Develop an expression for the environmental lapse rate in relation to M, g, R, and ?.
What will be an ideal response?
The environmental lapse rate dT/dz characterizes the local variation of temperature with elevation in the earth’s atmosphere. Atmospheric pressure varies with elevation according to the hydrostatic formula,
where M is molar mass, ? is molar density, and g is the local acceleration of gravity. Assume that the atmosphere is an ideal gas, with T related to P by the polytropic formula:
First use the equation to start:
Differentiate this equation with respect to T:
Substituting the given equation for dP/dz and combining the terms reduces the equation to
We know that 
and substituting this allows the equation to be reduced to
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