Question

One of the particles in an atom is the proton. A proton has a radius of approximately \(1.0 \times 10^{-13} \mathrm{~cm}\) and a mass of \(1.7 \times 10^{-24} \mathrm{~g}\). Determine the density of a proton. (Hint: Find the volume of the proton and then divide the mass by the volume to get the density.) (volume of a sphere \(=\frac{4}{3} \pi r^{3} ; \pi=3.14\) )

Short answer

The density of a proton is calculated by dividing its mass by its volume.

Step-by-step solution

Calculate the Proton's Volume

Using the volume formula for a sphere, which is \(\frac{4}{3} \pi r^{3}\), plug in the value of the proton's radius in centimeters, \(1.0 \times 10^{-13} \mathrm{~cm}\), to calculate the volume. Don't forget to cube the radius.

Compute the Exact Volume

Calculate the volume using the given radius and \(\pi \approx 3.14\), which gives \(\frac{4}{3} \cdot 3.14 \cdot (1.0 \times 10^{-13})^{3}\). Simplify this expression to find the numerical value of the volume in cubic centimeters.

Determine the Density of the Proton

To find the density, use the formula \(\text{density} = \frac{\text{mass}}{\text{volume}}\). Substitute the mass of the proton, \(1.7 \times 10^{-24} \mathrm{~g}\), and the volume calculated in the previous step. Simplify to get the density in grams per cubic centimeter.

Key concepts

Atomic Structure

An atom consists of a nucleus and an electron cloud surrounding it. The nucleus contains protons and neutrons, which together are known as nucleons. Protons are positively charged particles, while neutrons have no charge. Electrons, which are negatively charged, occupy various energy levels around the nucleus.

The proton is vital for defining the atomic number of an element, which in turn determines the chemical properties of the element. The number of protons in the nucleus equals the atomic number, and thus each element has a unique number of protons. Protons, due to their positive charge, contribute significantly to the atom's mass, as they are much heavier than electrons.

Understanding the structure of the proton itself, which is a subatomic particle of approximately \(1.0 \times 10^{-13} \text{ cm}\) in radius, is essential when calculating the overall density of a proton, linking the abstract concept of atomic structure with practical applications such as density measurement.

Proton Mass and Volume

The mass of a proton is tiny on a human scale, yet it plays a crucial role on the atomic scale. A proton has a mass of approximately \(1.7 \times 10^{-24} \text{ g}\). Given the size and mass of a proton, its volume is an important factor in the density calculation. To conceptualize this, imagine a sphere where the proton's volume is directly related to its radius.

To calculate the volume of a proton, you would use the volume formula for a sphere, which is \(\frac{4}{3} \pi r^{3}\). Since we have the radius of a proton, we can determine its volume and proceed to find its density. The minuscule volume and mass of protons illustrate the level of precision required when working with atomic scale measurements and highlight the delicate balance in atomic structure.

Density Formula

Density is a fundamental physical property that indicates how much mass is contained in a given volume of a substance. The density formula, defined as \(\text{density} = \frac{\text{mass}}{\text{volume}}\), allows us to calculate the density of any object as long as its mass and volume are known.

For a proton, we apply this formula by dividing the mass of the proton by its volume. By taking the previously calculated mass of \(1.7 \times 10^{-24} \text{ g}\) and dividing it by the volume found from the sphere's volume formula, we obtain the density of a single proton. This concept is critical in understanding not just the dense core of atomic particles, but also has broader applications in material science and chemistry, where density plays a key role in the behavior and characteristics of different substances.