Problem Statement
If the atmosphere pressure at sea level is 10.143 N/cm², determine the pressure at a height of 2500 m assuming the pressure variation follows (i) Hydrostatic law, and (ii) Isothermal law. The density of air is given as 1.208 kg/m³.
Given Data
- Pressure at sea-level, \(p_0 = 10.143 \, \text{N/cm}^2 = 101430 \, \text{N/m}^2\)
- Height, \(Z = 2500 \, \text{m}\)
- Density of air at sea-level, \(\rho_0 = 1.208 \, \text{kg/m}^3\)
- Acceleration due to gravity, \(g = 9.81 \, \text{m/s}^2\)
Solution
(i) Pressure by Hydrostatic Law
The hydrostatic law assumes the fluid density (\(\rho\)) is constant. The governing equation is:
Integrating this from sea level (\(Z_0=0, p=p_0\)) to height \(Z\) gives:
Substitute the given values:
Converting back to N/cm²:
(ii) Pressure by Isothermal Law
The isothermal law assumes the temperature (\(T\)) is constant. The governing equation is:
Using the ideal gas law, \(p_0 = \rho_0 R T\), we can substitute for \(RT = p_0 / \rho_0\).
Calculate the value of the exponent:
Now, calculate the final pressure:
Converting back to N/cm²:
(i) Pressure by Hydrostatic Law: \( \approx 7.18 \, \text{N/cm}^2 \)
(ii) Pressure by Isothermal Law: \( \approx 7.57 \, \text{N/cm}^2 \)
Explanation of Atmospheric Laws
This problem compares two common models for how atmospheric pressure changes with altitude:
1. Hydrostatic Law: This is the simplest model. It assumes that the density of the air remains constant at all altitudes. While this is accurate for liquids or for very small changes in height for gases, it becomes inaccurate over large altitudes because air is compressible and its density decreases as you go higher.
2. Isothermal Law: This model assumes that the temperature of the air remains constant. According to the ideal gas law, if temperature is constant, density becomes directly proportional to pressure (\(\rho = p/RT\)). This model accounts for the fact that air becomes less dense at higher altitudes, making it a more realistic approximation for the atmosphere than the simple hydrostatic law.
Physical Meaning
The results show that the pressure predicted by the Isothermal Law (\(7.57 \, \text{N/cm}^2\)) is higher than that predicted by the Hydrostatic Law (\(7.18 \, \text{N/cm}^2\)). This difference is significant and meaningful.
The Hydrostatic Law overestimates the pressure drop because it assumes the air remains just as heavy (dense) at 2500 m as it is at sea level. In reality, the air becomes thinner (less dense). The Isothermal Law accounts for this thinning of the air. Because the column of air above 2500 m is lighter than the hydrostatic model assumes, it exerts less weight, leading to a smaller pressure drop and therefore a higher remaining pressure at that altitude. For real-world atmospheric calculations, even more complex models (like the Standard Atmosphere model, which accounts for temperature changes) are used.
