# B Density and pressure -- How to calculate atmospheric pressure at higher elevations?

#### LT72884

Ok, we all know that density is mass/volume. So if air is 1.22kg/m3,If we increase pressure, volume will change, therfore density can change.

So how does one calculate density of air at higher elevations. Where i currently live, at 4700ft above sea lvl, im guessing air density is not 1.22kg/m3. So how would i calculate it?

Thanks

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#### Christopher Grayce

The easiest possible way is to assume the atmosphere is an isothermal ideal gas in a constant gravity field. Then the energy of each gas molecule E = 1/2 mv^2 + mgz, where g= 9.8 m/s^2 and z is height above some reference altitude z0, e.g. sea level, where you know a reference pressure, e.g. p0 = 1 atm. Boltzmann statistics gets you the distribution of molecules, therefore the density, therefore the pressure, is then proportional to exp(-E/kT). You don't care about the kinetic energy part, and it separates. You're left with an exponential distribution of altitudes, and all you need to do is get the mass and units straight, and do an integration to establish the constant in front of the exponential in terms of your reference pressure, the average mass of the gas molecules, k and T. Pick some T, e.g. the average temperature of the troposphere, and you're all set.

The most obvious flaws in this are that the atmosphere is not isothermal and that the gravity field isn't constant, but I don't think they would be huge flaws at this level of calculation. Neither T nor g vary much over 5000 ft.

#### gleem

For an ideal gas which air approximates use the ideal gas law

PV = nRT

where P is pressure in atmospheres (or millibars), V is volume in liters, is the number of moles in the volume, R is the universal gas constant = 0.0821 liters-atm /gram-mole-deg-Kelvin. and T is temperature in deg Kelvin ( 273° + °C)

using ρ =m/V with m the mass (in the volume) = gram molecular weight for air x the number of moles.

You get ρ = m×P/RT

from that you can compute the ratio of densities at different pressure and temp.which eliminates the mass and the constant R.

ρel = ρsl ×Pel×Tsl/(Psl×Tel)

You can calculate the pressure at elevation ( see https://www.mide.com/pages/air-pressure-at-altitude-calculator ) or you can get a barometer calibrated at sea level and bring it to you elevation and read the pressure. The pressure at sea level is usually 1 atmosphere (1013.2 millibars) ±2% at 0 °C (273°K).

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#### Chestermiller

Mentor
The barotropic formula gives the vertical pressure variation as $$p=p_0\exp{\left(-\frac{Mg}{R}\int_0^z{\frac{dz'}{T(z')}}\right)}$$where z is the altitude, $p_0$ is 1 bar, M is the molecular weight of air, g is the acceleration of gravity (essentially constant up to your altitude), T(z) is the absolute temperature at altitude z, and R is the universal gas constant. If z is in meters, the temperature in the troposphere decreases with altitude as $$T(z)=288.15-6.5z/1000$$ The density at altitude z is calculated from $$\rho=\frac{p(z)M}{RT(z)}$$

#### LT72884

Those are interesting formulas. Now i just need to find all the other numbers and calculate density in my area haha.

I know pressure can effect volume, but where is that in the basic density equation? What substitutes in for volume that has P involved?

Thanks for all the jelp

"Density and pressure -- How to calculate atmospheric pressure at higher elevations?"

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