Friday 13 November 2015

ideal gas - Calculating air density lapse with altitude (specifically, pressures)


This might be a bit more of an engineering question, but I'm calculating air density drop-off with altitude, and I'm having some problems calculating the pressure (I'll run through my method). This has been very useful in explaining, but the last bit lost me a little.


So we start with an ideal gas, then:


$$ pV = nRT $$ and using $$\frac{\rho V}{n} = M$$ where M is molar mass, you can calculate density to be:


$$ \rho = \frac{pM}{RT} $$ which implies a solution dependent only on pressure (p) and temperature (T).


Then define temperature using the Universal Standard Atmosphere lapse rate, $$T = T_0 - Lh$$ where L = 0.0065K/m and h is height in metres


Now at this point I'm a bit stuck. Wikipedia suggests the following equation:


$$p = p_0 \left(1 - \frac{L h}{T_0} \right)^\frac{g M}{R L}$$



Which, when calculated, provides values within a 5% tolerance - but where has it come from? I can't find any reference to its source or how it was derived. Can anyone help?




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