Pressure versus height: barometric formula

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Pressure versus height: barometric formula
Pressure versus height: barometric formula

Many people know that as altitude increases, air pressure decreases. Consider the question of why air pressure decreases with height, give the formula for the dependence of pressure on height, and also consider an example of solving the problem using the resulting formula.

What is air?

Air is a colorless mixture of gases that makes up our planet's atmosphere. It contains many different gases, the main ones being nitrogen (78%), oxygen (21%), argon (0.9%), carbon dioxide (0.03%) and others.

From the point of view of physics, the behavior of air under existing conditions on Earth obeys the laws of an ideal gas - a model according to which the molecules and atoms of a gas do not interact with each other, the distances between them are huge compared to their sizes, and the speed of movement at room temperature temperatures are about 1000 m/s.

Air pressure

A device for measuring pressure

Considering the question of the dependence of pressure on altitude, you should figure out what representsis the concept of "pressure" from a physical point of view. Air pressure is understood as the force with which the air column presses on the surface. In physics, it is measured in pascals (Pa). 1 Pa means that a force of 1 newton (N) is applied perpendicularly to a surface of 1 m22. Thus, a pressure of 1 Pa is a very small pressure.

At sea level, the air pressure is 101,325 Pa. Or, rounding off, 0.1 MPa. This value is called the pressure of 1 atmosphere. The above figure says that on a platform of 1 m2 air presses with a force of 100 kN! This is a great force, but a person does not feel it, since the blood inside him creates a similar pressure. In addition, air refers to fluid substances (liquids also belong to them). And this means that it exerts the same pressure in all directions. The last fact suggests that the pressure of the atmosphere from different sides on a person is mutually compensated.

Dependence of pressure on altitude

Change in pressure with height

The atmosphere around our planet is held by the earth's gravity. Gravitational forces are also responsible for the drop in air pressure with increasing altitude. In fairness, it should be noted that not only the earth's gravity leads to a decrease in pressure. And also lowering the temperature also contributes.

Since air is a fluid, then for it you can use the hydrostatic formula for the dependence of pressure on depth (height), that is, ΔP=ρgΔh, where: ΔP is the amount of pressure changewhen changing height by Δh, ρ - air density, g - free fall acceleration.

Given that air is an ideal gas, it follows from the ideal gas equation of state that ρ=Pm/(kT), where m is the mass of 1 molecule, T is its temperature, k is Boltzmann's constant.

Combining the above two formulas and solving the resulting equation for pressure and height, the following formula can be obtained: Ph=P0 e-mgh/(kT) where Ph and P0 - pressure at height h and at sea level, respectively. The resulting expression is called the barometric formula. It can be used to calculate atmospheric pressure as a function of altitude.

Sometimes for practical purposes it is necessary to solve the inverse problem, that is, to find the height, knowing the pressure. From the barometric formula, you can easily get the dependence of altitude on the pressure level: h=kTln(P0/Ph)/(m g).

Example of problem solving

The Bolivian city of La Paz is the "highest" capital in the world. From various sources it follows that the city is located at an altitude of 3250 meters to 3700 meters above sea level. The task is to calculate the air pressure at the altitude of La Paz.

La Paz, Bolivia

To solve the problem, we use the formula for the dependence of pressure on height: Ph=P0e-mg h/(kT), where: P0=101 325 Pa, g=9.8 m/s2, k=1.3810-23 J/K, T=293 K (20 oC), h=3475 m (average between 3250 m and3700 m), m=4, 81710-26 kg (taking into account the molar mass of air 29 g/mol). Substituting the numbers, we get: Ph=67,534 Pa.

Thus, the air pressure in the capital of Bolivia is 67% of the pressure at sea level. Low air pressure causes dizziness and general weakness of the body when a person climbs into mountainous areas.

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