Sometimes in chemistry lab a series of solutions of widely differing concentrations of the same solute is needed. This is often the case when making standard solutions for atomic absorption experiments or in other instrumental methods of analysis. In some instances, the technique of "serial dilutions" is used as an efficient way to make up the solutions. Rather than pipetting increasing amounts of a standard solution into a series of volumetric flasks, the most concentrated solution is made first, and some of it is used to make the second most concentrated solution. Then, some of that solution is used to make the third most concentrated solution. This is continued until the desired number of solutions has been made. This is also a good way to make very dilute solutions. In fact, it is the method used the controversial field of homeopathic "medicine". As will be shown in the subsequent calculations, solutions so dilute that they, on the average, contain NO molecules of the active ingredient, are sold to gullible consumers as medicines!

**Calculations**

The usual methods for dilution calculations can be used for each step in a series of serial dilutions. In cases where several repetitions of the same type of dilution are made, it is easy to extend this method so that only one calculation is needed.

Example:In an atomic absorption experiment devised by Dr. Schaumloffel for CHM 161, the students make up a series of standard solutions containing both Na^{+}and K^{+}for use in analysis of fake "urine" for its sodium and potassium content. The students started out by making a standard sodium solution with a concentration of 400 ppm in Na^{+}and 0.40 ppm in K^{+}.a. If they needed 100 mL each of solutions with concentrations of about 4.0 ppm , 2.0 ppm and 0.40 ppm in Na

^{+}for the AA experiment, what serial dilutions would provide them?The students first needed to calculate how much of the 400 ppm Na^{+}solution the students they needed to use to make 100.0 mL of a solution with a concentration of 4.0 ppm Na^{+}.400 ppm x c

_{i}= 4.0 ppm x 100.0 mLc

_{i}= 1.0 mLSo, they needed to pipet 1.0 mL of the stock solution into a 100.0 mL volumetric flask and fill to the line. If they were going to make the 2.0 ppm solution from the stock solution, it is clear that they would have to pipet 0.50 mL of it into a 100 mL volumetric flask. The relative error in pipetting such a small volume is quite large, so it would probably be better to use the 4.0 ppm solution they just made to make the 2.0 ppm solution.

If the 4.0 ppm solution were used to make a 2.0 ppm solution in a 100.0 mL volumetric flask, 50.0 mL of it would be needed since it would require a dilution by a factor of two. You might want to check this using the "magic" equation, but you should be able to do simple problems like this in your head.

Question 1:How could you make a 0.40 ppm Na^{+}solution from the 2.0 ppm one?

b. What will the concentrations of K

^{+}be in those solutions?

The calculations of the KA table of the results of these calculations is given in^{+}are done in the same way. Since the original stock solution had a K^{+}concentration of 0.40 ppm, and 1.00 mL of it was diluted to a volume of 100.0 mL, the concentration after the first serial dilution would be:

(0.40 ppm x 1.00 mL)/100.0 mL = 4.0 x 10^{-3}ppm K^{+}.The second dilution used 50.0mL of the 4.0 x 10

^{-3}ppm K^{+}solution which was diluted to a final volume of 100.0 mL. Its concentration would be:

(4.0 x 10^{-3}ppm x 50.0 mL)/100.0 mL = 2.0 x 10^{-3}ppm in K^{+}.The final dilution involoved taking 20.0 mL of the 2.0 x 10

^{-3}ppm K^{+}solution and diluting it in a 100.0 mL volumetric flask.

(2.0 x 10^{-3}ppm x 20.0 mL)/100.0 mL = 4.0 x 10^{-4}ppm

Fig. 1.A.3.

**Notation for Serial Dilutions in Homeopathic "Remedies" **

Homeopathic "remedies" are labeled according to the number of dilutions which were made to produce them. The designation consists of two parts, a number which refers to the number of serial dilutions done, and a letter (Roman numeral) which refers to the dilution factor for each dilution. For example, a product labeled "20X" would have had 20 serial dilutions by a factor of 10 done on it. One of the examples in the "Consumer Reports" article included a discussion of a belladonna extract which was labeled "30C"

Example:One of the examples in the "Consumer Reports" article included a discussion of a belladonna* extract (active ingredient is the poisonous substance atropine, 289.4 g/mole) which was labeled "30C". What does this mean, and by what factor would the final product be more dilute than the initial extract?The label "30C" means that 30 sequential 100-fold dilutions were done on the initial extract. The overall dilution factor would be (1/100) multiplied times itself 30 times, or (1/10

^{2})^{30}or 10^{-60}. Given the fact that Avogadro's number is only 6.02 x 10^{23}, it can be seen that alotof the diluted material would need to be used to get evenone moleculeof the active ingredient. In fact, if you did the calculations, you would find that you would have to ingest a volume of the stuff equal to more than 400 billion times the volume of the earth to get (maybe, if you're lucky) get just ONE molecule of the active ingredient!

Question 2:What volume of "30C" belladonna extract would contain one molecule of atropine?The impurities in the solvents used to make these formulations would likely be billions of times more concentrated than that of the "active ingredients" in the final product, even if the purest available solvents were used. (Environmental contamination in the manufacture of the products also is likely to be of a much higher concentration than that of the "active ingredient".) So, any beneficial results obtained when these formulations are used are likely the result of the placebo effect, which can be considerable in some people. (It seems that these "remedies" would be most effective in treating psychosomatic illnesses!) It is interesting also to note that in many cases, ethyl alcohol is used as part of the solvent in making homeopathic remedies.

*Type the word "belladonna" into a search engine on the Web to get some interesting links!

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On: Jan. 1, 2001

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