Model the rise of a balloon of mass M, which rises from the ground into the stratosphere [149]. At ground level, the balloon has a volume V₀, which is much smaller than the volume V_max it has when it is fully inflated. Nonetheless, the volume V₀ is sufficient to lift the payload. The ballom is filled with helium, which is considered an ideal gas. Use Archimedes’ principle given in exercise 10.4.5 and the model of Earth’s atmosphere established in exercise 10.5.
a) Find the maximum height z_max reached by the balloon.
b) Show that Archimedes’ force F_A exerted on the balloon is uniform as long as the balloon is not fully inflated

Data from exercise 10.5

Model the temperature profile T(z) of the Earth’s atmosphere as a function of height z. Ignore winds, clouds and the many effects due to the presence of moisture in the air and treat the air as an ideal gas. Assume that the Earth’s atmosphere reaches an equilibrium mostly by matter transfer. Suppose that when an air mass moves up or down, it does so adiabatically because the air thermal conductivity is small

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h h Oil Water Q