Bell shaped curve

A frequently encountered pH-rate profile exhibits a bell-like shape or hump, with two inflection points. This graphical feature is essentially two sigmoid curves back-to-back. By analogy with the earlier analysis of the sigmoid pH-rate curve, where the shape was ascribed to an acid-base equilibrium of the substrate, we find that the bell-shaped curve can usually be accounted for in terms of two acid-base dissociations of the substrate. The substrate can be regarded, for this analysis, as a dibasic acid H2S, where the charge type is irrelevant we take the neutral molecule as an example. The acid dissociation constants are 

The fractions of solute in each form are given by = [H2S]/5 F s = [HS ]/5 and Fs = [S ]/S where 5, the total molar concentration of substrate is [H2S] + [HS ] + [S ]. Combining these leads to expressions for the fractions of solute as functions of the dissociation constants and the hydronium ion concentration. 

The intersection of the curves Fhjs and Fhs occurs at pH = pA i, as can be found by setting Eqs. (6-73) and (6-74) equal. Also, when Fhs = Fs, pH = pA 2. These are general relationships. We note, at this point, that the function Fhs has the previously mentioned bell shape, and it is this function that will be of later kinetic interest. 

In order to find the inflection points in a plot of Fhs against pH, the second derivative d Fns/dpH is set to zero. The result is a quartic in [H ], which is not reproduced. When Ki is much larger than K2 (by at least three orders of magnitude), the location of the inflection points becomes particularly simple. Then at low pH. in the region of the left inflection point, [H ] is much greater than the quantity (K,K2) and 

Similarly, when (A iA 2) [H ], the location of the right inflection point is given by 

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Normal Distribution of Observations Many types of data follow what is called the gaussian, or bell-shaped, curve this is especially true of averages. Basically, the gaussian curve is a purely mathematical function which has very specif properties. However, owing to some mathematically intractable aspects primary use of the function is restricted to tabulated values. 

Procedures for curve fitting by polynomials are widely available. Bell-shaped curves, however, are fitted better and with fewer constants by ratios of polynomials. For figuring chemical conversions, the... 

Fuzzy sets represented by symmetrical triangles are commonly used because they give good results and computation is simple. Other arrangements include non-symmetrical triangles, trapezoids, Gaussian and bell shaped curves. 

Equation (6-78) has the same form as Eq. (6-74) for Fhs, so the simple scheme embodied in Eq. (6-77) can account for a bell-shaped curve when k is plotted against pH. 

Kinetic schemes other than that embodied in Eq. (6-77) can give rise to a bellshaped curve. As in Eq. (6-77), however most of these involve two ionizations. Thus Scheme V, where HS is a monoprotic acid and B is a base (or the kinetic equivalent of S + BH ) yields a bell-shaped curve. 

A less obvious scheme that can lead to a bell-shaped curve has been recognized by Zemer and Bender. The substrate is a weak acid or base, but possesses only one ionizable group, and no other ionizable reactant is involved (other than water). The second inflection in the curve is ascribed to an ionizable group created in an intermediate. An example has been discovered in the hydrolysis of o-carboxy-phthalimide ... 

In tlie case of a random sample of observations on a continuous random variable assumed to have a so-called nonnal pdf, tlie graph of which is a bell-shaped curve, tlie following statements give a more precise interpretation of tlie sample standard deviation S as a measure of spread or dispersion. 

The bell-shaped curve in part (c) of the figure combines both of these behaviors, so that activity first increases, then decreases, as pH is increased. This is consistent with the involvement of two ionizable groups—one with a low pA that acts as a base above its pK, and a second group with a higher pK that acts as an acid below its pK. ... 

Kinetic studies with pepsin have produced bell-shaped curves for a variety of substrate peptides see below, (a). 

If a large number of replicate readings, at least 50, are taken of a continuous variable, e.g. a titrimetric end-point, the results attained will usually be distributed about the mean in a roughly symmetrical manner. The mathematical model that best satisfies such a distribution of random errors is called the Normal (or Gaussian) distribution. This is a bell-shaped curve that is symmetrical about the mean as shown in Fig. 4.1. 

However, dendrimeric and hyperbranched polyesters are more soluble than the linear ones (respectively 1.05, 0.70, and 0.02 g/mL in acetone). The solution behavior has been investigated, and in the case of aromatic hyperbranched polyesters,84 a very low a-value of the Mark-Houvink-Sakurada equation 0/ = KMa) and low intrinsic viscosity were observed. Frechet presented a description of the intrinsic viscosity as a function of the molar mass85 for different architectures The hyperbranched macromolecules show a nonlinear variation for low molecular weight and a bell-shaped curve is observed in the case of dendrimers (Fig. 5.18). 

It has been shown that the rate of formation of oximes is at a maximum at a pH that depends on the substrate but is usually 4, and that the rate decreases as the pH is either raised or lowered from this point. We have previously seen (p. 425) that bell-shaped curves like this are often caused by changes in the rate-determining step. In this case, at low pH values step 2 is rapid (because it is acid catalyzed), and step 1... 

Fig. 4. EPR redox titration of ZJ. vulgaris Fepr protein at pH 7.5 of S = J components with dithionite and ferricyanide in the presence of mediators, [from (ZZ)]. ( , ) The Fepr protein-fingerprint signal (the 3+ state) monitored at g = 1.825 (O, ) signal with aU < 2 (the 5+ state) monitored atg = 1.898 ( , ) Titration in two directions starting from the isolated protein, which corresponds approximately to the top of the bell-shaped curve. ( , O) A titration starting from the fully preoxidized state. EPR conditions microwave frequency, 9.33 GHz microwave power, 13 mW modulation amplitude, 0.63 mT temperature, 15 K.

Several findings support this model. For instance, an early report suggested that there is a positive correlation between the density of (postsynaptic) jS-adrenoceptors in rat cortex and behavioural resistance to a mild environmental stress (novelty and frustration) but a negative correlation between these parameters when the stress is intensified (Stanford and Salmon 1992). More recently, it has been proposed that the phasic response of neurons in the locus coeruleus (which governs attentiveness ) depends on their tonic activity (which determines arousal). Evidence suggests that the relationship between these two parameters is described by a bell-shaped curve and so an optimal phasic response is manifest only at intermediate levels of tonic activity (Rajkowski et al. 1998). 

What we have is the familiar "Bell-Shaped" curve. This distribution has been variously called ... 

Fig. 3.42 represents the symmetric bell shape curve of 7, i.e., the genuine fundamental harmonic ac polarogram, which means the curve of only 7F discriminated for 7C, e.g., by means of phase-selective ac polarography. The term "fundamental is related to the character of the polarographic cell as a non-linearized network whose response is not purely sinusoidal but consists of the sum of a series of sinusoidal signals at first harmonic (o>) response, besides that of the second harmonic (2a>), the third harmonic (3a>), etc. 

Fig. 3.47 is comparable to Fig. 3.41 for sinusoidal ac polarography if the tilted shape provides a net compensation of the charging current one obtains a symmetric bell-shaped curve of I in the square-wave polarogram, similar to that depicted in Fig. 3.42. In fact, virtually all of the statements made before on the sinusoidal technique are valid for the square-wave mode except for the rigid shape of its wave this conclusion is according to expectation, especially as Fourier analysis reveals the square wave to be a summation of a series of only...

Some compounds exhibit pH behavior in which a bell-shaped curve is obtained with maximum instability at the peak [107]. The peak corresponds to the intersection of two sigmoidal curves that are mirror images. The two inflection points imply two acid and base dissociations responsible for the reaction. For a dibasic acid (H2A) for which the monobasic species (HA-) is most reactive, the rate will rise with pH as [HA-] increases. The maximum rate occurs at pH = (pA) + pK2)/2 (the mean of the two acid dissociation constants). Where an acid and base react, the two inflections arise from the two different molecules. The hydrolysis of penicillin G catalyzed by 3,6-bis(di-methylaminomethyl)catechol [108], is a typical example. For a systematic interpretation of pH-degradation profiles, see the review papers by van der Houwen et al. [109] and Connors [110]. 

Bolus administration is usually characterized by a normalized injection function that effectively inputs the dose in a bell-shaped curve manner. Since bolus administration is not instantaneous, this approach is considered more realistic. I(t) for bolus administration is... 

The effect of non-participating ligands on the copper catalyzed autoxidation of cysteine was studied in the presence of glycylglycine-phosphate and catecholamines, (2-R-)H2C, (epinephrine, R = CH(OH)-CH2-NHCH3 norepinephrine, R = CH(OH)-CH2-NH2 dopamine, R = CH2-CH2-NH2 dopa, R = CH2-CH(COOH)-NH2) by Hanaki and co-workers (68,69). Typically, these reactions followed Michaelis-Menten kinetics and the autoxidation rate displayed a bell-shaped curve as a function of pH. The catecholamines had no kinetic effects under anaerobic conditions, but catalyzed the autoxidation of cysteine in the following order of efficiency epinephrine = norepinephrine > dopamine > dopa. The concentration and pH dependencies of the reaction rate were interpreted by assuming that the redox active species is the [L Cun(RS-)] ternary complex which is formed in a very fast reaction between CunL and cysteine. Thus, the autoxidation occurs at maximum rate when the conditions are optimal for the formation of this species. At relatively low pH, the ternary complex does not form in sufficient concentration. 

Procedures for curve fitting by polynomials are widely available. Bell shaped curves usually are fitted better and with fewer constants by ratios of polynomials. Problem P5.02.02 compares a Gamma fit with those of other equations, of which a log normal plot is the best. In figuring chemical conversion, fit of the data at low values of Ett) need not be highly accurate since those regions do not affect the overall result very much. 

The other plots are made with the software TABLECURVE. The special function F2 used there is a log-normal relation and F3 is a sine-wave function. Usually a ratio of low degree polynomials also provides a good fit to bell-shaped curves here five constants are needed. The Gamma distribution needs only one constant, but the fit is not as good as some of the other curves. The peak, especially, is missed. 

Group 1 can be seen to approximate a normal distribution (bell-shaped curve) we can proceed to perform the appropriate parametric tests with such data. But group 2 clearly does not appear to be normally distributed. In this case, the appropriate nonparamctric technique must be used. 

For an infinite data set (in which the symbols ft. and o as defined in Section 1.7.2 apply), a plot of frequency of occurrence vs. the measurement value yields a smooth bell-shaped curve. It is referred to as bell-shaped because there is equal drop-off on both sides of a peak value, resulting in a shape that resembles a bell. The peak value corresponds to /l, the population mean. This curve is called the normal distribution curve because it represents a normal distribution of values for any infinitely repeated measurement. This curve is shown in Figure 1.3. 

Now these expressions describe the frequency dependence of the stress with respect to the strain. It is normal to represent these as two moduli which determine the component of stress in phase with the applied strain (storage modulus) and the component out of phase by 90°. The functions have some identifying features. As the frequency increases, the loss modulus at first increases from zero to G/2 and then reduces to zero giving the bell-shaped curve in Figure 4.7. The maximum in the curve and crossover point between storage and loss moduli occurs at im. 

The range of frequencies used to calculate the moduli are typically available on many instruments. The important feature that these calculations illustrate is that as the breadth of the distributions is increased the original sigmoidal and bell shaped curves of the Maxwell model are progressively lost. A distribution of Maxwell models can produce a wide range of experimental behaviour depending upon the relaxation times and the elastic responses present in the material. The relaxation spectrum can be composed of more than one peak or could contain a simple Maxwell process represented by a spike in the distribution. This results in complex forms for all the elastic moduli. 

In a situation whereby a large number of replicate readings, not less than 5 0, are observed of a titrimetric equivalence point (continuous variable), the results thus generated shall normally be distributed around the mean in a more or less symmetrical fashion. Thus, the mathematical model which not only fits into but also satisfies such a distribution of random errors is termed as the Normal or Gaussian distribution curve. It is a bell-shaped curve which is noted to be symmetrical about the mean as depicted in Figure 3.2. 

Many rate constants in aqueous solutions are pH or pD sensitive. In particular, enzyme catalyzed reactions often show maxima in plots of pH(pD) vs. rate. The example in Fig. 11.5 is constructed for a reaction with a true isotope effect, kH/kD = 2, and with maxima in the pH(pD)/rate dependences as shown by the bell shaped curves. These behaviors are typical for enzyme catalyzed reactions. When the isotope effect is obtained (incorrectly) by comparing rates at equal pH and pD, the values plotted along the steep dashed curve result. If, however, the rate constants at corresponding pH and pD (pD = pH + 0.5) are employed, a constant and correct value is obtained, kH/kD = 2. Thus for accurate measurements of the isotope effects one must control pH and pD at appropriate values (pD = pH + 0.5 in our example) using a series of buffers. In proton inventory experiments (see below) buffers should be employed to insure equivalent acidities across the entire range of solvent isotope concentration (0 < xD < 1), xD is the atom fraction of deuterium [D]/([H] + [D]). 

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