Problem — Fermi estimate: molecules in a room

Problem

Estimate the number of air molecules in a room measuring $4$ m $\times$ $5$ m $\times$ $3$ m at $T = 293$ K and $p = 1$ atm. Compare with Avogadro’s number.

Solution

I use the ideal gas law in its microscopic form $pV=Nk_B T$, from which $N=\frac{pV}{k_B T}$ Volume of the room: $V=4\cdot5\cdot3=60\ \text{m}^3$ Substituting the values ($p=1{,}013\cdot10^5$ Pa, $k_B=1{,}38\cdot10^{-23}$ J/K, $T=293$ K): $N=\frac{1{,}013\cdot10^5\cdot 60}{1{,}38\cdot10^{-23}\cdot 293}$ $N=\frac{6{,}08\cdot10^6}{4{,}04\cdot10^{-21}}=1{,}5\cdot10^{27}$ Comparison with Avogadro. $N_A=6{,}02\cdot10^{23}$, so $\frac{N}{N_A}=\frac{1{,}5\cdot10^{27}}{6{,}02\cdot10^{23}}\approx 2{,}5\cdot10^3$ the room contains roughly 2500 moles of air, i.e. thousands of times Avogadro’s number. $\ev{N\approx 1{,}5\cdot10^{27}\ \text{molecules}}$

Connections

Topics: Teoria cinetica dei gas Concepts: Legge dei gas perfetti Skills: Stime di Fermi