#$2.$ Let's compare a hydrogen balloon to a helium balloon, in terms of ability to lift a weight. Intuitively, since helium has twice the density, we might expect it to lift only half as much weight as a hydrogen balloon. Is this true? Let's do the calculation:

a$)$ First, consider a spherical hydrogen balloon with a radius of $3\mathrm{.}0$ meters. How much hydrogen $($in $kg)$ do you need in order to inflate the balloon to $1$ atm pressure at $25C?($assume ideal gas behavior, remember it's $H_2,$ not $H.$

b$)$ Using the density of air of $1\mathrm{.}22k\frac{g}{m^3},$ calculate the payload mass that the balloon can lift. Hint: neglect the mass of the balloon material itself. To lift the payload, the density of payload plus the hydrogen in the balloon must be less than the density of the surrounding air.

c$)$ Recalculate the payload mass using helium instead of hydrogen in the balloon.

d$)$ Recalculate the payload mass using methane in the balloon.

e$)$ Now, go back to the original question: does switching from hydrogen to helium cut the payload in half?