Steady-state terms

Stern-Volmer equation may be introduced by the transient component of dynamic quenching. If the quencher molecules are present near the fluorescent molecule at the moment of excitation, the initial quenching before a steady state is achieved, leads to the non-steady state term in the quenching expression. A sphere of transient quenching of volume r, may be defined as... 

The problem (91) is splitted into a steady state term s [R] and transient term 5 [R, t]. Since both boundary conditions (92 b, 92 c) are of the second kind, the average temperature has to be included into the splitting formula [20] ... 

Upon cancellation of the steady-state terms and division by the term... 

Because of the rather short half life of Rn, there can he significant changes with time in its activity, compared with the gas exchange time scale, and therefore the non-steady-state term (the last one in Eq. (10.26)) must be evaluated to determine accurate values of Gro. This has been done both by measuring a series of radon profiles at a single location and by averaging many individual profiles in different ocean basins using steady-state assumptions. 

The numerator of (5.2.26) is the thickness of the diffusion layer thus the importance of the steady-state term, which manifests spherical diffusion, depends mainly on the ratio of that thickness to the radius of the electrode. When the diffusion layer grows to a thickness that is an appreciable fraction of ro, it is no longer appropriate to use equations for linear diffusion, and one can expect the steady-state term to contribute significantly to the measured current. 

By analogy to the rigorous result for the spherical system, one can estimate the current at the disk as the simple linear combination of the Cottrell and steady-state terms ... 

This expression is the sum of a transient term and a steady-state term, where r is the radius of the sphere. At short times after the application of the potential step, the transient term dominates over the steady-state term, and the electrode is analogous to a plane, as the depletion layer is thin compared with the disc radius, and the current varies with time according to the Cottrell equation. At long times, the transient current will decrease to a negligible value, the depletion layer is comparable to the electrode radius, spherical diffusion controls the transport of reactant, and the current density reaches a steady-state value. At times intermediate to the limiting conditions of Cottrell behaviour or diffusion control, both transient and steady-state terms need to be considered and thus the full expression must be used. However, many experiments involving microelectrodes are designed such that one of the simpler current expressions is valid. 

The terms and symbols used in this equation have the following meaning. Cj is the molar concentration of species j (kmol/m fluid), so that dCj/dt is the non-steady-state term expressing accumulation or depletion. V is the nabla or del operator. In a rectangular coordinate system, x, y, z with unit vectors Sy, and 8, the gradient of a scalar function / is represented by V/ and the divergence... 

It can be seen that, unlike the case of a planar electrode, the current for a spherical electrode contains two terms, a time dependent and a steady state term. It should be noted... 

Equation (11.2.7) is known as the Cottrell equation (4) and a typical current transient is shown in Figure 11.1b. For spherical electrodes, the current transient (11.2.14) contains a steady-state term accounting for the effect of radial diffusion. This term wiU dominate at larger t when the Cottrell term tends to zero and the current reaches a steady-state value. The time required to establish the steady state is a function of the electrode radius. The smaller the electrode, the sooner the radial diffusion term becomes dominant (see Section 11.2.4 in this Chapter and Section 2.4 in Chapter 2, and Chapters 6 and 19 on Microelectrodes). To test whether the current response is controlled by diffusion, one plots i vs. For both geometries, the graph should be linear but its intercept will be zero for a planar electrode and equal to the steady-state current for a spherical electrode. For planar electrodes, the current is therefore expected to decay to zero at long times. In practice, this cannot be observed because of the onset of natural convections after 30 s. 

Dynamic models which are used for obtaining more stable process control. These will contain non-steady state terms, and should be very reliable. They are usually quite complicate. 

Expanding the sum to first order in h and removing the steady state terms (that cancel out) we have... 

The relation between level and inflow is the sum of two out-of-phase components. The derivative term lends the steady-state term by 90°, just as integrating produced a 90° phase lag. The gain of the derivative term to a signal of period To is exactly the inverse of the gain of an integrator ... 

---