observations of nature into two fundamental postulates or laws. Chapter 2 addressed the first law— the energy of the universe is conserved. We camiot prove this statement, but based on over a hundred years of observation, we believe it to be true. In order to use this law quantitatively— that is, to make numerical predictions about a system—we cast it in terms of a thermodynamic property internal energy, u. Likewise, the second law summarizes another set of observations about nature. We will see that to quantify the second law, we need to use a different thermodynamic property entropy, s. Like internal energy, entropy is a conceptual property that allows us to quantify a law of nature and solve engineering problems. This chapter examines the observations on which the second law is based explores how the property s quantifies these observations illustrates ways we can use the second law to make numerical predictions about closed systems, open systems, and thermodynamic cycles and discusses the molecular basis of entropy. 

To decide whether we need to worry about AS0 with regard to any particular reaction, we have to have some idea what physical meaning entropy has. To be very detailed about this subject is beyond the scope of this book, but you should try to understand the physical basis of entropy, because if you do, then you will be able to predict at least qualitatively whether AH° will be about the same or very different from AG°. Essentially, the entropy of a chemical system is a measure of its molecular disorder or randomness. Other things being the same, the more random the system is, the more favorable the system is. 

Equation (16-2) allows the calculations of changes in the entropy of a substance, specifically by measuring the heat capacities at different temperatures and the enthalpies of phase changes. If the absolute value of the entropy were known at any one temperature, the measurements of changes in entropy in going from that temperature to another temperature would allow the determination of the absolute value of the entropy at the other temperature. The third law of thermodynamics provides the basis for establishing absolute entropies. The law states that the entropy of any perfect crystal is zero (0) at the temperature of absolute zero (OK or -273.15°C). This is understandable in terms of the molecular interpretation of entropy. In a perfect crystal, every atom is fixed in position, and, at absolute zero, every form of internal energy (such as atomic vibrations) has its lowest possible value. 

Equation (48) is of the expected form for relations between reaction velocities and activation energies on the one hand, and between equilibrium constants and heats of reaction on the other. However, there are difficulties in the way of using (48) as a quantitative basis-for the BrOnsted relation. In the first place, the quantities E and e in the diagram refer strictly to the behavior of the system at absolute zero since no account is taken of the internal thermal energy of the molecules. In the second place experiment shows that even in a series of similar reactions the observed velocities and equilibrium constants often involve variations in entropies of activation and of reaction, and not only energy changes. These difficulties are not yet fully resolved, but there seems little doubt that diagrams such as Fig. 1 represent the essential molecular basis of the Bronsted relation. 

It is often claimed, with some justification, that statistical theory explains the basis of the second law and provides a mechanistic interpretation of thermodynamic quantities. For example, the Boltzmann expression for entropy is S = ks In W, where W is the number of ways the total energy is distributed among the molecules. Thus entropy in statistical theory is connected to the availability of microstates for the system. In classical theory, on the other hand, there are no pictures associated with entropy. Hence the molecular interpretation of entropy changes depends on statistical theory. 

Ref. 205). The two mechanisms may sometimes be distinguished on the basis of the expected rate law (see Section XVni-8) one or the other may be ruled out if unreasonable adsorption entropies are implied (see Ref. 206). Molecular beam studies, which can determine the residence time of an adsorbed species, have permitted an experimental decision as to which type of mechanism applies (Langmuir-Hinshelwood in the case of CO + O2 on Pt(lll)—note Problem XVIII-26) [207,208]. 

In equation (1.17), S is entropy, k is a constant known as the Boltzmann constant, and W is the thermodynamic probability. In Chapter 10 we will see how to calculate W. For now, it is sufficient to know that it is equal to the number of arrangements or microstates that a molecule can be in for a particular macrostate. Macrostates with many microstates are those of high probability. Hence, the name thermodynamic probability for W. But macrostates with many microstates are states of high disorder. Thus, on a molecular basis, W, and hence 5, is a measure of the disorder in the system. We will wait for the second law of thermodynamics to make quantitative calculations of AS, the change in S, at which time we will verify the relationship between entropy and disorder. For example, we will show that... 

Polymer catalysts showing interactions with the substrate, similar to enzymes, were prepared and their catalytic activities on hydrolysis of polysaccharides were investigated. Kinetical analyses showed that hydrogen bonding and electrostatic interactions played important roles for enhancement of the reactions and that the hydrolysis rates of polysaccharides followed the Michaelis-Menten type kinetics, whereas the hydrolysis of low-molecular-weight analogs proceeded according to second-order kinetics. From thermodynamic analyses, the process of the complex formation in the reaction was characterized by remarkable decreases in enthalpy and entropy. The maximum rate enhancement obtained in the present experiment was fivefold on the basis of the reaction in the presence of sulfuric acid. 

The fundamental driving force behind the remarkable elastic properties of the elastin polymer is believed to be entropic, where stretching decreases the entropy of the system and elastic recoil is driven by a spontaneous return to maximum entropy. The precise molecular basis for elasticity has not been fully elucidated and a number of models exist. Two main categories of structure-function models have been proposed those in which elastin is considered to be isotropic and devoid of structure, and those which consider elastin to be anisotropic with regions of order (Vrhovski and Weiss, 1998). 

A thermodynamic treatment is first suggested on the basis of which one can explain the inclined phase transition that occurs in monolayers of insoluble surfactants. By minimizing the Helmholtz free energy of the monolayer, the equilibrium radius and the equilibrium area fraction of the LC islands are obtained as functions of the average molecular surface area A. The mixing entropy provides a negligible effect on... 

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