Neutron stars are composed of solid nuclear matter, primarily neutrons. Assume the radius of a neutron is approximately 1.0×10 –13 cm. Calculate the density of a neutron. [Hint: For a sphere V = (4/3)πr 3 .] Assuming that a neutron star has the same density as a neutron, calculate the mass (in kg) of a small piece of a neutron star the size of a spherical pebble with a radius of 0.10 mm.

Hi everyone today. We have a question telling us the full loring is an al trope of carbon whose molecule consists of carbon atoms connected by single and double bonds. So as to perform a closed structure With few strings of 5 - seven atoms. Buckminster fuel ring is a type of food coloring that has a density of 1.65 g per cm cubed. The diameter of one molecule is 1.1 nanometers. And our goal is to calculate the mass of one molecule of Buckminster fuller ring. So we need to remember that density equals mass divided by volume and the volume of a sphere equals four thirds times pi times are radius cube. So we're going to take this and plug it in for our volume here which gives us density equals mass Divided by 4/3. Hi times are radius cube. Now our diameter is 1. nmeter To get our radius from that. We simply divide by two which equals 0.55 nm. And we need that in centimeters. So we're going to take our 0.55 nanometers and multiply by one m over 10 to the ninth nanometers times 10 squared centimeters over one m. And our nanometers are going to cancel out and our meters are going to cancel out. And that's going to give us 5.5 Times 10 to the negative eight cm. So now we can plug that in. So mass is going to equal density times four thirds pie are cute and mass Will equal the 1.65 grams per centimeter cubed. Given to us in the prompt times four thirds pie times 5.5 Times to the negative eight cm Q, And that equals 1.15 times to the -21 g. And that is our final answer. Thank you for watching. Bye.

Ch.2 - Atoms & Elements

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Tro 5th Edition

Ch.2 - Atoms & Elements

Problem 106

Chapter 2, Problem 106

Neutron stars are composed of solid nuclear matter, primarily neutrons. Assume the radius of a neutron is approximately 1.0×10^–13 cm. Calculate the density of a neutron. [Hint: For a sphere V = (4/3)πr³.] Assuming that a neutron star has the same density as a neutron, calculate the mass (in kg) of a small piece of a neutron star the size of a spherical pebble with a radius of 0.10 mm.

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Calculate the volume of a neutron using the formula for the volume of a sphere: $V = \frac{4}{3}\pi r^3$, where $r$ is the radius of the neutron.

Substitute the given radius of the neutron, $1.0 \times 10^{-13}$ cm, into the volume formula to find the volume of a neutron.

Use the mass of a neutron, approximately $1.675 \times 10^{-24}$ g, to calculate the density of a neutron using the formula: $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$.

Convert the radius of the pebble from millimeters to centimeters (0.10 mm = 0.010 cm) and calculate the volume of the pebble using the sphere volume formula: $V = \frac{4}{3}\pi r^3$.

Multiply the volume of the pebble by the density of a neutron to find the mass of the pebble in grams, and then convert the mass to kilograms.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Density

Density is defined as mass per unit volume, typically expressed in grams per cubic centimeter (g/cm³) or kilograms per cubic meter (kg/m³). It is a crucial property that helps determine how much matter is packed into a given space. In this context, understanding density allows us to calculate the mass of a neutron based on its volume and the relationship between mass and density.

Volume of a Sphere

The volume of a sphere is calculated using the formula V = (4/3)πr³, where r is the radius of the sphere. This formula is essential for determining the volume of both a neutron and a small piece of a neutron star, as it allows us to quantify the space occupied by these spherical objects. Accurate volume calculations are necessary to find the corresponding mass when density is known.

Mass Calculation

Mass can be calculated using the formula mass = density × volume. This relationship is fundamental in physics and chemistry, as it connects the amount of matter (mass) to how densely it is packed (density) and the space it occupies (volume). In the context of the question, this formula will be used to find the mass of a neutron star fragment by applying the density of a neutron to the calculated volume of the pebble-sized sphere.