When surrounded by air at a pressure of 1.0 ×10^5 N / m^2, a baskethall has a radius of 0.12 m,

step 1 Well, the volume of the ball is given by volume v is equal to 4 divided by 3, pi r cube, by r cu

When surrounded by air at a pressure of 1.0 ×10^5 N / m^2, a...

When surrounded by air at a pressure of $1.0 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$, a baskethall has a radius of $0.12 \mathrm{m},$ Compare its volume at this condition with the volume that it would have if you took it $15 \mathrm{m}$ below the water surface, where the pressure is $2.5 \times 10^{3} \mathrm{N} / \mathrm{m}^{2}$. What assumptions did you make?

Key Concepts

Boyle's Law

This principle states that for a fixed amount of an ideal gas kept at constant temperature (an isothermal process), the pressure and volume are inversely proportional. In other words, if the external pressure increases, the volume of the gas decreases proportionally, and vice versa.

Ideal Gas Law

The ideal gas law, given by PV = nRT, relates the pressure, volume, number of moles, and temperature of a gas. In many practical situations, especially when temperature and the amount of gas remain constant, this law simplifies to Boyle's law. It is essential for understanding how a gas responds to changes in environmental pressure.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the gravitational force acting on the fluid. It increases with depth below the surface. This concept is important when comparing conditions at different depths, as the pressure experienced by an object in a fluid will vary with its depth.

Spherical Geometry

When dealing with objects that can be approximated as spheres, the volume is determined by the formula V = (4/3)?r³. This geometric relationship is crucial for calculating the volume changes when the dimensions of a spherical object are altered, either through pressure changes or deformation.

Assumptions in Gas Behavior

In many problems involving gas under pressure, several key assumptions are made: the gas behaves ideally (meaning intermolecular forces and the volumes occupied by the gas molecules themselves are neglected), the process occurs isothermally (constant temperature), and the shape of the object (like a sphere) remains geometrically unchanged. These assumptions simplify the analysis but must be recognized because they may limit the accuracy of the model in real-world scenarios.

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