A dilution factor named "e"

Some people may estimate that the final concentration in A will be 0.5 g/L, corresponding to a lowering of the original concentration of 1 g/L by a factor of 2. But the dilution factor is  2.71828 . Those who had never used  e  in practice did it just now. The name "e" was given by Leonard Euler in 1748.  e  appears in cases where a lot of very small steps are used, in particular an indefinite sum of indefinite small steps. The term "continuous" is used for such a process and continuous dilution is a very graphic example. The final concentration after filling an equal-sized vessel B is  1/e = 0.368.   The next figure shows a model with two vessels B and C to be filled:

After filling of vessel B the concentration was 1 / e  =  0.368, during the filling of vessel C the concentration is again lessened by a factor e and thus the final result is:  1 / e / e  =  0.135. Within the step-wise calculation of  0.368, the exponential form   0.753  was used instead of repeated multiplication  0.75 * 0.75 * 0.75. For repeated division the same advantage will be drawn from  1 * e-3  instead of  1 /e /e /e.   With a start concentration c0 = 1 the following formulas result for the concentration c:

Vessels	Repeated dilution	Total dilution factor		c formula	c (c0 = 1)
1	1 /e	e1	2.72	c0 * e-1	0.368
2	1 /e /e	e2	7.39	c0 * e-2	0.135
3	1 /e /e /e	e3	20.09	c0 * e-3	0.050
4	1 /e /e /e /e	e4	54.60	c0 * e-4	0.018
5	1 /e /e /e /e /e	e5	148.41	c0 * e-5	0.007
6	.........

Detour