Functions Passing Through the Origin

If the linear function through the origin y = mx were obeyed with perfect precision by an experimental data set [xi, yi], we would have 

This is never the case for a real data set, which displays a deviation di for each data point owing to experimental error. For the real case. 

If the experimental error is random, the method of least squares applies to analysis of the set. Minimize the sum of squares of the deviations by differentiating with respect to m. 

Using a hand calculator, find the slope of the linear regression line that passes through the origin and best satisfies the points 

Scanning the data set, it is evident that the slope should be slightly less than 1.0. 

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We shall begin with the simplest case of a linear function passing through the origin to intr oduce the method and set up the ground rules. The more complicated... 

Typically, this equation applies to low concentrations of the adsorbate, where k may be predicted or derived from low concentration data. If k is to be determined by a least squares fit of q versus p data, and also ensure the function passes through the origin (p = q = 0), then k can be determined as... 

While the above examination considered the implications of the linear failure function of a simple powder, it is abundantly clear that the same arguments apply to any non-linear failure function passing through the origin. Hence, any dry hard-particle powder will flow out of a small outlet subject to the intervening very low-stress mechanisms. 

Expand the three detemiinants D, Dt, and for the least squares fit to a linear function not passing through the origin so as to obtain explicit algebraic expressions for b and m, the y-intercept and the slope of the best straight line representing the experimental data. 

The intensity of the emitted fluorescence In is, therefore, directly proportional to the amount of substance applied a This relationship is much simpler than the Kubelka-Munk function and always leads to a linear calibration curve passing through the origin If this is not true then interference is occurring [5]... 

The order in [B] is determined by varying [B]o- It is best to use a wide concentration range, to be certain of the correct functional form. The case v = fc[A][B] is easily recognized, since the plot of k versus [B] is a straight line that passes through the origin. 

Figure El 3.4b shows two different forms for the annualized capital cost of the compressors. Line A indicates the cost is a linear function of horsepower [ 70.00/(hp)(year)] with the line passing through the origin, whereas line B assumes a linear function of horsepower with a fixed initial capital outlay [ 70.00/(hp)(year) + 10,000] to take into account installation costs, foundation, and so on. For fine A, the objective function in dollars per year for the example problem is...

The work requirement for any vapor-compression process is a linear function of the temperature difference across the evaporating surface, at least up to a At of 20° F. This straight lines does not pass through the origin because of the effect of BPE. 

If plots in Fig. 10.3 are not straight lines the reaction is not of first order. For second order reaction, instead of- tog (1 - a ), a different function of a needs to be plotted along the Y-axis. This function is 1/(1 - a ). This time the plots would be straight lines, but not passing through the origin. 

For solutes the standard state and the activity usually must be defined in terms of behavior under conditions of infinite dilution, where by definition the activity of a solute is set equal to its concentration. Thus at infinite dilution the ratio of activity to concentration (in whatever units) is unity, and y, = 1. When the value of some physical property of a solution is plotted as a function of concentration, a curve like those in Figure 2-2 is obtained. If the asymptote passing through the origin on the concentration scale is extrapolated to higher concentrations, we obtain the standard state of unit activity for the property in question. This hypothetical solution, labeled S, of unit concentration exhibits the same type of behavior as the infinitely dilute solution. The extent to which the real value of the physical property measured differs from the hypothetical value at a specific concentration is expressed by the activity coefficient, a coefficient that is simply the ratio between two measurable quantities. In Figure 2-2 the activity coefficient yj is the ratio BC/AC and is defined by... 

The interest of using reciprocal values is that the experimental plot of as a function of must be a straight line parallel to e line corresponding to Levich s result for the mass-transfer-limited case, which passes through the origin. The ordinate value of the intercept of this straight line at = q is 4 -f-... 

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