Altitude and air density

sgstudent

As we go higher up a mountain, the air pressure decreases. But why would that cause the air density to decrease? Since pressure is hpg, so if h decreases pressure decreases but what causes the air density to decrease as well?

Thanks for the help :)

A.T.

Gases are compressible. More pressure means they are compressed into a smaller volume. Liquids not so much. Water has about the same density in a column.

sgstudent

Oh so at the surface of the earth, because the pressure is greater so it is more compressed while higher up the pressure is less so the compression is smaller?

But then again now with only one formula, P=hpg now that h and p decrease so the pressure will decrease disproportionately?

SteamKing

It's a direct consequence of PV = nRT.

sgstudent

How would you apply that formula here?

Thanks

cepheid

If you define N as number of particles (as opposed to number of moles), the ideal gas law is PV = NkT where k is the Boltzmann constant, a fundamental physical constant. Divide both sides by V and you get

P = nkT

Where n = N/V is the number of particles per unit volume. Multiply n by m, the mass per particle, and you get the mass per unit volume (aka density) rho. Hence n = rho/m and the ideal gas law becomes

P = (rho/m)kT

For air, there is more than one type of particle so m is a weighted average of the masses of the different molecules.

sgstudent

Oh so from this equation how will we show that the density and pressure changes with altitude?

cepheid

This equation shows that P is proportional to rho so that if P decreases at a constant T, then rho must as well. However, even if T is not constant, as long as it decreases more slowly than P does, then the reduction in pressure leads to a reduction in density. This is true in the troposphere, where temperature decreases linearly with altitude, but pressure decreases as a power law.