Atmospheric pressure at sea level is 76 cm of mercury. Calculate the vertical height of the air column exerting the above pressure. Assume the density of air is 1.29 kg/m³ and that of mercury is 13600 kg/m³. Why the height calculated by you is far less than the actual height of the atmosphere?

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Solution

Step 1. Given data
Height of the mercury, $h_{m e r} = 76 c m = 0.76 m$
Density of air, $ρ_{a i r} = 1.29 k g / m^{3}$
Density of mercury, $ρ_{m e r} = 13600 k g / m^{3}$
Height of the air column, $h_{a i r} = ?$
Step 2. Formula used
Since pressure is equal
$h_{a i r} \times ρ_{a i r} = h_{m e r} \times ρ_{m e r}$ …………………..(a)
Step 3. Calculation of the height of the air column
Substituting the given values in equation (a), we get
$h_{a i r} \times 1.29 = 0.76 \times 13600 h_{a i r} = \frac{0.76 \times 13600}{1.29} h_{a i r} = 8012.40 m h_{a i r} = 8.0124 k m$
Hence, the height of the air column is equal to 8.0124 km.
Step 4. The explanation for why the height calculated is less than the actual height of the atmosphere
While solving the numerical, we are assuming that the density of air is constant which is not the actual case because the density of air in the atmosphere is not constant.
With an increase in height, the density of the air decreases.
Hence, the height calculated is very less compared to the height of the atmosphere

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