Calculating Average Density: How Dense Is the Air?

Strategy

If we solve $P = hρg$ for density, we see that

\bar{ρ} = \frac{P}{hg} .

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We then take $P$ to be atmospheric pressure, $h$ is given, and $g$ is known, and so we can use this to calculate $\bar{ρ}$ .

Solution

Entering known values into the expression for $\bar{ρ}$ yields

\bar{ρ} = \frac{1 . 01 \times 10^{5} {N/m}^{2}}{(120 \times 10^{3} m) (9.80 {m/s}^{2})} = 8 . 59 \times 10^{- 2} {kg/m}^{3} .

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Discussion

This result is the average density of air between the Earth’s surface and the top of the Earth’s atmosphere, which essentially ends at 120 km. The density of air at sea level is given in Table 11.1 as $1.29 {kg/m}^{3}$ —about 15 times its average value. Because air is so compressible, its density has its highest value near the Earth’s surface and declines rapidly with altitude.