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A typical titration curve of a diprotic acid, oxalic acid, titrated with a strong base, sodium hydroxide.Both equivalence points are visible. Titrations are often recorded on graphs called titration curves, which generally contain the volume of the titrant as the independent variable and the pH of the solution as the dependent variable (because it changes depending on the composition of the ...
Inverted logistic S-curve to model the relation between wheat yield and soil salinity. Many natural processes, such as those of complex system learning curves, exhibit a progression from small beginnings that accelerates and approaches a climax over time. When a specific mathematical model is lacking, a sigmoid function is often used.
These approximations are only valid near, but not at, the end point, and so the method differs from end point estimations by way of first- and second-derivative plots, which require data at the end point. Gran plots were originally devised for graphical determinations in pre-computer times, wherein an x-y plot on paper would be manually ...
The first potentiometric titration was carried out in 1893 by Robert Behrend at Ostwald's Institute in Leipzig. He titrated mercurous solution with potassium chloride, potassium bromide, and potassium iodide. He used a mercury electrode along with a mercury/mercurous nitrate reference electrode. He found that in a cell composed of mercurous ...
Plot of y = x 3 with an inflection point at (0,0), which is also a stationary point. The roots, stationary points, inflection point and concavity of a cubic polynomial x 3 − 6x 2 + 9x − 4 (solid black curve) and its first (dashed red) and second (dotted orange) derivatives.
This plot, which is often smoothed, is referred to as a TGA curve. The first derivative of the TGA curve (the DTG curve) may be plotted to determine inflection points useful for in-depth interpretations as well as differential thermal analysis. A TGA can be used for materials characterization through analysis of characteristic decomposition ...
The smoothed curve (red line) and 1st derivative (green) were calculated with 7-point cubic Savitzky–Golay filters. Linear interpolation of the first derivative values at positions either side of the zero-crossing gives the position of the peak maximum. 3rd derivatives can also be used for this purpose. Location of an end-point in a titration ...
The derivative of () at the point = is the slope of the tangent to (, ()). [3] In order to gain an intuition for this, one must first be familiar with finding the slope of a linear equation, written in the form = +. The slope of an equation is its steepness.