Negative logarithm

Values of are small for weak acids and they range very widely (Table 4.1). It is common practice to quote values as the negative logarithm to the base ten, i.e. — logjo K.. since such numbers are less cumbersome and positive when Aj < 1. The symbol for -logio is by convention "p/ fhus -logjo becomes pK,. Table 4.1 shows some typical pAg values. 

To find appropriate empirical pair potentials from the known protein structures in the Brookhaven Protein Data Bank, it is necessary to calculate densities for the distance distribution of Ga-atoms at given bond distance d and given residue assignments ai,a2- Up to a constant factor that is immaterial for subsequent structure determination by global optimization, the potentials then ciiiergo as the negative logarithm of the densities. Since... 

Characteristic of a Common Logarithm of a Number. Every real positive number has a real common logarithm such that f a < b, log a < log b. Neither zero nor any negative number has a real logarithm. 

The values listed in Tables 8.7 and 8.8 are the negative (decadic) logarithms of the acidic dissociation constant, i.e., — logj, For the general proton-transfer reaction... 

Since the 0 s are fractions, the logarithms in Eq. (8.38) are less than unity and AGj is negative for all concentrations. In the case of athermal mixtures entropy considerations alone are sufficient to account for polymer-solvent miscibility at all concentrations. Exactly the same is true for ideal solutions. As a matter of fact, it is possible to regard the expressions for AS and AGj for ideal solutions as special cases of Eqs. (8.37) and (8.38) for the situation where n happens to equal unity. The following example compares values for ASj for ideal and Flory-Huggins solutions to examine quantitatively the effect of variations in n on the entropy of mixing. 

Here, yj is the measured value, <7 is the standard deviation of the ith measurement, and Ay is needed to say a measured value —Ay has a certain probabihty. Given a set of parameters (maximizing this function), the probabihty that this data set plus or minus Ay coiild have occurred is R This probability is maximized (giving the maximum hke-lihood) if the negative of the logarithm is minimized. 

Acidity is defined in terms of the pH scale, where pH is the negative logarithm of the hydrogen ion [H ] concentration. 

PK. — the negative logarithm of the equilibrium constant for acids or bases. This parameter is an indicator of the strength of an acid or base. Strong acids, such as H2SO4, and HCl, have low pK s (i.e., -1.0) while strong bases such as KOH and NaOH, have pK s close to 14.0. Weak acids and weak bases fall in the intermediate range. 

Table 2.2 gives the pH scale. Note again the reciprocal relationship between [H ] and [OH ]. Also, because the pH scale is based on negative logarithms, low pH values represent the highest H concentrations (and the lowest OH concentrations, as K, specifies). Note also that... 

Ta 1.5 X 10 2, K3 2.1 X 10 and 2.4 x and the corresponding negative logarithms are pA" 1.0, pA"2 1.8, pA"3 6.57 and pA"4 9.62. The P—O—P linkage is kinetically stable towards hydrolysis in dilute neutral solutions at room temperature and the reaction half-life can be of the order of years. Such hydrolytic breakdown of polyphosphate is of considerable importance in certain biological systems and has been much studied. Some factors which affect the rate of degradation of polyphosphates are shown in Table 12.10. 

For convenience, acid strengths are normally expressed using pXa values rather than Ka values, where the pKa is the negative common logarithm of the Ka. 

The numbers whose natural logarithms are 4.606 and —6.908 are 100 and 0.001, respectively. Notice that if the inverse logarithm is positive, the corresponding number is larger than 1 if it is negative, the number is smaller than 1. 

Nano- The prefix on a metric unit indicating a multiple of 10-9,7 Naphthalene, 206t, 590 Naproxen, 601 Natural gas, 215,583 Natural logarithms, 645 Negative integer, 643 Neon, 32... 

---