Naperian logarithms

This number forms the basis of the natural (or Naperian) logarithmic scale. For each positive number x, there is unique number y, such x = e, and y is called the natural logarithm of x. We can therefore write... 

If the solid phase at equilibrium comprises only the two species, SR C(s) and SR(s), then chemical thermodynamics can be applied to derive a general relationship between either fSR-c or fSR and the conditional equilibrium constant, K. .5 To simplify notation, let 1 = SR(s) and 2 s SR C(s), such that SR fsR C and Kads = K12. Because K12 depends only on temperature and pressure, infinitesimal changes in the mole fractions, x, (= xSR) and x2 (= xSR.c), that are isothermal and isobaric are constrained by a condition, analogous to Eq. 3.33a, but applied to Eq. 4.8 after taking the base e (naperian) logarithm of both sides ... 

If further data manipulation is envisaged Naperian logarithms are preferred (i.e. to base e rather than base 10) since data in this form lend themselves to computer manipulation, particularly if the notation popular in Germany is used [95]. If a log to the base 10 abscissa is used the area under the frequency curve decreases from 100 to 43.4 units in this case [see equations (2.24)]. 

The base of natural logarithms, < , is named after Leonhard Euler, 1707-1783, a great Swiss mathematician. Naperian logarithms are named after John Napier, 1550-1617, a Scottish landowner, theologian, and mathematician, who was one of the inventors of logarithms. 

On the right-hand graph, the salinity scale is taken as the Naperian logarithm of salinity. It is seen that the centerline of the three-phase region becomes a straight line so that the expression In 5 vs. ACN is linear, with a slope indicated as K. All the points along this line correspond to an optimum formulation where / = 1. 

TABLE III. Naperian logarithm of the solubility, ci j iCU),s TiCLj vs absolute temperature T and solubilities at a fixed temperature in various electrolytes. C(y,(7/ ) in mole dm . ... 

The optimum thickness for measuring the spectrum of any solute depends on the absorptivity of the solvent It is well known from many textbooks on instrumental analysis that the optimum pathlength of a cell should be such that the transmittance of the solvent should be about He, where e is the base of Naperian logarithms. Thus, the optimum absorbance, Aopt, of the solvent should be logioc, or 0.4... 

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