A spherical balloon filled with helium gas must lift a \(1.50-\mathrm{kg}\) scientific payload off the ground, and the mass of the empty balloon is \(0.500 \mathrm{~kg}\).

(a) What is the minimum diameter of the balloon? 

\((b)\) A helium shortage forces the scientists to consider using nitrogen gas in the balloon. What minimum balloon diameter is needed if nitrogen gas is used? The mass densities are \(1.286 \mathrm{~kg} / \mathrm{m}^{3}\) for air, \(0.179 \mathrm{~kg} / \mathrm{m}^{3}\) for helium gas, and \(1.251 \mathrm{~kg} / \mathrm{m}^{3}\) for nitrogen gas.