Question

The 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. For a sphere \(V=(4 / 3) \pi r^{3}\)

Short answer

The density of a proton is \( \frac{1.7 \times 10^{-24} \mathrm{g}}{\frac{4}{3} \pi (1.0 \times 10^{-13} \mathrm{cm})^3} \approx 4.13 \times 10^{14} \mathrm{g/cm}^3. \)

Step-by-step solution

Calculate the Volume of a Proton

Use the formula for the volume of a sphere, which is given by \( V = \frac{4}{3} \pi r^3 \). Plug in the radius of the proton \( r = 1.0 \times 10^{-13} \) cm to calculate the volume.

Compute the Density of the Proton

Density \( (\rho) \) is given by the mass divided by the volume. Use the mass \( m = 1.7 \times 10^{-24} \) g and the calculated volume from Step 1 to find the density of the proton.

Key concepts

Proton Mass

The mass of a proton is an essential value in many areas of physics and chemistry. It is the attribute that accounts for the majority of the proton's contribution to the atomic mass. When we refer to proton mass, we are considering a specific value: approximately \(1.67 \times 10^{-24}\text{ grams}\) (g). This may seem negligible, but protons are the building blocks of all known atoms; without this mass, we would not have the elements that make up the world.

Importance in Calculations: In calculations such as those determining proton density, the proton mass is crucial. We can compare this mass to the volume that the proton occupies to come to an understanding of its density. This fundamental particle has a standardized mass value in scientific calculations, providing consistency across research and application. Being one of the subatomic particles that make up an atom's nucleus, alongside neutrons, the proton's mass is significant in computations of atomic weight and isotopic characteristics.

Volume of a Sphere

The volume of a sphere is a measurement that describes the amount of space that is enclosed within that sphere. It's a fundamental geometric concept that applies to various scientific and mathematical contexts, particularly when dealing with spherical objects or particles like protons.

The formula to calculate the volume of a sphere is \( V = \frac{4}{3} \times \text{π} \times r^3 \) , where V represents the volume and r represents the radius of the sphere. To find the volume of a proton, one would plug in its radius, around \(1.0 \times 10^{-13}\text{ cm}\) , into the formula.

Understanding the Radius

The radius is half the diameter of the sphere and is a critical value needed to calculate volume. The formula itself conveys that the volume of a sphere increases at a cubic rate as the radius increases. In essence, even slight changes in the radius can greatly affect the volume, which in turn significantly impacts the calculated density of objects such as protons.

Density Formula

Density is a measure of how much mass is contained in a given volume. It is an important concept in many aspects of science, including material science, physics, and chemistry. The density of a substance can influence its buoyancy, stability and overall behavior in different environments.

The general density formula is made succinct by the equation \( \rho = \frac{m}{V} \) , where \rho is the density, m is the mass, and V is the volume. This equation helps us understand why objects of the same mass can have different densities, as it's directly proportional to the mass and inversely proportional to the volume.

Density of Protons:

In the context of protons, by using the known proton mass along with the volume formula for a sphere and inserting the values into the density formula, we can calculate the proton's density. This calculated density helps characterize the proton's physical properties and leads to a better understanding of atomic structures, nuclear physics, and even the basic principles behind particle physics.