Problem — Kitchen rice, Fermi style

Problem

Kitchen rice, Fermi style. Estimate how many grains of rice a $1\;\text{kg}$ bag contains. Proceed as follows: (a) estimate the mass of a single grain using a ruler (typical length $\sim 5\;\text{mm}$, cross-section $\sim 1\;\text{mm}^2$) and the density of starch ($\rho\approx 1{,}5\;\text{g/cm}^3$); (b) divide $1\;\text{kg}$ by the mass of a grain. Discuss the percentage uncertainty.

Solution

Step 1 — Volume of a grain (a). We approximate the grain to a small cylinder of length $\ell\approx 5\;\text{mm}$ and cross-section $A\approx 1\;\text{mm}^2$. The volume is

$V \approx \ell\cdot A = 5\;\text{mm}\cdot 1\;\text{mm}^2 = 5\;\text{mm}^3 = 5\cdot 10^{-9}\;\text{m}^3$

(recalling that $1\;\text{mm}^3 = (10^{-3}\;\text{m})^3 = 10^{-9}\;\text{m}^3$).

Step 2 — Mass of a grain. With the density of starch $\rho\approx 1{,}5\;\text{g/cm}^3 = 1{,}5\cdot 10^3\;\text{kg/m}^3$:

$m = \rho\, V = 1{,}5\cdot 10^3\cdot 5\cdot 10^{-9} = 7{,}5\cdot 10^{-6}\;\text{kg} = 7{,}5\;\text{mg}$

Step 3 — Number of grains (b). We divide the mass of the bag by the mass of a grain:

$N = \frac{1\;\text{kg}}{7{,}5\cdot 10^{-6}\;\text{kg}} = 1{,}3\cdot 10^{5}$

Step 4 — Order of magnitude. Being a Fermi estimate, retaining digits makes no sense: we conclude

$\ev{N \sim 10^{5}\text{–}10^{6}\ \text{grains}}$

Percentage uncertainty. Every linear dimension is known “by eye” with a relative error of $20\text{–}30\%$. The volume is a product of three lengths (or of length times area), so the relative error adds up: $\sim 3\times 25\% \approx 75\%$. In practice $N$ is reliable only to within a factor $2\text{–}3$: hence the width of a whole order of magnitude. The tabulated value for real rice is of the order of $10^4\text{–}10^5$ grains/kg (real grains are a bit heavier than solid starch because they are larger), consistent with this estimate within the unavoidable factor of a Fermi estimate.

Linked notes

Topics: Grandezze fisiche e misura Concepts: Notazione scientifica e ordini di grandezza Skills: Stime di Fermi · Conversione di unità