Avogadro's Number — Question 9

Question 9

An exact counting constant does not remove sample uncertainty

Use $N_A=6.02214076\times10^{23}\,\mathrm{mol^{-1}}$, exact. A sample mass is $m=2.00\pm.02$ g and its assigned molar mass is $M=40.0\pm.4$ g/mol. Treat these as independent hard bounds. The desired entity count is $N=mN_A/M$.

1. Find the central entity count and the tight allowed interval.

2. Compare the exact fractional bounds with the first-order sum of relative bounds. Why does the exact Avogadro constant not make the count exact?

Question 9 - Solution

An exact counting constant does not remove sample uncertainty

(a) Method and calculation

$$N_c=(2.00/40.0)N_A=.0500N_A=3.01107038\times10^{22}.$$$$N_{\min}=\frac{1.98}{40.4}N_A,\quad N_{\max}=\frac{2.02}{39.6}N_A.$$$$N/N_c\in[.99/1.01,1.01/.99]=[.980198,1.020202].$$

(b) Interpretation and further calculation

$$\left|\frac{\delta N}{N}\right|\lesssim\frac{.02}{2.00}+\frac{.4}{40.0}=.0200.$$$$\text{Exact deviations: }-1.98020\%,\ +2.02020\%.$$The first-order approximation is nearly symmetric; the exact inverse-mass relation is not. Uncertainty in measured inputs remains even when the proportionality constant is exact.

Physical check

The smallest count pairs the smallest mass with the largest molar mass, rather than pairing both lower endpoints.