Carbon-12 contains six protons and six neutrons. The radius of the nucleus is approximately 2.7 fm (femtometers) and the radius of the atom is approximately 70 pm (picometers). Calculate the volume of the nucleus and the volume of the atom.

Verified step by step guidance

Convert the radius of the nucleus from femtometers to meters. Recall that 1 fm = 1 x 10^{-15} m.

Use the formula for the volume of a sphere, V = \frac{4}{3} \pi r^3, to calculate the volume of the nucleus using the radius in meters.

Convert the radius of the atom from picometers to meters. Recall that 1 pm = 1 x 10^{-12} m.

Use the formula for the volume of a sphere, V = \frac{4}{3} \pi r^3, to calculate the volume of the atom using the radius in meters.

Compare the volumes of the nucleus and the atom to understand the scale difference between them.

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Key Concepts

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

Nuclear Volume Calculation

The volume of a nucleus can be estimated using the formula for the volume of a sphere, V = (4/3)πr³, where r is the radius of the nucleus. For Carbon-12, with a radius of approximately 2.7 femtometers, this formula allows us to calculate the nuclear volume, which is significantly smaller than that of the entire atom.

Atomic Volume Calculation

Similarly, the volume of an atom is calculated using the same spherical volume formula, but with the radius of the atom, which is about 70 picometers for Carbon-12. This calculation highlights the vast difference in scale between the atomic and nuclear volumes, illustrating the relative size of the nucleus compared to the entire atom.

Unit Conversion

Understanding unit conversion is crucial when calculating volumes in different units. In this case, converting femtometers to picometers (1 fm = 0.001 pm) is necessary to ensure consistency in measurements when calculating the volumes of the nucleus and the atom, allowing for accurate comparisons and results.

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Conversion Factors

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Textbook Question

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.

Textbook Question

Carbon-12 contains six protons and six neutrons. The radius of the nucleus is approximately 2.7 fm (femtometers) and the radius of the atom is approximately 70 pm (picometers). What percentage of the carbon atom's volume is occupied by the nucleus? (Assume two significant figures.)

Textbook Question

A penny has a thickness of approximately 1.0 mm. If you stacked Avogadro's number of pennies one on top of the other on Earth's surface, how far would the stack extend (in km)? [For comparison, the sun is about 150 million km from Earth and the nearest star (Proxima Centauri) is about 40 trillion km from Earth.]

Textbook Question

Consider the stack of pennies in the previous problem. How much money (in dollars) would this represent? If this money were equally distributed among the world's population of 7.0 billion people, how much would each person receive? Would each person be a millionaire? A billionaire? A trillionaire?

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