Thennal

The argument is sometimes given that equation (Al.6,29) implies that the ratio of spontaneous to stimulated emission goes as the cube of the emitted photon frequency. This argument must be used with some care recall that for light at thennal equilibrium, goes as BP, and hence the rate of stimulated emission has a factor... 

In the previous section we discussed light and matter at equilibrium in a two-level quantum system. For the remainder of this section we will be interested in light and matter which are not at equilibrium. In particular, laser light is completely different from the thennal radiation described at the end of the previous section. In the first place, only one, or a small number of states of the field are occupied, in contrast with the Planck distribution of occupation numbers in thennal radiation. Second, the field state can have a precise phase-, in thennal radiation this phase is assumed to be random. If multiple field states are occupied in a laser they can have a precise phase relationship, something which is achieved in lasers by a teclmique called mode-locking Multiple frequencies with a precise phase relation give rise to laser pulses in time. Nanosecond experiments... 

In real physical systems, the populations and h(0p are not truly constant in time, even in the absence of a field, because of relaxation processes. These relaxation processes lead, at sufficiently long times, to thennal... 

Tr(p ). For an initially thennal state the radius < 1, while for a pure state = 1. The object of cooling is to manipulate the density matrix onto spheres of increasingly larger radius. 

One may note, in concluding this discussion of the second law, that in a sense the zeroth law (thennal equilibrium) presupposes the second. Were there no irreversible processes, no tendency to move toward equilibrium rather than away from it, the concepts of thennal equilibrium and of temperature would be meaningless. 

Here p is the chemical potential just as the pressure is a mechanical potential and the temperature Jis a thennal potential. A difference in chemical potential Ap is a driving force that results in the transfer of molecules tlnough a penneable wall, just as a pressure difference Ap results in a change in position of a movable wall and a temperaPire difference AT produces a transfer of energy in the fonn of heat across a diathennic wall. Similarly equilibrium between two systems separated by a penneable wall must require equality of tire chemical potential on the two sides. For a multicomponent system, the obvious extension of equation (A2.1.22) can be written... 

Thennal equilibrium means free transfer (exchange) of energy in the fonn of heat, mechanical (liydrostatic) equilibrium means free transfer of energy in the fonn of pressure-volume work, and material equilibrium means free transfer... 

As shown in preceding sections, one can have equilibrium of some kinds while inhibiting others. Thus, it is possible to have thennal equilibrium (7 = T ) tln-ough a fixed impemieable diathemiic wall in such a case /i need not equal p, nor need /t equal It is possible to achieve mechanical equilibrium (p =p ) through a movable impemieable adiabatic wall in such a case the transfer of heat or matter is prevented, so T and p. 

Finally, in membrane equilibria, where the wall is pemieable to some species, e.g. the solvent, but not others, thennal equilibrium (7 = jP) and solvent equilibrium = /r )are found, but /iy yaiid p the difference oP-o" is the osmotic nressure. 

Now the system is thennally insulated and the magnetic field is decreased to zero in this adiabatic, essentially reversible (isentropic) process, the temperature necessarily decreases since... 

A thennal contact between two systems can be described in the following way. Let two systems with Hamiltonians Wj and be in contact and interact with Hamiltonian fi. Then the composite system (I -t II)... 

Consider two systems in thennal contact as discussed above. Let the system II (with volume and particles N ) correspond to a reservoir R which is much larger than the system I (with volume F and particles N) of interest. In order to find the canonical ensemble distribution one needs to obtain the probability that the system I is in a specific microstate v which has an energy E, . When the system is in this microstate, the reservoir will have the energy E = Ej.- E due to the constraint that the total energy of the isolated composite system H-II is fixed and denoted by Ej, but the reservoir can be in any one of the R( r possible states that the mechanics within the reservoir dictates. Given that the microstate of the system of... 

This has to be minimized with respect to E or equivalently m/N to obtain the thennal equilibrium result. The value of m/N that corresponds to equilibrium is found to be... 

Now eonsider two systems that are in thennal and diffiisive eontaet, sueh that there ean be sharing of both energy and partieles between the two. Again let I be the system and II be a mueh larger reservoir. Sinee the eomposite system is isolated, one has the situation in whieh the volume of eaeh of the two are fixed at V and V", respeetively, and the total energy and total number of partieles are shared Ej = + /i - -where / = (/, /")... 

Sinee E and A are independent variables, tlieir variations are arbitrary. Henee, for the above equality to be satisfied, eaeh of the two braeketed expressions must vanish when the (E, N) partition is most probable. The vanishing of the eoeffieient of dE implies the equality of temperatures of I and II, eonsistent with thennal equilibrium ... 

The thennal average of a physical quantity X can be computed at any temperature tlirough... 

For free particles, the mean square radius of gyration is essentially the thennal wavelength to within a numerical factor, and for a ID hamionic oscillator in the P ca limit. 

Neglecting derivatives of the third order and higher, we obtain Fourier s law of thennal conduction... 

Here the thennal diflfiisivity Dj. = tc/fp Cp. These two equations couple only to the longitudinal part T = V-iJ of the fluid velocity. From equation (A3.3.17) it is easy to see diat T satisfies... 

In the next section, we consider thennal fluctuations in an inliomogeneous system. 

For the system in thennal equilibrium, one can compute the time-dependent mean square displacement (ICr)... 

In the case of bunolecular gas-phase reactions, encounters are simply collisions between two molecules in the framework of the general collision theory of gas-phase reactions (section A3,4,5,2 ). For a random thennal distribution of positions and momenta in an ideal gas reaction, the probabilistic reasoning has an exact foundation. Flowever, as noted in the case of unimolecular reactions, in principle one must allow for deviations from this ideal behaviour and, thus, from the simple rate law, although in practice such deviations are rarely taken into account theoretically or established empirically. 

This rate coefficient can be averaged in a fifth step over a translational energy distribution P (E ) appropriate for the bulk experiment. In principle, any distribution P (E ) as applicable in tire experiment can be introduced at this point. If this distribution is a thennal Maxwell-Boltzmann distribution one obtains a partially state-selected themial rate coefficient... 

In a final, sixth step one may also average (sum) over a thennal (or other) quantum state distribution I (and F) and obtain the usual thennal rate coefficient... 

Simple collision theories neglect the internal quantum state dependence of a. The rate constant as a function of temperature T results as a thennal average over the Maxwell-Boltzmaim velocity distribution p Ef. 

Here one has the thennal average centre of mass velocity... 

We use the symbol for Boltzmaim s constant to distingiush it from tire rate constant k. Equation (A3.4.85) defines the thennal average reaction cross section (a). 

In principle, the reaction cross section not only depends on the relative translational energy, but also on individual reactant and product quantum states. Its sole dependence on E in the simplified effective expression (equation (A3.4.82)) already implies unspecified averages over reactant states and sums over product states. For practical purposes it is therefore appropriate to consider simplified models for tire energy dependence of the effective reaction cross section. They often fonn the basis for the interpretation of the temperature dependence of thennal cross sections. Figure A3.4.5 illustrates several cross section models. 

Assuming a thennal one-dimensional velocity (Maxwell-Boltzmaim) distribution with average velocity /2k iT/rr/tthe reaction rate is given by the equilibrium flux if (1) the flux from the product side is neglected and (2) the thennal equilibrium is retamed tliroughout the reaction ... 

The Lindemaim mechanism for thennally activated imimolecular reactions is a simple example of a particular class of compound reaction mechanisms. They are mechanisms whose constituent reactions individually follow first-order rate laws [11, 20, 36, 48, 49, 50, 51, 52, 53, 54, 55 and 56] ... 

Flow tube studies of ion-moleeule reaetions date baek to the early 1960s, when the flowing afterglow was adapted to study ion kineties [85]. This represented a major advanee sinee the flowing afterglow is a thennal deviee under most situations and previous instruments were not. Smee that time, many iterations of the ion-moleeule flow tube have been developed and it is an extremely flexible method for studying ion-moleeule reaetions [86, 87, 88, 89, 90, 91 and 92]. 

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