PRINCIPLES OF TRACER METHODOLOGY 19 



It is important to note that A" is actually the fractional rate of change of 

 A with time. In later applications A; will be given other specific mean- 

 ings. For small intervals of time this can be written as the differential 

 expression 



By integration between limits, various forms of the equation suitable for 

 use with experimental data are obtained: 



(1-12) 

 (1-13) 



A 



2.3 log -7- = -kt (1-14) 



where A = amount of substance A present at time t 



Ao = amount of substance .4 present at zero time 

 This behavior is often termed exponential or logarithmic removal and may 

 also be expressed as 



A = A 06-'^ ■ (1-15) 



where e = base of the natural logarithms. 



It is often convenient to use a constant ty,, which represents the time 

 for removal of half of substance A present at any given time and is some- 

 times called the half-value time. From Eq. (1-14) 



_ 2.3 log ^ 0.693 



hi -j: = — ^ (1-16) 



The following is a useful form of the equation for calculating ti,^ from meas- 

 urements at two different times or for calculating the amount of A present 

 at time t when Ao and ty, are known: 



^'' = log (AoM) (^-^^^ 



From Eq. (1-14) it will be noted that log A plotted against t will result 

 in a straight line. In practice, A is usually plotted on a log scale against 

 t on a linear scale. The constant k may be determined from 2.3 times the 

 slope of the line, the latter being (log Ao — log A)/t. 



It is often of interest to calculate the removal of substance A as a func- 



