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Find the order of magnitude of the density of the nucleus of an atom. What does this result suggest concerning the structure of matter? Model a nucleus as protons and neutrons closely packed together. Each has mass \(1.67 \times 10^{-27} \mathrm{kg}\) and radius on the order of \(10^{-15} \mathrm{m}\)

Short Answer

Expert verified
The order of magnitude of the density of the nucleus is approximately \(2.3 脳 10^{17}\) kg/m鲁. This suggests that an atom, and thus matter, is mostly empty space given that the small, dense nucleus contains almost all of the atom's mass, but occupies only a very small space.

Step by step solution

01

Determine the mass of the nucleus

The mass of a proton or a neutron is given as \(1.67 脳 10^{-27}\) kg. As the nucleus is comprised of protons and neutrons, the total mass of the nucleus will be the number of protons and neutrons multiplied by the mass of one proton (or neutron). Assuming there is one nucleon (either proton or neutron) in the nucleus to simplify calculations, the mass of the nucleus \(m\) will be \(1.67 脳 10^{-27}\) kg.
02

Calculate the volume of the nucleus

The nucleus can be modelled as a sphere, and the radius of this sphere is given as \(10^{-15}\) m. Since the formula for the volume \(V\) of a sphere is \((4/3)蟺r^{3}\), substituting \(r = 10^{-15}\)m will given the volume of the nucleus.
03

Calculate the density of the nucleus

The density \(蟻\) of an object is determined by the formula: \(蟻 = m/V\). Substituting the mass and volume obtained from Step 1 and Step 2 respectively, we can calculate the density of the nucleus.
04

Evaluate what the result suggests about the structure of matter

With the calculated density of the nucleus, we can then formulate an understanding about the structure of matter. Specifically, the nucleus of an atom contains almost all the mass of the atom but occupies a very small space, thus leading to the conclusion that matter is mostly empty space.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Atomic Structure
Atoms are the basic building blocks of matter. Each atom consists mainly of empty space, with a dense central region called the nucleus where nearly all its mass is concentrated. Surrounding the nucleus is a 'cloud' of electrons, which are negatively charged particles. The structure of an atom can be visualized as akin to a miniature solar system, where the nucleus represents the sun and the electrons represent planets orbiting around it.
Understanding atomic structure helps us to comprehend why atoms combine to form molecules and how elements behave in various chemical reactions. It's fascinating to think that the matter around us is composed of these tiny structures that we can't even see with the most powerful of microscopes. Atomic structure is a fundamental concept not only in chemistry but also in physics and material science.
Nucleus Modeling
Modeling the nucleus of an atom involves understanding it as a tightly packed collection of protons and neutrons. These subatomic particles are collectively called 'nucleons'. The nucleus is remarkably small and dense, with its radius on the order of \(10^{-15} \mathrm{m}\), which is thousands of times smaller than the overall size of the atom.
To model the nucleus, one could think of it as a sphere, where the striking density is mainly due to the nucleons being closely packed together almost side-by-side. The nuclear force, which is vastly stronger than the electromagnetic force, holds these particles together, overcoming the repulsion between like-charged protons.
  • Nuclear forces and binding energy contribute to the nuclear stability.
  • The number of protons defines the element while the number of neutrons affects its stability and isotope form.
Modeling the nucleus helps predict its behavior during nuclear reactions and how it might interact with other nuclei.
Proton Mass
The mass of a proton is a crucial factor in determining the properties of an atom. Each proton has a mass of approximately \(1.67 \times 10^{-27} \mathrm{kg}\).This mass might seem minuscule, but it's significant when considering the scale of an atom, where even tiny masses greatly affect the atom's behavior and properties.
Protons carry a positive electrical charge, and the number of protons in the nucleus defines the atomic number of an element. This, in turn, determines which element it is, as each element on the periodic table has a unique number of protons. Connectively, the proton mass is integral in calculating the overall mass of the nucleus and hence the atom's mass, albeit the entire atom's mass includes contributions from neutrons and the minimal mass of electrons.
  • Protons affect nuclear reactions, like those occurring in stars.
  • Mass and charge lead to crucial roles in chemical bonding and elemental identity.
Neutron Mass
Neutrons, like protons, are essential particles found in the nucleus of an atom. They have nearly the same mass as protons, about \(1.67 \times 10^{-27} \mathrm{kg}\),helping to make the nucleus very dense. However, neutrons carry no electrical charge, making them neutral.
The role of neutrons is crucial in understanding the stability of the nucleus. They effectively act as a buffer between protons, minimizing the repulsive forces from the positively charged protons. Neutrons change the isotope of an element without altering its chemistry and can influence the nucleus's stability and susceptibility to radioactive decay.
  • Neutrons play a key role in nuclear reactions, such as fission.
  • Their absence or presence in certain numbers leads to isotopes with different nuclear properties.
Understanding the neutron mass aids in studying nuclear energy, radioactive processes, and various applications in medicine and technology.

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Most popular questions from this chapter

A spherical aluminum ball of mass \(1.26 \mathrm{kg}\) contains an empty spherical cavity that is concentric with the ball. The ball just barely floats in water. Calculate (a) the outer \(\mathrm{ra}\) dius of the ball and (b) the radius of the cavity.

How many cubic meters of helium are required to lift a balloon with a \(400-\mathrm{kg}\) payload to a height of \(8000 \mathrm{m} ?\) (Take \(\rho_{\mathrm{He}}=0.180 \mathrm{kg} / \mathrm{m}^{3} .\) ) Assume that the balloon maintains a constant volume and that the density of air decreases with the altitude \(z\) according to the expression \(\rho_{\text {air }}=\rho_{0} e^{-z / 8,000}\) where \(z\) is in meters and \(\rho_{0}=1.25 \mathrm{kg} / \mathrm{m}^{3}\) is the density of air at sca level.

The water supply of a building is fed through a main pipe \(6.00 \mathrm{cm}\) in diameter. A \(2.00-\) cm-diameter faucet tap, located \(2.00 \mathrm{m}\) above the main pipe, is observed to fill a \(25.0-\mathrm{I}\) container in \(30.0 \mathrm{s}\). (a) What is the speed at which the water leaves the faucet? (b) What is the gauge pressure in the 6-cm main pipe? (Assume the faucet is the only "leak" in the building.)

A swimming pool has dimensions \(30.0 \mathrm{m} \times 10.0 \mathrm{m}\) and a flat bottom. When the pool is filled to a depth of \(2.00 \mathrm{m}\) with fresh water, what is the force caused by the water on the bottom? On each end? On each side?

The Bernoulli effect can have important consequences for the design of buildings. For example, wind can blow around a skyscraper at remarkably high speed, creating low pressure. The higher atmospheric pressure in the still air inside the buildings can cause windows to pop out. As originally constructed, the John Hancock building in Boston popped window panes, which fell many stories to the sidewalk below. (a) Suppose that a horizontal wind blows in streamline flow with a speed of \(11.2 \mathrm{m} / \mathrm{s}\) outside a large pane of plate glass with dimensions \(4.00 \mathrm{m} \times 1.50 \mathrm{m}\) Assume the density of the air to be uniform at \(1.30 \mathrm{kg} / \mathrm{m}^{3} .\) The air inside the building is at atmospheric pressure. What is the total force exerted by air on the window pane? (b) What If? If a second skyscraper is built nearby, the air speed can be especially high where wind passes through the narrow separation between the buildings. Solve part (a) again if the wind speed is \(22.4 \mathrm{m} / \mathrm{s}\), twice as high.

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