Page 830 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Appendix C: Miscellaneous Relations 785

Separating variables, Therefore,
dA 2:3026 log X ¼ ln X (C:25)
¼ r dt (C:13)
A
Example for X ¼ 5:
Integrating between the limits of A o and A and 0 and t,
respectively, 2:3026 log 5 ¼ ln 5 (C:26)

ln A 2:3026 0:69897 ¼ 1:6094 (C:27)
¼ rt (C:14)
A o
A logarithm to the base e, designated ln, may be
converted to a common log, or vice versa, by the
or,
relationship
A rt
Y
¼ e (C:15) X ¼ e ¼ 10 kY (C:28)
A o
Y=2:3026
¼ 10 (C:29)
which is the same as (C.11).
Suppose X ¼ 5, then
C.3.3.5 Laws of Exponents
The four laws of exponent are 1:6094 1:6094=2:3026 0:69897
5 ¼ e ¼ 10 ¼ 10 (C:30)
1. Multiplying
C.3.3.7 Application to Problems
y
x
10 10 ¼ 10 (xþy) (C:16) One of the common problems has to do with a semi-log
plot, with the log scale based on common logs. The coeffi-
2. Dividing cients, i.e., slope and intercept must yield data in terms of
natural logs.
10 x (x y)
¼ 10 (C:17)
10 y 1. Best-fit equation for semi-log plot. Suppose the best-
fit equation of data from an experimental plot is
3. Powers
log Y ¼ slope X þ log B (C:31)
x n
(10 ) ¼ 10 nx (C:18)
Multiply both side by 2.303:
4. Roots
2:303 log Y ¼ 2:303 slope X þ 2:303 log B
x 1=n
(10 ) ¼ 10 x=n (C:19)
(C:32)
ln Y ¼ 2:303 slope X þ ln B (C:33)
C.3.3.6 Natural Logarithms
Relationships between natural logs and common logs are The ‘‘slope’’ is from the log–log plot and is in log-
based on following a few fundamentals. cycles of Y per unit of X.
2. Illustration in terms of van’t Hoff equation. To illus-
1. Values of natural logs, ln, trate in terms of a common equation, i.e., the van’t
Hoff equation, for the effect of temperature on the
10 ¼ e 2:3026 (C:20) equilibrium constant, i.e.,

Taking natural logs, both sides,
C vh e DH
^

ð or, ln K ¼ DH =RT þ ln C vh Þ
RT
K ¼
ln 10 ¼ 2:3026 (C:21)
If the experimental data are plotted as a semi-
2. Conversion from log X and ln X. Let X ¼ 10. There-
log relationship, i.e., log K vs. 1=T, the slope is,
fore,
DH8=2.303R (DH8 is the standard state enthalpy
of reaction and R is the universal gas constant).
C log 10 ¼ ln 10 (C:22)
Therefore
C 1:0 ¼ 2:3026 (C:23)
DH
(C:34)
C ¼ 2:3026 (C:24) 2:303 slope ¼ R