Problem Statement
If the atmospheric pressure at sea-level is 10.143 N/cm², determine the pressure at a height of 2000 m assuming that the pressure variation follows: (i) Hydrostatic law, and (ii) Isothermal law. The density of air is given as 1.208 kg/m³.
Given Data & Constants
- Sea-level pressure, \(P_0 = 10.143 \, \text{N/cm}^2 = 101430 \, \text{N/m}^2\)
- Height, \(z = 2000 \, \text{m}\)
- Sea-level air density, \(\rho_0 = 1.208 \, \text{kg/m}^3\)
- Acceleration due to gravity, \(g = 9.81 \, \text{m/s}^2\)
Solution
Part (i): Pressure according to Hydrostatic Law
The hydrostatic law assumes that the density of the fluid (air) remains constant with altitude. This is a simplification.
Part (ii): Pressure according to Isothermal Law
The isothermal law assumes that the temperature of the air remains constant with altitude. This implies that density is proportional to pressure (\(P/\rho = \text{constant}\)).
Pressure by Hydrostatic Law: \( P \approx 77730 \, \text{N/m}^2 \) or \(7.773 \, \text{N/cm}^2\)
Pressure by Isothermal Law: \( P \approx 80300 \, \text{N/m}^2 \) or \(8.030 \, \text{N/cm}^2\)
Explanation of the Laws
(i) Hydrostatic Law: This model treats air as an incompressible fluid, like water. It assumes the density of air (\(\rho\)) is constant at all altitudes. This leads to a simple linear decrease in pressure with height. While easy to calculate, it's not physically accurate for a compressible gas like air, as air density clearly decreases with altitude.
(ii) Isothermal Law: This model assumes the temperature (\(T\)) is constant. According to the ideal gas law (\(P = \rho RT\)), if T is constant, then density (\(\rho\)) is directly proportional to pressure (\(P\)). This means that as you go higher and pressure drops, density also drops. This results in an exponential decrease in pressure with height, which is more realistic than the linear hydrostatic model.
Physical Meaning & Real-World Atmosphere
This problem demonstrates how different physical assumptions lead to different predictions. The Isothermal Law predicts a higher pressure (80.3 kPa) at 2000 m than the Hydrostatic Law (77.7 kPa). This is because the isothermal model accounts for the fact that the air becomes "thinner" (less dense) at higher altitudes, so the weight of the air column above 2000 m is less than what the constant-density hydrostatic model assumes.
In reality, neither model is perfectly accurate. The Earth's atmosphere is not isothermal; temperature generally decreases with altitude in the troposphere. The International Standard Atmosphere (ISA) model is a more sophisticated model that defines a standard temperature "lapse rate" (rate of decrease) with altitude, providing a much more accurate prediction of pressure and density for aviation and atmospheric science.
