Calculating Air density as a function of height?

[Noone1982]

Is it possible to calculate the air density as a function of height? We know that density is given as,

$$p\; =\; \frac{M}{V}$$

And that pressure is given as,

$$P\mbox{re}ssu\mbox{re}\; =\; pgh$$

But I am failing to see the connection to combine the two to get air density as a function of height. Any insight?

 

[radou]

It's a miracle that I remember this from my hydrology course (the most boring one): $$p(h) = p_{0}e^{-\frac{gh}{RT}}$$. This is the expression for air pressure at a height h. $$p_{0}$$ is the air pressure at the sea level, T is the average temperature at the height h, and R is a gas constant for dry air.

 

[OlderDan]

Is it possible to calculate the air density as a function of height? We know that density is given as,

$$p\; =\; \frac{M}{V}$$

And that pressure is given as,

$$P\mbox{re}ssu\mbox{re}\; =\; pgh$$

But I am failing to see the connection to combine the two to get air density as a function of height. Any insight?

The pressure of the atmosphere is the weight of the air above per unit area. If you move up slightly, the weight decreases by an amount that depends on the density and change in height. You can assume the density is proportional to the pressure. Your equation

$$P\mbox{re}ssu\mbox{re}\; =\; \rho gh$$

is the change in pressure for a small change in height for which the density may be assumed constant.