Avogadro's Number — Question 10

Question 10

A detector count can reveal an underlying molecular throughput

Use $N_A=6.02214076\times10^{23}\,\mathrm{mol^{-1}}$, exact. A beam detector registers $3.00\times10^6$ counts in 2.00 s. Each incident molecule is independently detected with probability .250, and each detection produces one count. Treat the observed count as its expected value for rate inference. Molecules have assigned molar mass 28.0 g/mol.

1. Infer incident molecular rate, molar rate and mass rate.

2. How long would this beam take to deliver $1.00\,\mu$g at unchanged rate? State the effect of mistakenly assuming 100% detector efficiency.

Question 10 - Solution

A detector count can reveal an underlying molecular throughput

(a) Method and calculation

$$\dot N=\frac{N_{\mathrm{det}}}{\eta\Delta t}=\frac{3.00\times10^6}{.250(2.00)}=6.00\times10^6\,\mathrm{s^{-1}}.$$$$\dot n=\dot N/N_A\approx9.96323\times10^{-18}\,\mathrm{mol/s},\quad\dot m=M\dot n\approx2.78971\times10^{-16}\,\mathrm{g/s}.$$

(b) Interpretation and further calculation

$$t=\frac{10^{-6}}{\dot m}\approx3.58461\times10^9\,\mathrm s.$$$$\dot N_{\eta=1}=.250\dot N,\quad t_{\eta=1}=4t.$$Ignoring missed molecules underestimates the rate by a factor of four and overestimates the delivery time by the same factor.

Physical check

Counting rate, amount rate and mass rate have different units and require separate conversions.