Avogadro's Number — Question 6

Question 6

Dilution controls an expected count rather than a guaranteed integer

Use $N_A=6.02214076\times10^{23}\,\mathrm{mol^{-1}}$, exact. A solution contains solute at $1.00\times10^{-9}$ mol/L. A well-mixed $1.00\,\mu$L aliquot is diluted by an overall concentration factor of $10^8$ before a new $1.00\,\mu$L aliquot is taken from the final solution. Assume ideal uniform mixing and unchanged solute entities.

1. Find the expected solute count in a 1.00 microlitre sample before and after dilution.

2. For an independently supplied Poisson sampling model $P(N=0)=e^{-\mu}$, find the empty-aliquot probability after dilution. What dilution factor would make the expected count exactly one?

Question 6 - Solution

Dilution controls an expected count rather than a guaranteed integer

(a) Method and calculation

$$1.00\,\mu\mathrm L=10^{-6}\,\mathrm L,\quad\mu_0=cVN_A=6.02214076\times10^8.$$$$\mu=\mu_0/10^8=6.02214076.$$

(b) Interpretation and further calculation

$$P(0)=e^{-6.02214076}\approx.00242447\quad(0.242447\%).$$$$D_{\mu=1}=\mu_0=6.02214076\times10^8.$$The expected count may be noninteger although every actual sample contains an integer number of solute entities.

Physical check

A mean of one does not mean exactly one in every aliquot; the sampling model must be stated separately.