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 over the infinitesimal change) can be expressed from Equation 14.4 as $d P=-\rho g d y .$ Also assume the density of air is proportional to the pressure, which, as we will see in Chapter 18 , is equivalent to assuming the temperature of the air is the same at all altitudes.

Key Concepts

Exponential Decay

The solution to the differential equation under the assumption that density is proportional to pressure is found through separation of variables and integration, leading to an exponential function. This exponential decay form, where pressure decreases exponentially with altitude, is a common feature in many natural decay processes and is fundamental to understanding atmospheric pressure variation.

Hydrostatic Equilibrium

This concept refers to the balance between the upward pressure force and the downward gravitational force in a fluid. In the context of the atmosphere, it describes how the pressure decreases with increasing altitude due to the weight of the air above. The relationship is often expressed via a differential equation that encapsulates this balance.

Isothermal Atmosphere Assumption

Assuming an isothermal atmosphere means that the temperature remains constant with altitude. This assumption simplifies the relationship between the density and pressure of the air, making it linear. As a result, the density becomes directly proportional to the pressure, which allows for easier integration of the differential equation governing the pressure variation.

Differential Equation Modeling

The process of setting up and solving the differential equation, where the infinitesimal change in pressure is related to the product of density, gravitational acceleration, and the infinitesimal change in altitude, is a key tool in modeling physical systems like the atmosphere. The equation dP = -?g dy encapsulates the physical idea of hydrostatic balance.

Transcript

00:01 Hi there, so for this problem, we need to show that the variation of atmospheric pressure and an altitude is given by the following expression, p -0, which is the atmospheric pressure, this times the exponential of minus alpha times y, where alpha is defined as the density times the acceleration due to gravity, and this divided by the atmospheric pressure.

00:31 So, to solve this problem, to show this expression, we know that the incremental version of the pressure minus the atmospheric pressure, which is the density times acceleration due to gravity, times the height, we can put the differential of this as just minus the density, the acceleration due to gravity, and this is the differential of the height.

00:58 We assume that the density of air is proportional to the pressure, or we can say that the pressure divided by the density is equal to the atmospheric pressure divided by the initial density.

01:12 So combining these two equations, we have the following.

01:15 Then we will have that the differential in the pressure can be written as minus the pressure times the density, the initial density, divided by the atmospheric pressure, these times acceleration due to gravity, and this.

01:28 This times the differential in the altitude.

01:32 Now we separate the variables from here.

01:35 So we can have that the differential in pressure, divided by the pressure, is equal to minus the acceleration due to gravity.

01:47 The initial density, this divided by the atmospheric pressure, and this times the differential in the height.

01:54 Now we integrate both sides of this expression.

01:56 So for the left side, we have have the integral from the atmospheric pressure to the pressure p of the differential in pressure divided by the pressure.

02:08 And then this is equal to minus the acceleration due to gravity.

02:12 The density, row sub zero.

02:15 This divided by the pressure p sub zero, and the integral from zero to y of the differential in y...

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