Question

A lead sphere has a mass of \(1.20 \times 10^{4} \mathrm{~g},\) and its volume is \(1.05 \times 10^{3} \mathrm{~cm}^{3}\). Calculate the density of lead.

Short answer

The density of the lead is approximately \(11.4 \mathrm{g/cm^{3}}\)

Step-by-step solution

Identify the given data

The mass \(m\) of the lead sphere is \(1.20 \times 10^{4} \mathrm{~g}\) and the volume \(V\) is \(1.05 \times 10^{3} \mathrm{~cm}^{3}\).

Apply the formula for density

The formula to calculate density \(D\) is \(D = \frac{m}{V}\). The inputs for this formula are obtained from the data given in the exercise.

Calculate the Density

Replace the values of mass and volume in the formula: \(D = \frac{1.20 \times 10^{4} g}{1.05 \times 10^{3} cm^{3}}\).

Simplify

Calculate the fraction to find the density. The answer will be approximately \(11.4 \mathrm{g/cm^{3}}\)

Key concepts

Density Formula

The concept of density is fundamental in understanding how matter is composed and distributed. Density is simply how much mass is contained in a given volume. The density formula is expressed as \( D = \frac{m}{V} \) where \( D \) represents density, \( m \) is the mass of the substance, and \( V \) is the volume it occupies.

When using the density formula, it's crucial to pay attention to the units of mass and volume. They should be compatible for the result to be accurate and meaningful. Common units for mass include grams (g) or kilograms (kg), and for volume, cubic centimeters (\rm{cm^3}\rm{}) or liters (l) could be used. Converting between units might be necessary to ensure the formula applies correctly.

To dive a bit deeper, if we have a substance with a high mass and a small volume, it will possess a high density, indicating that its particles are closely packed together. Conversely, if an object has a low mass relative to its volume, its density will be low, implying that the particles are more spread out. This is an essential concept in various scientific disciplines, such as physics, chemistry, and material science.

Lead Sphere Density

When we specifically look at a lead sphere's density, we are dealing with a material known for its high density. This characteristic means that despite its size, a lead object will be relatively heavy. For instance, the density of lead can be used to determine if an object is made of pure lead or only lead-coated.

The exercise provided illustrates this point with a lead sphere's mass and volume being given. By applying the density formula, the high numerical value of the density (in the case of lead, around \(11.4 \mathrm{g/cm^3}\)) confirms its dense nature. This value is historically significant as well because lead has been used for various purposes from ancient times to modern-day applications, including batteries, soundproofing, and radiation shielding, due to its high density.

Understanding the density of lead is also beneficial for those pursuing work in materials science or any field that involves analyzing substances and their properties.

Mass and Volume

Mass and volume are the two critical components that make up the density formula. Mass is a measure of how much 'stuff' or matter an object contains and is generally measured in grams or kilograms. Volume, on the other hand, is the amount of space that matter occupies, commonly expressed in cubic centimeters or liters.

An important aspect to remember is that mass is an intrinsic property of matter which means it does not change with the object's location. However, volume can change depending on various conditions such as temperature and pressure. For instance, a substance may expand and take up more volume when heated or contract and occupy less volume when cooled.

In our exercise scenario, the mass of the lead sphere is given in grams and the volume in cubic centimeters, perfectly setting up for use in the density formula. Always ensure that the units you are working with are consistent across the measurements for accurate calculations. When students grasp these concepts, they are well on their way to successfully solving a wide range of problems in physics and engineering.