Volume of Water in a Sphere: Understanding Density, Temperature, and Pressure

In this article, we delve into the fascinating world of calculating the volume of water in a spherical container. Specifically, we will explore the volume of water that can be held by a hollow sphere with a radius of 10 cm, and how various factors such as temperature, pressure, and density influence the actual volume of water it can contain.

Calculating the Volume of a Sphere

First, let’s start with a simple calculation. The formula for the volume of a sphere is given by:

Formula: Volume of a Sphere

The volume of a sphere with radius ( R ) is expressed as:

[ V frac{4}{3} pi R^3 ]

Given that ( R 10 ) cm, we convert it to decimeters (dm) since ( 10 ) cm is equivalent to ( 1 ) dm.

Step-by-Step Calculation

1. Convert the radius from cm to dm: R 10 text{cm} 1 text{dm}. 2. Substitute ( R 1 ) dm into the formula:

[ V frac{4}{3} pi (1)^3 text{dm}^3 approx 4.18879 text{dm}^3 ]

Since ( 1 text{dm}^3 1 text{liter} ), the volume of the sphere is approximately ( 419 ) liters.

Conversion to Gallons

If you prefer using gallons, the volume can be converted using the following conversion factor:

[ 1 text{liter} frac{1}{3.7854} text{gallons} approx 0.264172 text{gallons} ]

Therefore, the volume of water that a sphere with a radius of 10 cm can hold is approximately:

[ 419 text{ liters} div 3.7854 approx 111 text{ gallons} ]

Factors Affecting the Volume of Water in a Sphere

The volume of water a sphere can hold is not just dependent on its physical dimensions but also influenced by other environmental factors such as temperature and pressure. Understanding these factors is crucial for accurate measurements and calculations.

Density at Various Temperatures

Water density varies with temperature. At ( 25^circ text{C} ) and atmospheric pressure, the density of water is approximately ( 997 text{kg/m}^3 ).

Calculating Mass from Volume

Using the volume of the sphere (( 419 ) liters) and the density of water, we can calculate the mass of water it can hold:

[ text{Mass} text{Density} times text{Volume} ]

Converting ( 419 ) liters to cubic meters (( 1 text{liter} 0.001 text{m}^3 )):

[ 419 times 0.001 text{m}^3 0.419 text{m}^3 ]

Then,

[ text{Mass} 997 text{kg/m}^3 times 0.419 text{m}^3 approx 418.123 text{kg} approx 604.31 text{g} ]

Pressure and Volume Relationship

A higher pressure leads to a lower volume of water, while a higher temperature leads to a higher volume. This relationship is described by Boyle’s Law and Charles’s Law, respectively. Boyle’s Law states that the pressure of a gas is inversely proportional to its volume, while Charles’s Law states that the volume of a gas is directly proportional to its temperature.

Conclusion

Understanding the volume of water in a sphere is not only a matter of simple geometry but also involves considering the complex interplay of environmental factors. By calculating the volume using the given formula and applying the correct conversions, we can accurately determine the amount of water a sphere with a 10 cm radius can hold. Furthermore, recognizing the influence of temperature and pressure on water’s volume is vital for various applications, from practical everyday uses to scientific and industrial purposes.