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13.3 - SIZE OF THE NUCLEUS

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Rutherford's Experiments

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Teacher
Teacher

Today we're discussing the size of the nucleus, starting with Rutherford's groundbreaking experiments. Can anyone remind us what Rutherford figured out regarding the atomic nucleus?

Student 1
Student 1

He discovered that atoms have a nucleus that's very small and dense compared to the atom itself.

Teacher
Teacher

Exactly! His experiments with gold foil and alpha particles showed that the positive charge is concentrated in a very small area. What was the distance of closest approach for an alpha particle with 5.5 MeV energy?

Student 2
Student 2

About 4.0 × 10^-14 m.

Teacher
Teacher

Great job! This means we can infer that the nucleus has to be smaller than that distance. What does this infer about nuclear forces?

Student 3
Student 3

It means that at higher energies, other forces begin to play a role, not just the Coulomb force.

Teacher
Teacher

Correct! The point where this happens indicates how we can gather more information about the nucleus.

Nuclear Size Formula

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Teacher
Teacher

Let's look at how we can express the size of a nucleus mathematically. The formula we use to calculate the radius based on mass number is R = R0 A^(1/3). Can someone explain what R0 represents?

Student 2
Student 2

R0 is a constant equal to 1.2 times 10 to the power of -15 m!

Teacher
Teacher

Exactly! This formula shows how the size of different nuclei relates to their mass numbers. If the volume is proportional to A, what does this say about the nuclei's density?

Student 4
Student 4

It means the density remains constant no matter the size or mass of the nucleus.

Teacher
Teacher

Right! This high uniform density is what differentiates nuclear matter from other forms of matter. Can anyone give an example of this density comparison?

Student 1
Student 1

Nuclear density is around 2.3 × 10^17 kg m^-3 compared to water, which is around 1,000 kg m^-3.

Teacher
Teacher

Excellent! This helps in understanding why something like neutron stars have such extreme densities.

Application of Nuclear Density

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Teacher
Teacher

Now that we understand nuclear density, how can we relate this knowledge to real-world situations, like neutron stars?

Student 3
Student 3

Since neutron stars have extremely high density, it means that the matter is compressed a lot, similar to a massive nucleus.

Teacher
Teacher

Exactly! This analogy is important in astrophysics. What does this tell us about the nature of atomic structures as a whole?

Student 4
Student 4

It shows that most of the atom is actually empty space, which is why such dense matter feels solid to us.

Teacher
Teacher

Very insightful! Understanding the size of the nucleus helps contextualize many phenomena in both atomic and astrophysical scales.

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

This section discusses the size of atomic nuclei, particularly how experiments have determined their dimensions and the implications of this size.

Standard

The section explains how Rutherford's experiments led to the estimation of nuclear sizes using scattering of alpha particles, revealing that the radii are related to mass numbers. Additionally, it highlights the incredibly high density of nuclear matter compared to ordinary matter.

Detailed

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Audio Book

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Rutherford's Experiment and Nucleus Size

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As we have seen in Chapter 12, Rutherford was the pioneer who postulated and established the existence of the atomic nucleus. At Rutherford’s suggestion, Geiger and Marsden performed their classic experiment: on the scattering of a-particles from thin gold foils. Their experiments revealed that the distance of closest approach to a gold nucleus of an a-particle of kinetic energy 5.5 MeV is about 4.0 × 10–14 m.

Detailed Explanation

Rutherford's work was crucial in establishing that atoms possess a nucleus. His colleagues, Geiger and Marsden, conducted experiments where alpha particles were directed at thin gold foil. They found that the closest distance an alpha particle could approach the gold nucleus was approximately 4.0 × 10⁻¹⁴ meters. This significant finding indicated that the nucleus is very small compared to the overall size of the atom.

Examples & Analogies

Imagine trying to pinpoint the location of a tiny marble in a large soccer stadium. Just like the marble's size is small compared to the vast space of the stadium, the nucleus is a tiny part of the atom, surrounded by a much larger empty space, which constitutes the atomic structure.

Nucleus Size vs. Atomic Size

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The scattering of a-particle by the gold sheet could be understood by Rutherford by assuming that the coulomb repulsive force was solely responsible for scattering. Since the positive charge is confined to the nucleus, the actual size of the nucleus has to be less than 4.0 × 10–14 m.

Detailed Explanation

Rutherford deduced that the significant scattering observed was due to the electrostatic repulsion of the positively charged nucleus on the positively charged alpha particles. This implies that if an alpha particle comes too close to a nucleus, it experiences a repulsive force, which means the size of the nucleus must be even smaller than the measurement obtained from experiments.

Examples & Analogies

Think of the nucleus as a small, dense ball that is tightly wrapped in a balloon filled with air. As you try to push the air balloon (alpha particle) towards the ball (nucleus), the air inside pushes back, making it hard for the balloon to get too close. In the same way, the nucleus repels approaching particles.

Measurement of Nuclear Sizes

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If we use a-particles of higher energies than 5.5 MeV, the distance of closest approach to the gold nucleus will be smaller and at some point the scattering will begin to be affected by the short-range nuclear forces, and differ from Rutherford’s calculations. Rutherford’s calculations are based on pure coulomb repulsion between the positive charges of the a-particle and the gold nucleus. From the distance at which deviations set in, nuclear sizes can be inferred.

Detailed Explanation

When higher energy alpha particles are used, they can approach closer to the nucleus than lower energy particles. This close proximity will eventually lead to interactions governed not just by electrostatic force but also by the strong nuclear force, which operates at very short distances. By analyzing the point at which measurements differ from the expected data based solely on electric repulsion, more accurate estimates of nuclear sizes can be made.

Examples & Analogies

Imagine you are throwing marbles at a distant target. As you throw them harder, they get closer to hitting the bullseye, but once they are too close, the target begins to exert an unexpected force (like a spring), changing how the marbles move. This change tells you about the target's characteristics.

Formula for Nuclear Radius

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It has been found that a nucleus of mass number A has a radius R = R₀ A^(1/3), where R₀ = 1.2 × 10⁻¹⁵ m (1.2 fm; 1 fm = 10⁻¹⁵ m). This means the volume of the nucleus, which is proportional to R³, is proportional to A. Thus, the density of the nucleus is a constant, independent of A, for all nuclei.

Detailed Explanation

The established formula indicates that as the mass number (total number of protons and neutrons) of a nucleus increases, its radius also increases, specifically at a rate proportional to the cube root of A. Since volume relates to the cubic measurement of radius, larger nuclei will have larger volumes but will maintain a consistent density across different nuclei. This reflects the compact nature of nuclear matter.

Examples & Analogies

Think about filling balloons with air. If you blow up more balloons (adding more mass), each balloon gets slightly larger. However, if each balloon is made of the same stretchy material, no matter how many air molecules you add, they all maintain a similar flexibility and density.

Density of Nuclear Matter

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The density of nuclear matter is approximately 2.3 × 10¹⁷ kg/m³. This density is very large compared to ordinary matter, say water, which is 10³ kg/m³. This is understandable, as we have already seen that most of the atom is empty.

Detailed Explanation

Nuclear matter exhibits an extraordinarily high density due to the compact arrangement of protons and neutrons within a small volume. This density far exceeds that of typical matter, such as water, by many orders of magnitude, underlining the concept that atoms are mostly empty space. Hence, the nucleus, being the core of the atom, holds a disproportionate amount of mass in a tiny size.

Examples & Analogies

Imagine compressing a whole school (massive in size) into the dimensions of a small box. Just like the school’s resources and people are densely packed into that small box, the nucleus has an immense amount of mass crammed into a tiny space, leading to its extremely high density.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Rutherford's Experiments: Led to the discovery of the atomic nucleus.

  • Closest Approach: The distance where significant interactions occur between particles and nuclei.

  • Nuclear Size Formula: R = R0 A^(1/3), which relates the size of nuclei to their mass numbers.

  • Density of Nuclear Matter: Approximately 2.3 × 10^17 kg/m^3, which is significantly denser than ordinary matter.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • An example of nuclear size could be how the radius of an iron nucleus, with mass number 56, can be calculated using the formula to find its size.

  • Comparative examples discussing the difference in densities between nuclear matter and typical substances, like water.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • Rutherford's gold foil showed us the way, in finding nuclei, where the protons play.

📖 Fascinating Stories

  • Imagine a tiny, dense ball at the center of a huge empty room. This ball is our nucleus, surrounded by a great deal of space, emphasizing how small it is compared to the atom.

🧠 Other Memory Gems

  • To remember R = R0 A^(1/3), think of 'Radius Rings Around'.

🎯 Super Acronyms

NUCLEUS

  • Nucleus Unites Charges
  • Living Energy Under Space.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Atomic Nucleus

    Definition:

    The positively charged center of an atom, containing protons and neutrons.

  • Term: Alpha Particle

    Definition:

    A type of radiation consisting of two protons and two neutrons, emitted during some types of radioactive decay.

  • Term: Coulomb Force

    Definition:

    The electrostatic force of attraction or repulsion between charged particles.

  • Term: Mass Number (A)

    Definition:

    The total number of protons and neutrons in an atomic nucleus.

  • Term: Nuclear Density

    Definition:

    The mass per unit volume of atomic nuclei, approximately 2.3 × 10^17 kg m^-3.