The single proton that forms the nucleus of the hydrogen atom has a radius of approximately 1.0×10 −13 cm. The hydrogen atom itself has a radius of approximately 52.9 pm. What fraction of the space within the atom is occupied by the nucleus?

Hello everyone today. We are being given the following problem and asked to solve for it. It says the atom. Iron 56 has 26 protons and 30 neutrons. Its nucleus has a radius of about 4.6 km toe meters. While its atomic radius is around 156 PICO meters calculate the percentage of the volume of the iron atom that is occupied by the nucleus. So the first thing we can do is we can assume that this is a perfect sphere and so in doing so we can calculate the volume for the atom. So the volume of the sphere is four thirds times pi times r cubed. We then have to have our our in terms of meters. So we'll sit at the radius of the nucleus is equal to 4.6 p. M. To meters that we have in the problem. And use the conversion factor that for every one kilometer we have one times 10 to the negative 15th meters. We can then use another conversion factor that for one PICO meter we have one times 10 to the negative 12 m. Our units are going to cancel out and we are going to be left with 4. times 10 to the negative 3rd m. We can employ our values in to the equation. So we have the volume of sphere as you go to four thirds pi and we now have our radius which is 4.6 times to the negative third meters, cute which gives us 4.772 times 10 to the negative seven PK meters cubed. Now, the volume of the atom itself is going to be the same principle with our 4/3 pi with our radius of 1 56 PK meters. Instead giving us we're gonna cube that, giving us 1.6 times 10 to the seventh meters cubed to find the percentage of the volume of the nucleus. We simply have to take our volume of the sphere 4.7 times 10 to the negative seventh and divide that by the volume of our atom, giving us 2.6 times 10 to the -12%. A very small percentage, but a percentage nonetheless, I hope this helped until next time.

Ch.1 - Matter, Measurement & Problem Solving

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Tro 4th Edition

Ch.1 - Matter, Measurement & Problem Solving

Problem 125

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Chapter 1, Problem 125

The single proton that forms the nucleus of the hydrogen atom has a radius of approximately 1.0×10⁻¹³ cm. The hydrogen atom itself has a radius of approximately 52.9 pm. What fraction of the space within the atom is occupied by the nucleus?

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Convert the radius of the hydrogen atom from picometers to centimeters. Recall that 1 pm = 1.0 \times 10^{-12} cm.

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

Calculate the volume of the hydrogen atom using the same formula for the volume of a sphere, with the radius of the hydrogen atom.

Determine the fraction of the space occupied by the nucleus by dividing the volume of the nucleus by the volume of the hydrogen atom.

Express the result as a percentage to understand what fraction of the atom's space is occupied by the nucleus.

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Key Concepts

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

Atomic Structure

Atomic structure refers to the arrangement of protons, neutrons, and electrons within an atom. The nucleus, which contains protons and neutrons, is extremely small compared to the overall size of the atom, which includes the electron cloud. Understanding this structure is essential for calculating the relative sizes and volumes of the nucleus and the atom.

Volume Calculation

To determine the fraction of space occupied by the nucleus, one must calculate the volumes of both the nucleus and the atom. The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius. This calculation allows for a comparison of the volumes to find the fraction of the atom's space that the nucleus occupies.

Unit Conversion

Unit conversion is crucial when comparing measurements expressed in different units. In this case, the radius of the nucleus is given in centimeters, while the radius of the hydrogen atom is in picometers (pm). Converting these measurements to the same unit is necessary to accurately compute the fraction of space occupied by the nucleus.

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Conversion Factors

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The single proton that forms the nucleus of the hydrogen atom has a radius of approximately 1.0×10−13 cm. The hydrogen atom itself has a radius of approximately 52.9 pm. What fraction of the space within the atom is occupied by the nucleus?

Verified Solution

Key Concepts

Atomic Structure

Volume Calculation

Unit Conversion

The single proton that forms the nucleus of the hydrogen atom has a radius of approximately 1.0×10⁻¹³ cm. The hydrogen atom itself has a radius of approximately 52.9 pm. What fraction of the space within the atom is occupied by the nucleus?