Mass frequency curves

The Johnie Boy sample contained a considerable number of spherical particles in the three smallest size fractions. These particles have a size distribution which differs from that of the irregularly shaped debris particles, being lognormal with a mean below 1/x. They have not been included in the size and mass frequency curves for convenience of calculation since their effect on the curves is small. However, the fraction weights and hence the number of particles in the fractions were corrected for the presence of these spherical particles. 

The size and mass frequency curves of the complete cloud samples, derived from those of the fractions and the fraction weights, are shown in Figures 2 through 6. The sedimentation corrections have been applied. They are quite small for all but the largest particles found. 

Figure 5. Mass frequency curves for three Koon samples...

Figure 6. Mass frequency curves for a Bravo and a Zuni sample and for a...

Fig. 8.6 Converting the line-start attenuation curve into a mass frequency curve...

FIG. 4 Mass frequency curves during aggregation representation of the reduced concentration c(n, t) S2(r)//Vi as a function of the reduced variable n/S(t) for the system (polystyrene latex particles bearing carboxylic acid surface groups) suspended in 0.15 mol/L NaCl at pH 3.0. 

The median particle diameter is the diameter which divides half of the measured quantity (mass, surface area, number), or divides the area under a frequency curve ia half The median for any distribution takes a different value depending on the measured quantity. The median, a useful measure of central tendency, can be easily estimated, especially when the data are presented ia cumulative form. In this case the median is the diameter corresponding to the fiftieth percentile of the distribution. 

One point, which is often disregarded when nsing AFM, is that accurate cantilever stiffness calibration is essential, in order to calculate accurate pull-off forces from measured displacements. Althongh many researchers take values quoted by cantilever manufacturers, which are usually calculated from approximate dimensions, more accurate methods include direct measurement with known springs [31], thermal resonant frequency curve fitting [32], temporary addition of known masses [33], and finite element analysis [34]. 

As AD is made smaller, a histogram becomes a frequency distribution curve (Fig. 4.1) that may be used to characterize droplet size distribution if samples are sufficiently large. In addition to the frequency plot, a cumulative distribution plot has also been used to represent droplet size distribution. In this graphical representation (Fig. 4.2), a percentage of the total number, total surface area, total volume, or total mass of droplets below a given size is plotted vs. droplet size. Therefore, it is essentially a plot of the integral of the frequency curve. 

Average Particle Size A powder has many average sizes hence it is essential that they be well specified. The median is the 50 percent size half the distribution is coarser and half finer. The mode is a high-density region if there is more than one peak in the frequency curve, the distribution is said to be multimodal. The mean is the center of gravity of the distribution. The center of gravity of a mass (volume) distribution is defined by Xyu = 2) XdV/ Z dV where dV = X dN dV is the volume of dN particles of size X This is defined as the volume-moment mean diameter and differs from the mean for a number or surface distribution. 

Alternatively the recorder trace of attenuation against titne can be converted to attenuation against Stokes diameter and then normalized, i.e. the area under the curve is made equal to 100 (Figure 8.6). It is assumed that this curve is the mass frequency distribution (dJT/dc/yy versus d f) unconected for the breakdown in the law of geometric optics. The normalized curve of the product of dfT/dr/y and extiiiction coefilcient is the corrected distribution. This has the effect, in the above example, of reducing the measured median size of titanium dioxide from 0.49 pm to 0.45 pm. 

A more exact procedure is to solve the Bom-von Karman equations of motions 38) to obtain frequencies as a function of the wave vector, q, for each branch or polarization. These will depend upon unit-cell symmetry and periodicity, force constants, and masses. Thus, for a simple Bravais lattice with identical atoms per unit cell, one obtains three phase-frequency relations for the three polarizations. For crystals having two atoms per unit cell, six frequencies are obtained for each value of the phase or wave vector. When these equations have been solved for a sufficient number of wave-vectors, g hco) can, in principle, be obtained by direct count . Thus, a recent calculation (13) of g to) based upon a normal-mode calculation that included intermolecular forces gave an improved fit to the specific heat data of Wunderlich, and showed additional peaks of 140, 90 and 60 cm in the frequency distribution. Even with this procedure, care must be exercised, since it has been shown that significant features of g k(o) may be rormded out. Topological considerations have shown that significant structure in g hco) vs. ho may arise from extreme or saddle points in the phase-frequency curves (38). 

Figure 27.9 (a) Potential energy and ground-state adiabatic potential curves (solid curves) and SCTeffective mass (dashed curve) as a function of reaction coordinate for the intramolecular H-tranferin 1,3-pentadiene. (b) Harmonic frequencies for modes orthogonal to the reaction coordinate. 

Figure 20-1 presents the data from Table 20-1 in both cumulative and frequency format. In order to smooth out experimental errors it is best to generate the frequency curve from the slope of the cumulative curve, to use wide-size intervals or a data-smootning computer program. The advantage of this method of presenting frequency data is that the area under the frequency curve equals 100 percent, hence, it is easy to visually compare similar data. A typical title for such a presentation would be Relative and cumulative mass distributions of quartz powder by pipet sedimentation. 

Water quality monitoring consists of frequent analysis of the main constituents. The required data input consists of (1) mean composition of the influent (2) mean composition of native groundwater in each layer of the target aquifer (3) native geochemistry of each layer of the target aquifer (4) the cumulative frequency curve of detention times in each model layer or flow path as derived from either separately run hydrological model or tracer breakthrough data and (5) specific information derived from the mass balance of the water phase (the reactions that are needed, how O2 and NO3" distribute over the various redox reactions, etc.). 

Follow a procedure similar to that of Example 2.4, in order to elaborate a histogram, a frequency curve, and an undersize cumulative percentage graph, and determine the median size of the powder (note that the particle size distribution obtained will be by number, while in Example 2.4 the particle size distribution obtained was by mass). 

The two signals, inclination and resonance frequency, are acquired simultaneously, but are independent of each other. They yield the stress and the mass (Fig. 9A). We deposited a water drop on a silicon cantilever hydrophobized with a monolayer of hexamethyldisilazane (HMDS). The initial contact angle was 80°. It decreased nearly linearly during evaporation, and was 70° at the end. Tlie initial contact radius was 33 im, and decreased nearly linearly during evaporation. At the end it was below 10 xm. At present, we can record the inclination curve with a temporal resolution of 0.1 ms between data points, and the frequency curve with 5 ms. The mass calculated from the resonance frequency of the cantilever and from video microscope images is similar (Fig. 9B), although the time resolution ( 5 ms) and the sensitivity ( 50 pg) is much... 

The reason that does not change with isotopic substitution is that it refers to the bond length at the minimum of the potential energy curve (see Figure 1.13), and this curve, whether it refers to the harmonic oscillator approximation (Section 1.3.6) or an anharmonic oscillator (to be discussed in Section 6.1.3.2), does not change with isotopic substitution. Flowever, the vibrational energy levels within the potential energy curve, and therefore tq, are affected by isotopic substitution this is illustrated by the mass-dependence of the vibration frequency demonstrated by Equation (1.68). 

Direct Mass Measurement One type of densitometer measures the natural vibration frequency and relates the amplitude to changes in density. The density sensor is a U-shaped tube held stationaiy at its node points and allowed to vibrate at its natural frequency. At the curved end of the U is an electrochemical device that periodically strikes the tube. At the other end of the U, the fluid is continuously passed through the tube. Between strikes, the tube vibrates at its natural frequency. The frequency changes directly in proportion to changes in density. A pickup device at the cui ved end of the U measures the frequency and electronically determines the fluid density. This technique is usefiil because it is not affec ted by the optical properties of the fluid. However, particulate matter in the process fluid can affect the accuracy. 

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