A lead sphere of diameter \(48.6 \mathrm{~cm}\) has a mass of \(6.852 \times 10^{5} \mathrm{~g} .\) Calculate the density of lead.

Understanding the density of materials is a fundamental concept in chemistry and physics. Density is defined as the mass of a substance divided by its volume. This intrinsic property helps us understand how tightly matter is packed together. Specifically, when talking about a lead sphere's density, we're referring to how much mass is contained within a given volume of the sphere shaped by lead.

In the example provided, we're asked to calculate the density of a lead sphere with a known diameter and mass. The first step is to find the radius and then the volume, which we need to calculate the sphere's density. Knowing that lead is a dense metal, we expect our density value to be high compared to less dense materials like plastic or wood.

Volume measures how much space an object occupies, and it's essential for various scientific calculations. For a sphere, volume is determined by the formula \( V = \frac{4}{3}\pi r^{3} \), where \( r \) is the radius of the sphere. To put it simply, the larger the radius, the more space the sphere takes up, leading to a larger volume.

In our problem, after finding the radius by halving the diameter, we use the formula to calculate the volume. This step is critical because volume directly impacts the calculation of density, which is our ultimate goal in the problem. Remember, since the radius is cubed in the formula, even small changes in the radius can significantly affect the volume, and thus the density.

The mass-volume relationship is key to understanding density and how it relates to the intrinsic properties of materials. This fundamental concept simply states that density equals mass divided by volume \( (p = \frac{mass}{volume}) \) and represents how much 'stuff' is in a given space.

When applied to the lead sphere example, we take the mass (in grams) and volume (in cubic centimeters) to calculate density. In this case, given the high atomic weight of lead and its packed atomic structure, we expect a significant amount of mass per unit of volume, indicating a high density. This property is why lead is frequently used in applications requiring materials with high-density properties, like radiation shielding or weights.