Introducing Probabilistic Concepts in Chemistry The Preparation of 1o
- M~ Solution ~ as a Limiting Case
M. Sastre de Vicente
Departamento de Quimica Fundamental e Industrial, Avda da Zapateira sin, Univenidad de La Coruiia, E-15071 La Coruna, Spain The definitions of various concentration units are among the first few concepts taught in early chemistry courses. These unite are used to provide a quantitative idea of the amount of a given substance contained within another. This process is usually limited to providing the definition, which is followed by immediate usage and application of the defined concepts. In this short paper we discuss a straightforward dilution experiment that can he used as the starting point for establishing a link between chemistry and some concepts of probability theory. Experimental Take 1L of a 1M aqueous solution of a substance A and dilute it to 1:1000 using a pipet. Repeat the process eight times as shown in Figure 1. After the eighth dilution, the resulting solution should have an approximate concentration of M, that is, 0.6 molecules per liter. This means that, at the end of the process, the 1-Lflask should contain 0.6 molecules. What is the actual meaning of this result?
into account that the mole concentration is proportional to the mole fraction or the probability, according t o the'previous analogy for dilute solutions, the experiment illustrated in Figure 1can be discussed in simple terms from a statistical point of view. Discussion According to common sense, as the dilution process shown in Figure 1progresses, the probability of withdrawing solvent molecules only on pipetting the solution volume required to withdraw a mole fractionxi will gradually increase. If the withdrawal of solution molecules is assumed to conform to a binomial distribution ( I ) , then the associated variance will be given by
In addition, the sampling error made in substance Awill be
Mole Fraction and Probability For simplicity, let us consider a solution of two substances A and B. The mole fraction is defined as Interpretation
where i =A, B. Hence, Thus. the mole fraction ofa substance i denotes the number of &olecules of the substance contained in a vessel that holds n~ + n~ = n molecules. The previous definition is formally equivalent to the probability uf withdrawingn, molecules from a set that contains n. In fact. such a ~rohnbilitv P i s also occasionally expressed by eq l.1n addition, taking dilution
Figure 1. The dilution process to produce a 674
Journal of Chemical Education
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M solution.
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According to eq 4, e + when P + 0 a t constant n, where this last parameter is associated with the withdrawn sample volume. Hence the above dilution process can be interpreted as follows. Although there is a fairly high probability that the solution resulting from the first dilution will he lo3 M in A, the probability of preparing a loa4 M solution of A is virtually zero. As a result, the last flask shown in Figure 1will most certainly contain only solvent molecules rather than 0.6 solute molecules as one would expect from the conventional calculations.
An Alternatiue Dilution Process On the basis of the above reasoning, one could implement a n alternative dilution process that would probably allow a lod4 M solution to be prepared from the 1M starting solution with some guarantee. I n such a process (Fig. 2), the overall contents of the 1-L flask would be added to the solvent volume required for a final solution of loa4 M in A. This would involve using about loz3 L of solvent, which is obviously infeasible on the laboratory scale. Even if it were, how long would the solution take to reach equilibrium? This alternative procedure would be equivalent to pouring 1mol of Ainto the Atlantic Ocean and wondering "What is the mole fraction of A in the ocean?" At most, one could deal with local mole fractions, so this alternative procedure would be equally inviable.
Conclusion From the above-described dilution process, it follows that, even though solution concentrations can be calculated most readily in a n automatic fashion, preparing highly dilute solutions (e.g., M) is far fmm simple. Whether or not i t is possible is also arguable in practice. The above examples have been discussed with students in the author's class in order to improve available knowledge on such concepts a s the mole fraction, sampling error,
Figure 2. Alternative procedure forproducing a
M solution
probability, concentration, etc., all of which are quite indispensable and wmmonly used by chemists.
Volume 70 Number 8 August 1993
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