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Example 23.2 Adiabatic temperature gradient Find the value of dT/dz when energy flow is dominated by convection near the ground.

solution

We don't have to consider the whole atmosphere to do this. We only have to look at a small volume, or parcel, of air, and see how it behaves. Suppose we have a parcel of air with volume V0 rising from just above the ground, where the temperature is T0 and the pressure is P0. As the parcel of air rises, its tem perature, pressure and volume are T, P and V. For an adiabatic process, these quantities are related by where y is the ratio of the specific heats at constant pressure and constant volume. For a monatomic gas, y = 5/3. For a diatomic gas, y = 7/5. Since the Earth's atmosphere is mostly N2 and O2, we use 7/5. (If there is a lot of water vapor present, y will be different.)

We can eliminate the volume using the ideal gas law

where N is the number of molecules in the parcel of air. This gives p1-y ty = pi-y Ty

For convenience, we will call this constant quantity C. Solving for T gives

To obtain dT/dz, we differentiate both sides with respect to z, giving yTy-1 a f J=ciy -1 )py-2 a

dP J

dz J

where we have used the hydrostatic law dP/dz = -pg. Solving for dT/dz, and substituting for C, we have dT dz

T0 P0

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