Displayed Equations

UPD represents a potential dependent adsorption with great sensitivity and selectivity towards the nature of the metal surface and its termination. The complexity of the voltammetry features arises from the existence of one or more UPD ML superstructures^22,23 and/or one or more UPD MLs formed.^24,25

$$\Delta E = \Delta E^{0}_{\theta \to 0} - (RT / mF) \cdot [\ln(\theta/(1-\theta)) + f\theta + g\theta^{3/2}]$$ (1)

Here the $$\Delta E^{0}_{\theta \to 0}$$ term represents the underpotential of the most positive stripping peak of the UPD adlayer where its coverage approaches zero. The term f is the Temkin parameter describing the attractive UPD ML–substrate interactions, as given in Eq. (1).

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The UPD ML coverage can be controlled effectively down to a fraction of a monolayer by proper choice of an underpotential. The amount of deposited P is controlled by the reaction stoichiometry and the coverage of the UPD ML of metal M. The most general formulation of the SLRR reaction is given by the following equation:¹⁶

$$(\theta_{M} \cdot \rho_{M}^{\mathrm{UPD}}) \cdot M^{0}{\mathrm{UPD}}/S(h,k,l) + (\theta{M} \cdot \rho_{M}^{\mathrm{UPD}}) \cdot (m/p)P^{p+}{\mathrm{solv}} \Rightarrow (\theta{M} \cdot \rho_{M}^{\mathrm{UPD}}) \cdot M^{m+}{\mathrm{solv}} + (\theta{M} \cdot \rho_{M}^{\mathrm{UPD}}) \cdot (m/p)P^{0}_{s}/S(h,k,l)$$ (2)

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Velocity Spectra. Another method of presenting the same data is of interest here. This method involves the concept of a spectral velocity for the elasto-plastic system. The definition which is most convenient is to define the spectral velocity V by the relationship

$$V = pu_{y}$$ (2)

where p denotes the undamped circular natural frequency of the system. Then the quantity

$$\frac{1}{2} mV^{2}$$

represents the maximum recoverable strain energy of the system. It should be noted that this definition of the spectral velocity for the elasto-plastic system is consistent with that used for elastic systems, since in an elastic system the yield deformation may be taken equal to the maximum elastic deformation.

Relation Between Input Motion and Response Values. It is convenient to plot velocity spectra on a four-way logarithmic grid, so that displacement, velocity, acceleration and period may all be read from a single chart.

Calculation:

$$\frac{\left(\textbf{Inventory and Related Property (current year)} + \textbf{General Property, Plant and Equipment (current year)}\right) - \left(\textbf{Inventory and Related Property (prior year)} + \textbf{General Property, Plant and Equipment (prior year)}\right)}{\textbf{Inventory and Related Property (prior year)} + \textbf{General Property, Plant and Equipment (prior year)}}$$

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