(a). Explain why a heavy balloon inflated with helium-4 gives a lower reading on an accurate weighing scale than the same balloon when deflated, despite having a larger mass.
A party balloon of mass 10 g is inflated with helium-4.
(b). Show that, under typical atmospheric conditions, the volume of the balloon V should satisfy
V≳1.0×10−2 m3
if it is to float.
The balloon is initially inflated to a volume V0=2.0×10−2 m3, then released. It floats up gently before coming to rest on the ceiling. Unfortunately, there is a leak, and the balloon slowly deflates through a very small hole of area A.
(c). Show that the root mean-square momentum Prms of a particle in a pure ideal gas of temperature T is
Prms=3mkT,
where m is the particle mass. Hence reason why the typical momentum in any given direction is
P=mkT.
(d). The pressure of a gas can be loosely thought of as the total momentum flowing through unit area per unit time. Making the approximation that all particles travelling in a given direction have momentum P, explain why N, the number of atoms of helium-4 in the balloon, decreases over time as:
dtdN=−mkTpA.
(e). Hence obtain an expression for the rate of change of the volume, dV/dt, in terms of A, k, T, m.
(f). If after 1 hour the balloon has shrunk by 5% in volume, estimate the time until the balloon drops back to the floor, and estimate the area of the leaky hole. How does this compare to the initial surface area of the balloon?
(g). Suggest why the model presented above would break down if the hole was made significantly larger.
(h). The balloon was itself filled from a rigid, pressurised cylinder of helium-4 of fixed volume v. If this cylinder also develops a small leak (of area a), derive an algebraic expression for the gas pressure inside the cylinder as a function of time. Take the initial pressure to be p0. No numerical calculations are needed.