The measurement of entropies

The Third-Law entropy at any temperature, S T), is equal to the area under the graph of C/Tbetween T= 0 and the temperature T (Fig. 2.9). If there are any phase transitions (for example, melting) in that range, then the entropy of each transition at the transition temperature is calculated like that in eqn 2.4 and 

To implement the calorimetric procedure the heat capacity of the substance is measured (for instance, by using a differential scanning calorimeter (DSC)) down to as low a temperature as feasible and then using eqn 2.3. In practice, a polynomial in T is fitted to the experimental data and then Cp/T is integrated from the lowest temperature attainable up to the temperature of interest. Thus, if the function Cp T) = a + bT+ cT — is fitted (for instance, by using a least-squares procedure in a software package) to the data between Ti y,sst and trs where trs is the temperature of a phase transition, the entropy just before the phase transition is 

Then another polynomial is fitted to the heat capacities for the new phase up to the temperature of interest (or the next phase transition and a similar integral is evaluated). At each phase transition the enthalpy of transition is measured (once again, typically with a DSC), the entropy of transition is calculated as AtjjH(rtjj)/Ttjj by analogy with eqn 2.4, and this value is added to the value calculated by integrating the heat capacity. 

There remains the experimental problem of determining S(riowest). the entropy at the lowest attainable temperature. If very low temperatures (within a few kelvins of T = 0) can be reached and reliable measurements of Cp made, it is possible to use an extrapolation based on the observation that many non-metallic substances have a heat capacity that obeys the Debye T -law  

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This is frequently stated for an isolated system, but the same statement about an adiabatic system is broader.) A2.1.4.6 IRREVERSIBLE CHANGES AND THE MEASUREMENT OF ENTROPY... 

The measurement of entropy S° T) requires a measurement of heat capacity from absolute zero up to T, the temperature of interest. Heat capacities are measured in a calorimeter in which the temperature change pursuant to an input of a known quantity of heat (e.g. electrical energy) is monitored. Calculating Cp requires knowing the mass, the temperature increment and energy input, and the heat capacity of the empty apparatus. (In the case of liquids, sometimes what is measured is Cgat, the saturation heat capacity, defined by C at = T(dS/ST i, which differs from Cp by the relationship... 

So far, all we know about entropy is that it increases in spontaneous reactions in isolated systems, and that it appears in equations such as (4.55) and (4.56). Hidden in the equations we have derived so far is an important relationship between entropy and heat capacity, which we will see in Chapter 5 serves as a basis for the measurement of entropy. 

At such low temperatures, most matter is solid, and the best type of solid sample to study is a crystal. Studies of crystals showed some intriguing thermodynamic behavior. For instance, in the measurement of entropy it was found that absolute entropy approached zero as the temperature approached absolute zero. This is experimental verification of the third law of thermodynamics. But a measurement of the heat capacity of the solid showed something interesting The heat capacity of the solid approached zero as the temperature approached absolute zero, also. But for virtually all crystalline solids, the heat-capacity-versus-temperature plot took on a similar shape at low temperatures, typified by Figure 18.3 The curves have the distinct shape of a cubic function, that is, y = x. In this case, the variable is absolute temperature, so experimentally it was found that the constant-volume heat capacity Cy was directly related to T ... 

The second law of thermodynamics also consists of two parts. The first part is used to define a new thermodynamic variable called entropy, denoted by S. Entropy is the measure of a system s energy that is unavailable for work.The first part of the second law says that if a reversible process i f takes place in a system, then the entropy change of the system can be found by adding up the heat added to the system divided by the absolute temperature of the system when each small amount of heat is added ... 

Chapter 4 covers much of the same ground as chapter 3 but from a more formal dynamical systems theory approach. The discrete CA world is examined in the context of what is known about the behavior of continuous dynamical systems, and a number of important methodological tools developed by dynamical systems theory (i.e. Lyapunov exponents, invariant measures, and various measures of entropy and... 

In 1877, the Austrian physicist Ludwig Boltzmann proposed a molecular definition of entropy that enables us to calculate the absolute entropy at any temperature (Fig. 7.6). His formula provided a way of calculating the entropy when measurements could not be made and deepened our insight into the meaning of entropy at the molecular level. The Boltzmann formula for the entropy is... 

The procedure for generating a decision tree consists of selecting the variable that gives the best classification, as the root node. Each variable is evaluated for its ability to classify the training data using an information theoretic measure of entropy. Consider a data set with K classes, Cj, I = Let M be the total number of training examples, and let... 

When ammonium nitrate, NH jNOj, dissolves in water, it absorbs heat. Consequently, its standard enthalpy of solution must be positive. This means that the entropy change caused by ammonium nitrate going from solid to solution must increase for the process to proceed spontaneously. This is exactly what one would expect based on the concept of entropy as a measure of randomness or disorder. 

The work of Ludwig Boltzmann (1844-1906) in Vienna led to a better understanding, and to an extension, of the concept of entropy. On the basis of statistical mechanics, which he developed, the term entropy experienced an atomic interpretation. Boltzmann was able to show the connections between thermodynamics and the phenomenon of order and chance events he used the term entropy as a measure... 

Regardless of the relative importance of polar and nonpolar interactions in stabilizing the cyclohexaamylose-DFP inclusion complex, the results derived for this system cannot, with any confidence, be extrapolated to the chiral analogs. DFP is peculiar in the sense that the dissociation constant of the cyclohexaamylose-DFP complex exceeds the dissociation constants of related cyclohexaamylose-substrate inclusion complexes by an order of magnitude. This is probably a direct result of the unfavorable entropy change associated with the formation of the DFP complex. Thus, worthwhile speculation about the attractive forces that lead to enantiomeric specificity must await the measurement of thermodynamic parameters for the chiral substrates. 

Entropy, which has the symbol S, is a thermodynamic function that is a measure of the disorder of a system. Entropy, like enthalpy, is a state function. State functions are those quantities whose changed values are determined by their initial and final values. The quantity of entropy of a system depends on the temperature and pressure of the system. The units of entropy are commonly J K1 mole-1. If S has a ° (5°),... 

The majority of reported studies of formation of cyclodextrin inclusion complexes in solution have been mainly concerned with determination of the stability constants by using equilibrium spectroscopic techniques, and the measurement of the enthalpy and entropy changes characterizing the complexation reaction. The aim of much of this work has been to determine the driving force of complex-formation. Despite the amount of research in this area, however, no general agreement has been reached, and... 

The entropy of a substance is a measure of the amount of disorder within that system -the larger the value of the entropy, the greater the amount of disorder. Entropy is given the symbol S and the standard entropy of a substance, S°, is the entropy of 1 mol of the substance at a pressure of 1 atm and (usually) a temperature of 298 K. Standard entropy values, S°, for some selected substances are given on p. 17 of the SQA Data Booklet. Notice that the units of entropy are J K" mol". ... 

Extreme cases were reactions of the least stabilized, most reactive carbene (Y = CF3, X = Br) with the more reactive alkene (CH3)2C=C(CH3)2, and the most stabilized, least reactive carbene (Y = CH3O, X = F) with the less reactive alkene (1-hexene). The rate constants, as measured by LFP, were 1.7 x 10 and 5.0 X lO M s, respectively, spanning an interval of 34,000. In agreement with Houk s ideas,the reactions were entropy dominated (A5 —22 to —29e.u.). The AG barriers were 5.0 kcal/mol for the faster reaction and 11 kcal/ mol for the slower reaction, mainly because of entropic contributions the AH components were only —1.6 and +2.5 kcal/mol, respectively. Despite the dominance of entropy in these reactive carbene addition reactions, a kind of de facto enthalpic control operates. The entropies of activation are all very similar, so that in any comparison of the reactivities of alkene pairs (i.e., ferei)> the rate constant ratios reflect differences in AA//t, which ultimately appear in AAG. Thus, car-benic philicity, which is the pattern created by carbenic reactivity, behaves in accord with our qualitative ideas about structure-reactivity relations, as modulated by substiment effects in both the carbene and alkene partners of the addition reactions. " Finally, volumes of activation were measured for the additions of CgHsCCl to (CH3)2C=C(CH3)2 and frani-pentene in both methylcyclohexane and acetonitrile. The measured absolute rate constants increased with increasing pressure Ayf ranged from —10 to —18 cm /mol and were independent of solvent. These results were consistent with an early, and not very polar transition state for the addition reaction. 

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