Entropy absolute, experimental determination

It should be noted that what one measures in experiments is the difference in the entropy, and not the absolute entropy. Assuming that the entropy is zero at absolute zero in accordance with the Nemst-Planck postulate, one can determine the absolute entropy experimentally. However, it is well known that SCL is a metastable state, and there is no reason for its entropy to vanish at absolute zero [16]. Indeed, it has been demonstrated some time ago that the residual entropy at absolute zero obtained by extrapolation is a nonzero fraction of the entropy of melting [43 ], which is not known a priori. Therefore, it is impossible to argue from experimental data that the entropy indeed falls to zero, since such a demonstration will certainly require calculating absolute entropy though efforts continue to date [61, 62]. 

FIGURE 7.11 The experimental determination of entropy, (a) The heat capacity at constant pressure in this instance) of the substance is determined from close to absolute zero up to the temperature of interest, (b) The area under the plot of CP/T against T is determined up to the temperature of interest. 

The symbol 9 is called the characteristic temperamre and can be calculated from an experimental determination of the heat capacity at a low temperature. This equation has been very useful in the extrapolation of measured heat capacities [16] down to OK, particularly in connection with calculations of entropies from the third law of thermodynamics (see Chapter 11). Strictly speaking, the Debye equation was derived only for an isotropic elementary substance nevertheless, it is applicable to most compounds, particularly in the region close to absolute zero [17]. 

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... 

Two points concerning the evaluation of the first integral in Equation (15.9) require further discussion. In most experimental determinations of absolute entropies, the lowest temperature attained ranges from 1 to 15 K ... 

Over the years, many experiments have been carried out which confirm the third law. The experiments have generally been of two types. In one type the change of entropy for a change of phase of a pure substance or for a standard change of state for a chemical reaction has been determined from equilibrium measurements and compared with the value determined from the absolute entropies of the substances based on the third law. In the other type the absolute entropy of a substance in the state of an ideal gas at a given temperature and pressure has been calculated on the basis of statistical mechanics and compared with those based on the third law. Except for well-known, specific cases the agreement has been within the experimental error. The specific cases have been explained on the basis of statistical mechanics or further experiments. Such studies have led to a further understanding of the third law as it is applied to chemical systems. 

The condition discussed in the previous paragraph demands certain care in the experimental determination of absolute entropies, particularly in the cooling of the sample to the lowest experimental temperature. In order to approach the condition that all molecules are in the same quantum state at 0 K, we must cool the sample under the condition that thermodynamic equilibrium is maintained within the sample at all times. Otherwise some state may be obtained at the lowest experimental temperature that is metastable with respect to another state and in which all the molecules may not be in the same quantum state at 0 K. 

S 0 (2 98 K) for N2O, as experimentally determined from Cp/T measurements is some 6 JK-1 mol smaller than the spectroscopically derived result. This residual entropy arises from the fact that the true crystal lattice contains units arranged both as NNO as well as ONN close to the absolute zero of temperature. This residual disorder therefore contrasts with the perfectly ordered arrangement of either NNO NNO or ONN ONN which would satisfy requirements of perfect order at 0 K. 

Values of AG° for matty formation reactions are tabulated in standard references. The reported values of AG are not measured experimentally, but are calculated by Eq. (13.16). The detennination of A5 may be based on the tliird law of thennodynamics, discussed in Sec. 5.10. Combination of valnes from Eq. (5.40) for the absolute entropies of the species taking part in the reaction gives the valne of AS. Entropies (and heat capacities) are also commonly determined from statistical calcnlations based on spectroscopic data. ... 

Entropy measures the change in order a positive change in entropy measures increased disorder, and a negative change in entropy measures increased order. For the transition where ice melts to form water at 0°C (273°K on the absolute scale where motion ceases at 0°K), the increase in entropy is the experimentally determined heat absorbed during the transition divided by the temperature for the transition, that is, 273°K (see Chapter 5 for a more complete discussion). 

For simple molecules it is possible to calculate absolute entropies theoretically, and for most systems the theoretical value very closely agrees with third-law entropies calculated experimentally. However, in some situations the third-law entropy (determined using Equation 8.27 and accounting for any phase transitions) does not agree with the theoretical value. The origin of this discrepancy is generally due to defects or impurities that are frozen in the system at low temperatures. The third law does not apply for such a system because the system is not in its thermodynamically most stable state. The difference between the experimental entropy and the theoretical entropy for such a system is referred to as the residual entropy, which is the value of the entropy at 0 K for systems for which the third law is not applicable. 

In Section 5.5 we discussed how calorimetry can be used to measure AH for chemical reactions. No comparable, easy method exists for measuring AS for a reaction. By using experimental measurements of the variation of heat capacity with temperature, however, we can determine the absolute entropy, S, for many substances at any temperature. (The theory and the methods used for these measurements and calculations are beyond the scope of this text.) Absolute entropies are based on the reference point of zero entropy for perfect crystalline solids at 0 K (the third law). Entropies are usually tabulated as molar quantities, in units of joules per mole-Kelvin (J/mol-K). 

The ability to obtain the complete set of vibrational modes for large polyatomic systems is of considerable importance. Experimentally this information is very difficult to determine and, once available, it becomes possible to compute thermodynamics quantities such as absolute entropies. Where necessary, improvements on the harmonic approximation have been computed by introducing cubic and quartic terms in studies of a variety of organic molecules. ... 

The significance of the third law of thermodynamics is that it enables us to determine experimentally the absolute entropies of substances. Starting with the knowledge that the entropy of a pure crystalline substance is zero at 0 K, we can measure the increase in entropy of the substance as it is heated. The change in entropy of a substance, AS, is the difference between the final and initial entropy values ... 

The activation free energy of desorption may be computed from the rate of desorption as determined experimentally from the change in the surface potential with time. The theory of absolute rates has been applied to desorption by Eley (120) and Higuchi et cd. (107) to obtain energies and entropies of activation as a function of coverage. The rate of desorption is given by,... 

Some actual experimental data will be quoted below (sec Table XVI), but for the present sufficient indication has been given of the procedure used for determining the entropies of substances which are solid or liquid at ordinary temperatures. For such substances the standard states are the pure solid or pure liquid at 1 atm. pressure (cf. 12e), and the standard entropies, per g. atom (for elements) or per mole (for compounds), at 25 C, derived from heat capacity measurements are recorded in Table XV. As stated earlier ( 19h), entropies are usually expressed in terms of calories per degree, and the quantity 1 cal. deg. the temperature being on the usual absolute scale, in terms of the centigrade degree, is often referred to as an entropy unit and abbreviated to e.u. 

It is of importance to note that, except for hydrogen and deuterium molecules, the entropy derived from heat capacity measurements, i.e., the thermal entropy, as it is frequently called, is equivalent to the practical entropy in other words, the nuclear spin contribution is not included in the former. The reason for this is that, down to the lowest temperatures at which measurements have been made, the nuclear spin does not affect the experimental values of the heat capacity used in the determination of entropy by the procedure based on the third law of thermodynamics ( 23b). Presumably if heat capacities could be measured right down to the absolute zero, a temperature would be reached at which the nuclear spin energy began to change and thus made a contribution to the heat capacity. The entropy derived from such data would presumably include the nuclear spin contribution of R In (2i + 1) for each atom. Special circumstances arise with molecular hydrogen and deuterium to which reference will be made below ( 24n). 

The integration requires that all order structures and the enthalpies of all transformations in the complex natural material are known at low temperatures down to the vicinity of absolute zero, i. e. are determined by means of experimental investigations and theoretical calculations. This is frequently an extensive work, even with a single pure substance. For these reasons the entropies of natural materials cannot at present be given quantitatively. However, entropy balances for different process variants can be compared numerically with each other, if the same natural material is always used as the input. An example is shown in Section 7.7. 

ArG298 = -RT InK. This experimental scale of relative acidities was converted to a scale of absolute acidities by including certain compounds as anchor points. Thus, the gas-phase acidity of PH3 was determined to be ArG29s = 363 2 kcal/mol. The entropy change for the deprotonation process was evaluated by procedures using statistical mechanics as ArS = 24.9 2 cal - mol" K From these data the deprotonation enthalpy of PH3 at 298 K was calculated to be ArH298=PA(PHi) = 370.4 2 kcal/mol [1, 2]. 

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