Macroscopic observables

Figure XVIII-2 shows how a surface reaction may be followed by STM, in this case the reaction on a Ni(llO) surface O(surface) + H2S(g) = H20(g) + S(surface). Figure XVIII-2a shows the oxygen atom covered surface before any reaction, and Fig. XVIII-2h, the surface after exposure to 3 of H2S during which Ni islands and troughs have formed on which sulfur chemisorbs. The technique is powerful in the wealth of detail provided on the other hand, there is so much detail that it is difficult to relate it to macroscopic observation (such as the kinetics of the reaction).

Macroscopic observables, such as pressme P or heat capacity at constant volume C v, may be calculated as derivatives of thermodynamic functions. 

In general a macroscopic observable can be calculated as an average over a corresponding microscopic quantity weighted by the Boltzman probability distribution as in eq. (16.5). 

Students ability to connect observations at the macroscopic level with their descriptions using the submicro and symbolic types of representation improved as a consequence of the LON teaching approach. Teachers attributed the improvement to the consistent use of all three types of representation and to the use of visible models as a tool for bridging the gap between macroscopic observations and symbolic notations of chemical equations. 

The macroscopic observable free energy is the average overall possibilities of arranging r i.e.,... 

Chemical engineers have traditionally approached kinetics studies with the goal of describing the behavior of reacting systems in terms of macroscopically observable quantities such as temperature, pressure, composition, and Reynolds number. This empirical approach has been very fruitful in that it has permitted chemical reactor technology to develop to a point that far surpasses the development of theoretical work in chemical kinetics. 

The density of states is the central function in statistical thermodynamics, and provides the key link between the microscopic states of a system and its macroscopic, observable properties. In systems with continuous degrees of freedom, the correct treatment of this function is not as straightforward as in lattice systems - we, therefore, present a brief discussion of its subtleties later. The section closes with a short description of the microcanonical MC simulation method, which demonstrates the properties of continuum density of states functions. 

In strict terms, 0th order reactions do not really exist. They are always macroscopically observed reactions where the rate of the reaction is independent of the concentrations of the reactants for a certain time period. Formally, the ODE for a basic 0th order reaction is defined below ... 

In section 6.2, you explored the rate law, which defines the relationship between the concentrations of reactants and reaction rate. Why, however, does the rate of a reaction increase with increased concentrations of reactants Why do increased temperature and surface area increase reaction rates To try to explain these and other macroscopic observations, chemists develop theories that describe what happens as reactions proceed on the molecular scale. In this section, you will explore these theories. 

It is noteworthy that prior to the advent of scanning probe microscopy electrochemically driven reconstruction phenomena had been identified and studied using traditional macroscopic electrochemical measurements [210,211], However, STM studies have provided insight as to the various atomistic processes involved in the phase transition between the reconstructed and unreconstructed state and promise to provide an understanding of the macroscopically observed kinetics. An excellent example is provided by the structural evolution of the Au(lOO) surface as a function of potential and sample history [210,211,216-223], Flame annealing of a freshly elec-tropolished surface results in the thermally induced formation of a dense hexagonal close-packed reconstructed phase referred to as Au(100)-(hex). For carefully annealed crystals a single domain of the reconstructed phase... 

Liitzenkirchen, J. (1997) Ionic strength effects on cation sorption on oxides. Macroscopic observations on their significance in microscopic interpretation. J. Coll. Int. Sci. 195 149-155... 

List three processes that can limit the macroscopically observed rate of a chemical s biodegradation. 

To solve eqn. (359) requires knowledge of the form of the operator. Some useful formulae can be generated without any detailed knowledge of Ly however. The macroscopic observable related to the density is the probability that the pair survives at a time t. From eqn. (123), this is simply... 

This is the macroscopic or phenomenological equation for the macroscopic observable Q. 

Let q represent an observable quantity of a macroscopic system such as the circuit in Figure 1. Assuming that there are no other macroscopic observables, one can derive Eq. (15) from the equation of motion of all particles at the expense of a regrettable, but indispensable, repeated randomness assumption, similar to Boltzmann s Stosszahlansatz. 6 It then also follows that, provided q is an even variable, W has a symmetry property called detailed balancing. 6,7... 

Our conclusion is that Eqs. (30) and (32) are equivalent in those cases in which the formula makes sense at all. For it is clear that it only makes physical sense to speak about the autocorrelation function of a certain quantity, if this quantity can be measured at successive times without disturbing the system, that is, if it is a macroscopically observable quantity. 

All that remains to be done for determining the fluctuation spectrum is to compute the conditional average, Eq. (31). However, this involves the full equations of motion of the many-body system and one can at best hope for a suitable approximate method. There are two such methods available. The first method is the Master Equation approach described above. Relying on the fact that the operator Q represents a macroscopic observable quantity, one assumes that on a coarse-grained level it constitutes a Markov process. The microscopic equations are then only required for computing the transition probabilities per unit time, W(q q ), for example by means of Dirac s time-dependent perturbation theory. Subsequently, one has to solve the Master Equation, as described in Section TV, to find both the spectral density of equilibrium fluctuations and the macroscopic phenomenological equation. 

Alternatively, one may argue that Eq. (32) does not really correspond to the physical situation, because a measurement of Q only determines its value with a certain margin and, therefore, does not force the system in one single eigenfunction Xv This argument leads again to an equivalent expression, provided that Q is a macroscopic observable. 

Here y stands for the values of a set of macroscopic observables Y(q, p). We shall prove (6.1) under the following two conditions. 

For definiteness consider a closed, isolated physical system. If at t = 0 the quantity Y has the precise value y0 the probability density P(y, t) is initially 5(y — y0). It will tend to Pe(y) as t increases. If y0 is macroscopically different from the equilibrium value of Y it means that y0 is far outside the width of Pe(y), because macroscopically observed values are large compared to the equilibrium fluctuations. We also know from experience that the fluctuations remain small during the whole process. That means that P y, t), for each t, is a sharply peaked function of y. The location of this peak is a fairly well-defined number, having an uncertainty of the order of the width of the peak, and is to be identified with the macroscopic value y(t). For definiteness one customarily adopts the more precise definition... 

Next consider a macroscopic observable with its associated operator A. The precise characterization of macroscopic is one of the main tasks of statistical mechanics I merely say that A must have the properties used in the following. Its rate of change must be very small compared to the microscopic motion. That means that... 

When there are more macroscopic observables B,C,... the process can be continued. The end result is a collection of coarse-grained observables //, A, 5, C, ..., which all commute with one another. The Hilbert space H is decomposed in linear subspaces that are common eigenspaces of these observables. We shall call these subspaces phase cells and indicate them with a single label J. They correspond to definite values of the coarse-grained variables, which we shall now denote by Ej, AJy BJy Cj,. These phase cells are the macrostates. 

To justify this claim I show that the expectation value of any macroscopic observable A in a state ip can be expressed in the Pj alone. 

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