Problem — Fermi estimate, atoms in an apple

Problem

How many atoms does a $200$ g apple contain? Estimate the order of magnitude assuming the apple is mostly water (molar mass $\sim 18$ g/mol, $3$ atoms per molecule).

Solution

Step 1 — Number of moles. Treating the apple as water:

$n = \frac{m}{M} = \frac{200\;\text{g}}{18\;\text{g/mol}} \approx 11\;\text{mol}$

Step 2 — Number of molecules. With Avogadro’s number $N_A \approx 6{,}0\cdot 10^{23}$ mol$^{-1}$:

$N_\text{mol} = n\,N_A = 11\cdot 6{,}0\cdot 10^{23} \approx 6{,}7\cdot 10^{24}\;\text{molecules}$

Step 3 — Number of atoms. Each water molecule ($\text{H}_2\text{O}$) has $3$ atoms:

$N = 3\,N_\text{mol} = 3\cdot 6{,}7\cdot 10^{24} \approx 2\cdot 10^{25}\;\text{atoms}$

$\ev{N \sim 2\cdot 10^{25}\;\text{atoms}}$

A huge number: even an everyday object contains tens of thousands of billions of billions of atoms. It is this gigantic scale that makes the macroscopic “average” behaviour emerge from microscopic quantum laws.

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Topics: Fisica quantistica Skills: Stime di Fermi · Conversione di unità