Pressure as a function of altitude

Gases and liquids exert pressure on the surfaces of solids that are immersed in them, simply the force of the atoms bouncing off of the solid surface. The pressure gets lower as you climb higher in the atmosphere. As we ascend, we decrease the amount of fluid that is above us, decreasing the pressure that we feel.

Scuba divers know that diving 10 m deep increases the pressure by about 1 atm. Each 10 m of depth is the same 1 atm pressure increase: descending from 30 to 40 m would increase the pressure by the same 1 atm as descending from 0 to 10 m. We say that pressure is linear with depth (Fig. 5.2). The pressure can be calculated as

10 m where we are using variable z to denote the vertical position, as before, with positive numbers upward, as before. So a depth is a negative height. At the water surface, z = 0, and we have 1 atm pressure from the weight of the atmosphere. The increase in pressure with depth, from the weight of the water, is linear with depth below the surface because the factor ā€”1 atm/10 m of height is constant.

The pressure in the atmosphere is nonlinear with altitude, in that a climb of 1 m at sea level changes the pressure much more than 1 m of climb up at the tropopause. The equation to describe pressure as a function of height in the atmosphere is based on the exponential function, which is a number called e raised to a power. The value of e is approximately 2.71828... The exponential function was invented by bankers to calculate compound interest for bank accounts. The interest that a bank account pays is proportional to the amount of money in the account. The mathematical formula for the amount of money in the bank at any given time, if the payout from interest is continually deposited into the account to start earning interest itself, is the exponential function

Balance(t) = Balance (initial) ā–  ekt where Balance(t) is the bank balance at any time t in years, and k is an interest rate, like 10% per year or 0.1 year-1.

The exponential function comes up time and again in the natural sciences. Population growth and radioactive decay are two examples. In each case, the rate of change of the variable depends linearly on the value of the variable itself. Population growth is driven by the number of babies born, which depends on the number of potential parents to beget them. The rate of radioactive decay that you would measure with a Geiger counter depends on the number of radioactive atoms present. The growth rate of your bank account depends on its size.

The multiplier in the exponent describes the relative rate of change of the variable, in this case, the interest rate. If the exponent is positive, as for population, the growth accelerates as it progresses (Fig. 5.3). One could reasonably call this type of growth an explosion. For decay, the exponent is negative, and the quantity of radioactive atoms gets ever closer to zero with time, but mathematically never gets there.

The atmospheric pressure varies as a function of altitude according to an exponential decay type of equation in height

The height z is zero at sea level, leaving us with e0 which equals 1, so the pressure at sea level is 1 atm. At an altitude of 8 km, pressure is lower than at sea level by a factor of eā€”1 or 1/e, about 37%. We call that altitude the e-folding height. Most of the mass of the atmosphere is contained in the e-folding height. In fact, if the entire atmosphere were to be at 1 atm pressure, instead of smeared out in a decaying exponential function, it would fit into one e-folding height exactly. If we were tracking the decay of a radioactive chemical with time, the scaling factor in the exponential would be called an e-folding time. This quantity is similar to a half-life for radioactive decay but rather than decaying to half the initial quantity, we're waiting until we're 37%

Fig. 5.3 The exponential functions ex and e-x. For ex, the growth rate of the function is proportional to its value. Examples of this type of behavior include interest on a bank account and population growth. For e-x, the value of the function decays proportionally to its value. Radioactive decay does this, as does pressure as a function of altitude in the atmosphere.

Fig. 5.3 The exponential functions ex and e-x. For ex, the growth rate of the function is proportional to its value. Examples of this type of behavior include interest on a bank account and population growth. For e-x, the value of the function decays proportionally to its value. Radioactive decay does this, as does pressure as a function of altitude in the atmosphere.

lower than the original. It takes 44% longer to decay to 1/e than it does to 1/2 of the original number of atoms.

From the appearance of the exponential function in the equation for pressure, you could probably guess that the rate of change of pressure with altitude must depend on the pressure itself in some way. This would be astute. The rate of change of pressure depends on pressure because at high pressure, gas is compressed, and so a climb of 1 m through gas at high pressure would rise above more molecules of gas than would a climb through a meter of gas at low pressure. Imagine a wall made of compressible bricks (Fig. 5.4). A row of bricks is thinner at the bottom of the wall because they are compressed. Batman climbing up the wall would pass more bricks per step at the bottom than at the top. For incompressible bricks (the normal kind), the rows are all of the same height and the mass of the wall above you is a linear function of height.

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Solar Panel Basics

Solar Panel Basics

Global warming is a huge problem which will significantly affect every country in the world. Many people all over the world are trying to do whatever they can to help combat the effects of global warming. One of the ways that people can fight global warming is to reduce their dependence on non-renewable energy sources like oil and petroleum based products.

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Responses

  • jari
    How pressure varies with altitude exponential?
    8 years ago
  • don
    What is the function of the gases at altitude?
    11 months ago

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