Volume and Capacity

FIELDS OF STUDY: Classical Mechanics; Fluid Mechanics

ABSTRACT: Volume is the three-dimensional space occupied by an object. The internal volume of a container determines its capacity. The volume of various three-dimensional shapes can generally be calculated using simple formulas based on the area of one side multiplied by the height of the shape. Volume is a fundamental parameter for understanding density and other physical properties.

PRINCIPAL TERMS

• buoyancy: upward force exerted by a fluid on a submerged or floating object.
• density: the amount of mass per unit volume.
• ideal gas law: a law that describes the relationship between the temperature (T), pressure (p), volume (V), and number of particles (n) of an ideal gas, written as pV = nRT.
• pressure: the force exerted per unit area.
• surface area: the total area of the outward-facing surface of a three-dimensional object.
• temperature: the average kinetic energy of particles that make up a substance.
• wavelength: the distance between crests of a wave; all electromagnetic radiation is transmitted as waves, with longer wavelengths corresponding to lower frequencies and less energy and vice versa.

A Three-Dimensional Measurement

The volume of an object is the amount of three-dimensional space it occupies. Volume is measured in units of cubic distance, such as cubic meters or cubic feet. It is a necessary parameter for determining spatial relationships. When volume is used to determine storage capability, it is often called capacity.

Volume is also necessary for determining the density of an object. Density quantifies how much matter is contained in a certain area. High-density items, such as solid lead, have lots of atoms or molecules in a very small space. Loosely arranged particles, such as oxygen molecules in the atmosphere, have a low density.

The dynamics of fluids (gases and liquids) are distinct from those of solids. Fluid dynamics are highly contingent on volume and density, making these two measurements important when engineering ways to contain, transfer, and use fluids.

Fluids, Buoyancy, and the Ideal Gas Law

All fluids have buoyancy, which is a force that counters gravity and pushes upward on submerged objects. An object’s ability to float depends on the densities of both the object and the fluid. If the object is less dense than the fluid in which it is submerged, it will float.

All fluids exert pressure on their surroundings, whether those surroundings are the container that holds the fluid or an object submerged in it. Pressure is distinct from buoyancy; rather than pushing upward, for example, it pushes inward on a submerged object from all sides. The strength of this pressure is influenced by the density of the fluid. Denser fluids tend to exert more pressure.

The two types of fluids have slightly different relationships to volume and pressure. A liquid is able to vary its shape but not its volume, while a gas is able to vary both its shape and its volume. This means that a gas will expand to fill whatever container it is in. The unique ability of a gas to vary its density is described by the ideal gas law. This law relates temperature (T) in kelvins (K), pressure (p) in pascals (Pa), volume (V) in cubic meters (m³), and number of gas particles (n) in moles (mol). The law is written as

pV = nRT

where R is the gas constant, equal to 8.3144621 joules per mole-kelvin (J/mol·K).

The ideal gas law assumes that all the molecules of a gas bounce around and off of each other and the container with perfect elasticity, completely uninfluenced by the attractive forces that hold molecules of solids and liquids together. While there is no such thing as a truly ideal gas, the relationships the law describes are still useful:

• Pressure is directly proportional to the number of gas particles and the temperature of the gas. Tightly packed particles press outward with more force than do loosely packed particles. Similarly, the temperature of a gas is a measure of the kinetic energy of the particles. Heating up a gas causes the particles to move more quickly. When they bounce off each other and their container, they exert more force.
• Pressure is inversely proportional to volume. If a gas squeezes into a smaller space, its particles will bounce off each other and the container more frequently, thus increasing the pressure.

Calculating Capacity

The volumes of common three-dimensional shapes can be determined using simple measurements and the following formulas:

• The volume of a cylinder is equal to pi (π) times the radius of one circular surface squared (r²) times the height of the cylinder (h):

V_c = πr²h

• The volume of a rectangular prism, such as a box, is equal to the width of one side (w) times the length of that side (l) times the height of the prism (h):

V_r = wlh

• The volume of a sphere is equal to four-thirds times pi (π) times the radius of the sphere cubed (r³):

Using these equations, it is possible to calculate the capacity of many everyday objects.

Sample Problem

A cylindrical oil-based home heating tank is 3 meters tall. The radius of each circular end of the cylinder is 0.5 meter. What is the heater’s capacity in cubic meters?

Answer:

Plug the values above into the formula for the volume of a cylinder, and solve:

V_c = πr²h

V_c = π × (0.5 m)² × 3 m

V_c = 2.36 m³

The volume of the tank is approximately 2.36 cubic meters. Note that for many three-dimensional shapes, calculating the volume is as simple as calculating the area of one end and multiplying this value by the height of the shape. In the above example, the quantity πr² is simply the area of one of the circles that serves as a base for the cylinder.

Surface Area vs. Volume

The surface area of a three-dimensional shape bears a unique relationship to its volume. Surface area increases as a squared value (m²), whereas volume increases as a cubed value (m³). Thus, increasing the overall size of a shape causes its volume to increase much more quickly than its surface area. This is significant because surface area represents the area of interaction between an object and its environment. A very large object has a proportionately very small surface area, meaning the environment’s influence on its interior may be slow. For example, an elephant is very large but has relatively little surface area, so its skin takes a long time to heat up in the sun and a long time to cool down. Because of this, elephants have evolved large ears to provide a surface that they can use to regulate their temperature. Conversely, very small objects have a very high surface-area-to-volume ratio. This is why babies are more susceptible than adults to chills and overheating.

Volume in Everyday Life

Volume quantifies space, an important and limited resource in everyday life. It is also a basic measurement for determining other characteristics of objects. Volume and density are necessary measurements for determining the behavior of a gas and the force it exerts on its environment. Furthermore, the mathematical ratio of surface area to volume plays a major role in biology and chemistry by influencing the rate of interaction between an object and its environment.

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