Logarithms to Base

The other logarithms commonly used in chemistry are those to base 10. Returning to the basic definition introduced in the previous chapter, i.e if 

The logarithm of a number x is normally abbreviated to log x, i.e. if the subscript denoting the base is omitted then we assume that the base is 10. 

The table below shows the values of In x and log x for several values of x. 

In each case the ratio In xAog x has the value 2.303. It thus follows that  

This is usually measured using pH, which is defined as 

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In = Natural logarithm to base e MTD = Mean Temperature Difference, °F, see Figure 10-33 = Log mean temperature difference LTD = Atj = Least terminal temperature difference GTD = Atj = Greater terminal temperature difference... 

In the Nernst equation the term RT/nF involves known constants, and introducing the factor for converting natural logarithms to logarithms to base 10, the term has a value at a temperature of 25 °C of 0.0591 V when n is equal to 1. Hence, for an ion M+, a ten-fold change in ionic activity will alter the electrode potential by about 60 millivolts, whilst for an ion M2 +, a similar change in activity will alter the electrode potential by approximately 30 millivolts, and it follows that to achieve an accuracy of 1 per cent in the value determined for the ionic concentration by direct potentiometry, the electrode potential must be capable of measurement to within 0.26 mV for the ion M+, and to within 0.13 mV for the ion M2 +. ... 

Many calculations using Equation refer to standard temperature, 298.15 K. Furthermore,. eq often has a very large or very small value that is expressed using power-of-ten notation. For such values, calculations using the logarithm to base 10 (log) rather than In are more convenient log X-2.302 585 In X. We can substitute these values and the value for F and then evaluate the multiplier for the log term at standard temperature ... 

The use of scientific notation to describe ion concentrations is somewhat cumbersome, and chemists have agreed to employ a pH scale to state the concentration of hydrogen ions (the capital H in pH stands for hydrogen). They define pH to be the negative logarithm to base 10 of the hydrogen ion concentration ... 

The pH value is defined as the negative of the logarithm to base 10 of the concentration of the hydrogen ion. The acidity or basicity of a substance is defined most typically by the pH value. 

Ratios of power in electronic circuits are measured in decibels. A hypothetical unit of a Bel would describe the logarithm to base 10 of the ratio of two power levels the decibel (dB) is a unit of a tenth of a Bel. Hence the ratio of two power levels Pi and P2 may be described as a = 10 x log10(P[/P2) dB. Conversely, Pi = P2 x 10 /10. The decibel is dimensionless and, since it is a logarithmic unit, amplifications and attenuations expressed in decibels can... 

Taking logarithms (to base 10 or base e) of both sides gives... 

Note-, logarithms are to base 10 because the Debye-Hiickel equation is given in logarithms to base 10. 

A logarithmic plot using logarithms to base 10 of the concentration against time is a straight line with a slope of 0.4584. Assuming a first order elimination process, the value of kei (see Figure 8.6) calculated from this slope is... 

Since logarithms to base 10 (as in logioKp/po) are sometimes quoted we give the alternative (old fashioned) form of the equation ... 

The Partition Coefficient itself is a constant. It is defined as the ratio of concentration of compound in aqueous phase to the concentration in an immiscible solvent, as the neutral molecule. The partition coefficient (P) therefore is the quotient of two concentrations and is usually given in the form of its logarithm to base 10 (log P). The Log P will vary according to the conditions under which it is measured and the choice of partitioning solvent. 

A plot of Aj versus tl would produce a curve with an exponential decay, approaching [AJ = 0 asymptotically (Fig. 7.1). Taking logarithms to base 10 on both sides of Equation 7.16 gives... 

The angle brackets in Eq. 4.15 represent molar concentrations of the particular species. In practice, one may select 1 mol of water for study, which means (H2O) = 1 in Eq. 4.5. Eurthermore, it is customary to represent the negative logarithm to base 10 of the molar concentrations in Eq. 4.5 ... 

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