The Work Term

So far we have said little about the work term in Eq. 4-18. Suppose our system is the 1-kg steel ball described previously. The system is practically rigid, and the work done on it generally consists of something, e.g., a hand, pushing it. This work is shown by Eq. 4.1. j 

Now consider the system shown in Fig. 4.1. Let us assume that the material in the tank is something easily compressed, such as air or steam. In this case we can do work on the system by moving the volume-changing piston the magnitude of this work is shown by Eq. 4.1. However, the force required to move the piston is equal to the piston s cross-sectional area times the pressure in the tank. Further, the product of the piston s cross-sectional area and the distance traveled is equal to the change in vjolume of the tank so 

When we move the piston inward, dV is negative so positive. When 

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The nonconvective energy flux across the boundary is composed of two terms a heat flux and a work term. The work term in turn is composed of two terms useful work deflvered outside the fluid, and work done by the fluid inside the control volume B on fluid outside the control volume B, the so-called flow work. The latter may be evaluated by imagining a differential surface moving with the fluid which at time 2ero coincides with a differential element of the surface, S. During the time dt the differential surface sweeps out a volume V cosdSdt and does work on the fluid outside at a rate of PV cos dS. The total flow work done on the fluid outside B by the fluid inside B is... 

The work term IF is restricted to the mechanical work deflvered to the outside via normal and shear forces acting on the boundary. Electrochemical work, ie, by electrolysis of the fluid, is excluded. Evaluation of the integral requires knowledge of the equation of state and the thermodynamic history of the fluid... 

If the work term W( is expanded to breakdown shaft work done to or from the system and the work done by the system, then... 

Work interchange between a system and its surroundings can take on any of a variety of forms including mechanical shaft work, electrical work, magnetic work, surface tension, etc. For many applications, the only work involved is that of compression or expansion against the surroundings, in which case the work term in Equation 2-102 becomes... 

Substitution into Eq. (10-83), still ignoring the work terms, gives... 

When R is CH3 the process is called methyl transfer. For such reactions, the work terms and are assumed to be very small compared to AG° and can be neglected, so that the Marcus equation simplifies to... 

Here, E is the total energy of the system, Ej is the energy per mole of component i, F is the volumetric flow rate, C is the molar concentration and S is the number of components (reactants and inerts). The work term can be separated into flow work and other work Ws, according to... 

Gibbs energy of electron transfer (AGe ) and the work terms for precursor formation (Wp) and successor dissociation (ws),... 

In subsequent works, Marcus developed his theory further in a series of papers providing expressions for the work terms, the reorganization energy and the macroscopic ET rate constants [3 6]. Assuming a sharp liquid-liquid boundary, the solution of the mean molar volume of reactants yields an expression for of the form... 

In order to fully evaluate Eq. (17), expressions involving the outer-sphere reorganization term A,q, as well as the work terms, were also established [5],... 

Electrical, and the mechanical forms of energy, are included in the work term in an energy balance. Electrical energy will only be significant in energy balances on electrochemical processes. 

For many processes the work term will be zero, or negligibly small, and equation 3.7 reduces to the simple heat balance equation ... 

The work term in an energy balance is unlikely to be significant unless a gas is expanded or compressed as part of the process. To compute the pressure work term ... 

In the context of the Marcus formulation, the lowering of the activation barrier in an inner-sphere process could arise from the reduction of the work term wp as a result of the strong interaction in the ionic products, e.g., [RitSn+ IrCU3 ] and [RitSn+TCNE ]. The electrostatic potential in such an ion pair is attractive and may cause the tetraalkyltin to achieve a quasi five-coordinate configuration in the precursor complex, reminiscent of a variety of trigonal bipyramidal structures already well-known for tin(IV) derivatives, i.e.,... 

The extent to which steric effects adversely affect the attainment of such intimate ion-pair structures would be reflected in an increase in the work term and concomitant diminution of the inner-sphere rate. This qualitative conclusion accords with the reactivity trend in Figure 16. However, Marcus theory does not provide a quantitative basis for evaluating the variation in the work term of such ion pairs. To obtain the latter we now turn to the Mulliken theory of charge transfer in which the energetics of ion-pair formation evolve directly, and provide quantitative informa-... 

Mechanistic Formulation of Electron Transfer. The Importance of the Work Term. Accordingly, the electron transfer mechanism can be considered in the light of the standard potentials E° for each redox couple, i.e., E x for the oxidation of the donor (D D+ + e ) and E ed for the reduction of the acceptor (A + e" A"). Thus the general reaction scheme for an irreversible process is represented by (20) ... 

AG° corresponds to the standard free energy change of the redox process D + A D+ + A , and the work terms wp and wr represent the energy required to bring together the products and reactants,... 

Evaluation of the Work Term from Charge Transfer Spectral Data. The intermolecular interaction leading to the precursor complex in Scheme IV is reminiscent of the electron donor-acceptor or EDA complexes formed between electron donors and acceptors (21). The latter is characterized by the presence of a new absorption band in the electronic spectrum. According to the Mulliken charge transfer (CT) theory for weak EDA complexes, the absorption maximum hv rp corresponds to the vertical (Franck-Condon) transition from the neutral ground state to the polar excited state (22). 

The charge transfer Scheme V is akin to the adiabatic electron transfer cycle in Scheme IV. In this case the work term Wp required to bring the products D+ and A together to the mean separation rp in the CT excited state is given by ... 

We associate such variations in the work term wp with changes in the mean separation r in the EDA complexes. Qualitatively, such changes may be viewed as steric effects which hinder the close approach of the acceptor and the donor. For example, the constancy of wp for the substituted-anthracene donors accords with the minor steric perturbation a substituent is expected to exert... 

Figure 18, Relationship between the activation free energy AG including the work term Awp and the driving force for electron transfer AG° to TCNE (left) and IrClt (right) following Equation 30,...

Figure 19, Variation of the work term Awp evaluated for various charge-transfer...

Figure 19 (bottom) shows that the cleavage of alkylmetals by mercuric chloride follows an apparent negative Br nsted slope a (left figure). However, when the work term Wp is included with EqX, the curved free energy relationship with negative slopes is transformed into the linear correlation shown on the right. 

Figure 20, Unified free energy relationship for ion-pair formation. Key left, free energy relationships between the rates of reaction (log kobJ and the oxidation potential Eox° of the donor and right, after inclusion of the work term following Equation 31, (Keys to symbols are located at the far left.)...

By employing the variation of the work term Awp in eq 31, we are actually focussing in large part on the electrostatic interaction in the ion pair, since the other structural factors drop out by cancellation in the comparative procedure used in eq 28. Thus the contribution to Awp consists largely of a coulombic potential (i.e., -e2/rpA) the m an separation r in the... 

Plan. Evaluate Angas gaseous products- gaseous reactants The sign of the work term is opposite that of Awgas since w = -PAY = -AngasRT at constant P and T. 

The discussion above has been directed principally to thermally induced spin transitions, but other physical perturbations can either initiate or modify a spin transition. The effect of a change in the external pressure has been widely studied and is treated in detail in Chap. 22. The normal effect of an increase in pressure is to stabilise the low spin state, i.e. to increase the transition temperature. This can be understood in terms of the volume reduction which accompanies the high spin—dow spin change, arising primarily from the shorter metal-donor atom distances in the low spin form. An increase in pressure effectively increases the separation between the zero point energies of the low spin and high spin states by the work term PAV. The application of pressure can in fact induce a transition in a HS system for which a thermal transition does not occur. This applies in complex systems, e.g. in [Fe (phen)2Cl2] [158] and also in the simple binary compounds iron(II) oxide [159] and iron(II) sulfide [160]. Transitions such as those in these simple binary systems can be expected in minerals of iron and other first transition series metals in the deep mantle and core of the earth. 

We consider the work term to be composed of that pertaining to forces associated with turning paddlewheel shafts and shear stress (Ws), and work associated with normal pressure forces (Wp). For most applications in fire, Ws will not apply and therefore we will ignore it here. For the piston of face area, A, the work due to pressure (p) on the surroundings, moving at a distance (x2 x ), is... 

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