Calculating temperature given altitude

[moonman239]

Homework Statement

I am trying to figure out how to calculate the temperature at any given altitude in the troposphere.

Homework Equations

Let P=air pressure, V=volume of air above a given altitude, n=number of moles of gas, R=the ideal gas constant, T=temperature, and h=height above sea levelX₁ and X₂ = the value of X at geographic point 1 and point 2.

PV=nRT (the ideal gas law)
Volume of a sphere = 4/3*pi*radius³
Surface area of a sphere = 4*pi*radius²

The Attempt at a Solution

P₁/ P₂ = V₁nRT₁ / V₂nRT₂.

Now, we don't need to know what n and R are, because they are canceled out of the equation. So ignore them. We are now left with P₁/P₂ = V₁/V₂ * T₁/T₂.

V = 4/3*pi*radius of troposphere - (4/3 * pi * (6371.5+h))
V₁ / V₂ is pretty obvious and can't be simplified.

 

[JohnRC]

The main problem that I see with this is that fall off in pressure is governed more by the barometric equation p= p0 exp (–h/A) where A is roughly 7 km, than by an increase in available volume at greater height.

I do not know how meteorologists do this calculation, but the recommendation of the ICAO Standard atmosphere is that the temperature will fall at a constant rate of 0.0065 °C/m up to 11 km altitude in the temperate zone. Obviously that is an "other things being equal" calculation that cannot take account of local atmospheric conditions.