Free Energy Worked examples

Thus, while the probability of a particular work value and the distribution of work values can give some estimate of the relative error in the free energy (for example by performing bootstrap or subsampling analysis over the full data set), there is no inherent way to extrapolate from the full finite data set to a larger (better converged) estimate. 

The free energy of the crystal corresponding to other ensembles may be computed through inclusion of additional work terms in the equation for free energy, for example ... 

If there are other kinds of work, similar expressions apply. For example, with electromagnetic work (equation (A2.1.8)1 instead of pressure-volume work, one can write for the Helmholtz free energy... 

Many researchers have applied similar approaches to develop or apply linear free energy relationships, when the substituent is directly attached to the double bond, with some success. Two of the more notable examples can be found in the Patterns of Reactivity Scheme (Section 7.3.4) and the works of Giese and coworkers.16 19... 

The agreement is remarkably good, which shows that the lead accumulator is almost theoretically reversible, and the example is all the more interesting in that it contains direct measurements of the maximum work (i.e., the diminution of free energy) of an isothermal process carried out in two entirely different ways. 

Another simple approach assumes temperature-dependent AH and AS and a nonlinear dependence of log k on T (123, 124, 130). When this dependence is assumed in a particular form, a linear relation between AH and AS can arise for a given temperature interval. This condition is met, for example, when ACp = aT" (124, 213). Further theoretical derivatives of general validity have also been attempted besides the early work (20, 29-32), particularly the treatment of Riietschi (96) in the framework of statistical mechanics and of Thorn (125) in thermodynamics are to be mentioned. All of the too general derivations in their utmost consequences predict isokinetic behavior for any reaction series, and this prediction is clearly at variance with the facts. Only Riietschi s theory makes allowance for nonisokinetic behavior (96), and Thorn first attempted to define the reaction series in terms of monotonicity of AS and AH (125, 209). It follows further from pure thermodynamics that a qualitative compensation effect (not exactly a linear dependence) is to be expected either for constant volume or for constant pressure parameters in all cases, when the free energy changes only slightly (214). The reaction series would thus be defined by small differences in reactivity. However, any more definite prediction, whether the isokinetic relationship will hold or not, seems not to be feasible at present. 

We have seen several examples of a technique for separation of gas mixtures which, in contrast with most commercial processes, requires no physical transfer of solvent, handling of solids, or cycling of temperature or pressure. The energy requirements can also be far lower The thermodynamic minimum work of separation is, under isothermal conditions, the free energy difference between the process stream and byproduct, or permeate, stream. When this difference is due only to the partial pressure difference of component 1, it becomes ... 

In practice, this simple formula will hardly ever work, especially if the free energy changes appreciably with . Consider, for example, two states of the systems, , and j such that A,1 f <) - AA( j) = 5/, BT. Then, on average, the former state is sampled only seven times for every 1,000 configurations sampled from the latter state. Such nonuniform sampling is undesirable, as it leads to a considerable loss of statistical accuracy. For the free energy profile shown in Fig. 3.1, transitions between... 

In summary, using work collected from forward and backward paths greatly improves the accuracy of the estimates, and for the symmetric system studied here eliminates the bias. In our particular example, the cumulant estimators using forward and backward work data produce the most precise free energy estimates, followed by Bennett s optimal estimator. However, this somewhat poorer performance of the optimal estimator is caused in part by the high degree of symmetry of the system studied. 

As discussed in the example from the work of Hodel et al. [3], one of the most efficient ways to improve the accuracy of free energy calculations with a given force field is to enhance the conformational sampling. Thus, it is important to assess the extent to which phase space is covered. 

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