Show that the variation of atmospheric pressure with...

Show that the variation of atmospheric pressure with altitude is given by $P=P_{0} e^{-\alpha y},$ where $\alpha=\rho_{0} g / P_{0}, P_{0}$ is atmospheric pressure at some reference level $y=0$ , and $\rho_{0}$ is the atmospheric density at this level. Assume the decrease in atmospheric pressure over an infinitesimal change in altitude (so that the density is approximately uniform) is given by $d P=-\rho g d y$ and that the density of air is proportional to the pressure.

Key Concepts

Hydrostatic Equilibrium

This principle states that at each point of a fluid at rest, the downward gravitational force is exactly balanced by the upward pressure gradient force. In the context of an atmosphere, it leads to the differential relation dP = -?g dy, where dP is the infinitesimal change in pressure with an infinitesimal change in altitude dy, ? is the local density, and g is the gravitational acceleration.

Isothermal Atmosphere

An isothermal atmosphere is one in which the temperature remains constant with altitude. Under this assumption, the relationship between pressure and density simplifies because the ideal gas law implies that density is directly proportional to pressure. This proportionality permits the substitution of ? in terms of P in the hydrostatic equilibrium equation, facilitating its integration.

Exponential Atmospheric Pressure Variation

When the density is assumed to be proportional to pressure, and the temperature remains constant, integrating the hydrostatic equilibrium equation results in an exponential decay of pressure with altitude. This relationship, P = P? e^(??y), where P? is the pressure at a reference level and ? is a constant that encapsulates the proportionality factors, describes how atmospheric pressure decreases exponentially with height.

Separation of Variables in Differential Equations

This is a mathematical method used to solve differential equations by isolating the variables on opposite sides of the equation. In the derivation of the atmospheric pressure variation, separating the terms involving pressure and those involving altitude allows one to integrate each side independently, ultimately leading to the exponential solution for pressure as a function of altitude.

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