Hydrostatic balance and lapse rate for air subject to gravity: assume that the atmosphere has a uniform composition, with molecules having mass m, and that it can be treated as an ideal gas. also assume that the acceleration of gravity can be treated as independent of height. (a) for a small volume of air at equilibrium, of height dz, a vertical difference in pressure on the top and bottom face provides a supporting force to balance the weight of the gas within the volume. based on this idea, derive a relation for dp/dz, showing that it is proportional to p/t (b) the temperature variation with height is established for the most part by convection which proceeds adiabatically. starting with the first law, for an adiabatic process, consider the vertical displacement of a quantity of air, and find a relation between du and pdv for the gas in this process. replace du with a corresponding relation using c, and use the ideal gas law to express dv in terms of dp and dt. as a result, find an expression for dt/dp. finally, use your result from part (a) to obtain an expression for dt/dz, and show that the result is dt/dz -mg/cp.