A spherical helium balloon of 30 ft in diameter is at ambient T and P, 60 F and 14.69 psia. How

step 1 So in this problem we are given temperature is equal to 60 frame height which is equal to 519 .7

A spherical helium balloon of 30 ft in diameter is at...

A spherical helium balloon of $30 \mathrm{ft}$ in diameter is at ambient $T$ and $P, 60 \mathrm{F}$ and 14.69 psia. How much helium does it contain? It can lift a total mass that equals the mass of displaced atmospheric air. How much mass of the balloon fabric and cage can then be lifted?

Key Concepts

Buoyancy and Archimedes' Principle

Buoyancy is the upward force exerted on an object immersed in a fluid, equivalent to the weight of the fluid displaced by the object. Archimedes' Principle underpins this concept, stating that the buoyant force is equal to the weight of the displaced fluid. For a helium balloon, the lifting capability is determined by the difference between the weight of the air displaced and the weight of the helium, allowing one to deduce how much additional mass (e.g., from the balloon’s fabric and cage) can be lifted.

Density Differences Between Gases

The difference in density between helium and atmospheric air is critical in calculating the net lift generated by the balloon. While helium is much lighter than air, the actual lift is derived from the difference in mass between the displaced volume of air and the helium itself. This concept is essential in determining the balloon’s payload capacity, ensuring that the lighter-than-air lift provided by helium overcomes the weight of the enclosed gas and any attached structures.

Spherical Geometry

This concept involves understanding how to calculate the volume of a sphere using its diameter or radius. In practical applications such as the helium balloon problem, the formula V = (?/6)·d³ (or equivalently V = (4/3)?·r³) is used to determine the total volume of gas enclosed, which is a crucial parameter in further calculations regarding gas properties and buoyancy.

Ideal Gas Law

The Ideal Gas Law, expressed as PV = nRT, relates the pressure, volume, temperature, and the amount of gas. In the context of the helium balloon, this law is used to relate the ambient conditions of temperature and pressure to the number of moles (and thereby the mass) of helium contained within the balloon. It helps understand how the gas behaves under different environmental conditions.

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