Linear condition

Equation (4-13) is nonlinear in the concentration jc. We, therefore, impose the linearization condition, namely, that the perturbation is sufficiently small that the term in is negligible. The result is... 

We apply the linearization condition by negleeting the terms. The equilibrium conditions are... 

The sensitivity of the equilibrium constant to temperature, therefore, depends upon the enthalpy change AH . This is usually not a serious limitation, because most reaction enthalpies are sufficiently large and because we commonly require that the perturbation be a small one so that the linearization condition is valid. If AH is so small that the T-jump is ineffective, it may be possible to make use of an auxiliary reaction in the following way Suppose the reaction under study is an acid-base reaction with a small AH . We can add a buffer system having a large AH and apply the T-jump to the combined system. The T-jump will alter the Ka of the buffer reaction, resulting in a pH jump. The pH jump then acts as the forcing function on the reaction of interest. 

Figure 10.6. Internal profile (simulated) obtained for the separation under linear conditions.

Time-of-flight (linear) conditions (at reduced scan rates, RP = 15,000 or greater is achievable over limited mlz ranges) 3 1000-5000... 

Figure 6, on the other hand, illustrates the differences between operating an SMB under linear and non-linear conditions. In particular, this figure illustrates the effect of the overall concentration on the region of complete separation re-... 

This system of equalities is equivalent to the following linear conditions on the fe-matrix ... 

Selecting the isotherm model and initial estimations for the values of its parameters. For instance, in the case of the Langmuir isotherm (Equation 10.39), an estimation for the a parameter can be achieved by an injection made under linear conditions (through Equation 10.50). At this point, the least-squares estimation of b—starting from an initial guess— does not present any difficulty. 

The maximum production rate, however, often results in nnacceptable recovery yields. Low recovery yield requires further processing by recycling the mixed fractions. The recovery yield at the maximum production rate strongly depends on the separation factor. In the cases of difficnlt separations, when the separation factor under linear conditions is aronnd or lower than a= 1.1, the recovery yield is not higher than 40%-60%. Even in the case of a=1.8, the recovery yield at the maximum production rate is only about 70%-80%. The situation is still less favorable in displacement chromatography, particularly if the component to be purified is more retained than the limiting impurity. In this case, from one side the impurity, whereas from the other side the displacer, contaminates the product. 

The instrument yields the absorbance by ratioing the transmitted intensities in the presence and absence of sample. Linearity is only observed for weak concentrations (typically below 3 ppm). The methods used, comparable to those used in molecular absorption spectrophotometry, involve classical protocols methods using a calibration curve or standard additions, as long as the range of concentrations stays within the linear conditions of absorbance. 

Figure 2.5 Diffusion from a region of finite concentration into a region of zero concentration. Diffusional response to a step-function excitation under linear conditions.

It is important to recognize the unique relationship that exists between the responses to an impulse and step change in concentration. The derivative of the step response (Eq. 2.14) is identical to the impulse response (Eq. 2.4), and the integral of the impulse response is identical to the step response. This reciprocity is an important property of linear systems in general. The reader should now appreciate that under linear conditions, the time dependence of any concentration profile can be treated by adding the response functions for its component impulses. 

In the same vein as Eq. (6), Rule 2 can be generalized to a linearity condition, namely ... 

To convert an optical signal into a concentration prediction, a linear relationship between the raw signal and the concentration is not necessary. Beer s law for absorption spectroscopy, for instance, models transmitted light as a decaying exponential function of concentration. In the case of Raman spectroscopy of biofluids, however, the measured signal often obeys two convenient linearity conditions without any need for preprocessing. The first condition is that any measured spectrum S of a sample from a certain population (say, of blood samples from a hospital) is a linear superposition of a finite number of pure basis spectra Pi that characterize that population. One of these basis spectra is presumably the pure spectrum Pa of the chemical of interest, A. The second linearity assumption is that the amount of Pa present in the net spectrum S is linearly proportional to the concentration ca of that chemical. In formulaic terms, the assumptions take the mathematical form... 

A time-dependent quantity X t) responding to the sine wave modulation of some low level perturbation dKcos a>t and frequency a>/2n, is generally written in linear conditions as ... 

Presence of an applied electric field The IEF formalism can be extended to take into account the presence of an applied field, [18], In this case, under linear conditions, Equation (2.259) becomes ... 

We examine a solute zone migrating under linear conditions. For the moment we confine our attention to the solute in the mobile phase. If complete equilibrium between phases existed at all points, the mobile-phase solute would form a Gaussian-like concentration profile as indicated by the shaded profile in Figure 10.7. However the actual profile for the mobile phase, shown by the dashed line, is shifted ahead of the shaded (equilibrium) profile due to solute migration. The basis of the profile shift is explained as follows. 

Isocratic elution under linear conditions at various mobile phase salt concentrations can be employed to determine v and K26 by the following expression ... 

Equation (5) is valid under only linear conditions it is mandatory to check that the test has been performed under linear condition. To accomplish this, it is best to perform two injections of different concentration the obtained retention time should not be affected by the concentration. 

For the two-dimensional case considered in the preceding, one linear condition defines a line which divides the plane into two half-planes. For a three-dimensional case, one linear condition defines a set plane which divides the volume into two half-volumes. Similarly, for an n-dimensional case, one linear condition defines a hyperplane which divides the space into two halfspaces. 

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