The ideal gas equation:

$\overline{){\mathbf{P}}{\mathbf{V}}{\mathbf{=}}{\mathbf{n}}{\mathbf{R}}{\mathbf{T}}}$

The density of an ideal gas:

$\overline{){\mathbf{\rho}}{\mathbf{=}}\frac{\mathbf{n}\mathbf{m}}{\mathbf{V}}}$

Weight of the hot air = m_{h}g = ρ_{h}V_{1}g

Total weight, W, of the ballon including the hot air inside:

W = m_{b}g + ρ_{h}V_{1}g

The Buoyant force on the balloon is equal to the weight of cold air displaced:

Hot air balloons float in the air because of the difference in density between cold and hot air. Consider a balloon in which the mass of the pilot basket together with the mass of the balloon fabric and other equipment is *m*_{b}. The volume of the hot air inside the balloon is *V*_{1} and the volume of the basket, fabric, and other equipment is *V*_{2}. The absolute temperature of the hot air at the bottom of the balloon is *T*_{h} (where *T*_{h} > *T*_{c}). The balloon is open at the bottom, so that the pressure inside and outside the balloon is the same. Assume that we can treat air as an ideal gas. Use *g* for the magnitude of the acceleration due to gravity. For the balloon to float, what is the minimum temperature *T*_{min} of the hot air inside in terms of the variables given?

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