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Next: Ideal Gases Up: The Boltzmann Distribution Previous: The Isothermal Atmosphere

How Big are Atoms?

Wait a minute! How did I calculate  h0?  I had to know  m  for the different molecules, and that requires some knowledge of the sizes of atoms - information that has not yet been set forth in this book! In fact, empirical observations about how fast the pressure of the atmosphere does drop off with altitude could give a pretty good idea of his big atoms are; this isn't how it was done historically, but let's give it a try anyway:

Suppose that, by climbing mountains and measuring the density of oxygen molecules (O2) as a function of altitude, we have determined empirically that  h0  for O2 is about 8,000 m. Then, according to this simple model, it must be true that the mass  m  of an O2 molecule is about

\begin{displaymath}m \, \approx \, { \tau \over h_0 \, g }
\, = \, { 300 \time . . . 
 . . . s 10^{-23}
\over 8 \times 10^3 \times 9.81 } \hbox{\rm ~kg}
\end{displaymath}


\begin{displaymath}\hbox{\rm or} \quad
m \, \approx \, 5.3 \times 10^{-26} \hbox{\rm ~kg}
\end{displaymath}

This is a mighty small mass!

Now to mix in just a pinch of actual history: Long ago, chemists discovered (again empirically) that different pure substances combined with other pure substances in fixed ratios of small integers times a certain characteristic mass (characteristic for each pure substance) called its molecular weight  A.  People had a pretty good idea even then that these pure substances were made up of large numbers of identical units called "atoms,"15.24 but no one had the faintest idea how big atoms were -- except of course that they must be pretty small, since we never could see any directly. The number  N0  of molecules in one molecular weight of a pure substance was (correctly) presumed to be the same, to explain why chemical reactions obeyed this rule. This number came to be called a "mole" of the substance. For oxygen (O2), the molecular weight is roughly 32 grams or 0.032 kg.

If we now combine this conventional definition of a mole of O2 with our previous estimate of the mass of one O2 molecule, we can estimate

\begin{displaymath}N_0 \; \approx \; { 0.032 \over 5.3 \times 10^{-26} }
\; \approx \; 6 \times 10^{23}
\end{displaymath}

The exact number, obtained by quite different means, is

\begin{displaymath}N_0 \equiv 6.02205 \times 10^{23}
\end{displaymath} (15.17)

molecules per mole. This is known as AVOGADRO'S NUMBER.

Turning the argument around, the mass of a molecule can be obtained from its molecular weight  A  as follows: One mole of any substance is defined as a mass   $A \times 1$ gram, and contains  N0  molecules (or atoms, in the case of monatomic molecules) of the substance. Thus helium, with  A = 4,  weighs 4 gm (or 0.004 kg) per mole containing  N0  atoms, so one He atom weighs   (0.004/N0) kg  or   $6.6 \times 10^{-27}$ kg.


next up previous
Next: Ideal Gases Up: The Boltzmann Distribution Previous: The Isothermal Atmosphere
Jess H. Brewer - Last modified: Mon Nov 16 16:12:45 PST 2015