What Is the Size of the Nucleus Formula Derivation and Dependence on Mass Number
The smallest unit of any matter is an atom with the properties of a chemical element and is the basic building block of chemistry. Most of the part inside an atom is empty space with its centre having positively charged particles called protons and neutral particles called neutrons. These protons and neutrons constitute the nucleus of the atom. The nucleus is surrounded by negatively charged particles called electrons which form a cloud around it.
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In this article, we will look into the theory behind the size of the nucleus and see what the Rutherford gold foil experiment proved.
History of Nucleus and Electron
The nucleus is also called “atomic nucleus” and comes from the Latin word “nucleus” which is another word for “nux” (it means kernel or nut). Michael Faraday coined this term in 1844 when he was trying to describe the centre of an atom. The sciences which study the composition and characteristics of a nucleus are called nuclear physics and nuclear chemistry.
J.J.Thompson carried out his cathode ray tubes experiment and found out that all atoms contain negatively charged particles called electrons.
In the “Plum pudding model” given by Thompson, an atom had negatively charged electrons meshed inside a positively charged “soup.”
Rutherford gold foil experiment discovered that most parts of an atom are empty and its centre has a positively charged nucleus.
Facts about Nucleus
There is a strong electric force between protons and neutrons within the nucleus which holds them together.
Electrons are attracted towards the nucleus due to the positive proton in it, but they are moving so fast that they either fall around it or orbit it at a distance.
The positive charge of the nucleus comes from protons in it.
Protons and neutrons weigh much more than the tiny electrons; hence almost all the mass of a nucleus is centred around the nucleus.
The proton count of a nucleus determines it as an atom of a specific element.
The neutron count of a nucleus determines which isotope of an element is that atom.
Rutherford Gold Foil Experiment
J.J. Thompson had proposed the model of an atom, but it was Ernest Rutherford’s model that was finally accepted as the correct nuclear model. The final model was given after the Rutherford Gold Foil Experiment. Rutherford wanted to know how electrons were arranged within an atom. To do this, he decided to carry out an experiment and made fast-moving particles (alpha particles ɑ) fall on a thin gold foil.
Alpha particles are helium ions (doubly charged) with a mass of 4μ and possess a considerable amount of energy.
The gold foil was selected since he needed an extremely thin layer. The thickness of this gold foil was around that of 1000 atoms.
ɑ particles are much heavier compared to protons; hence he was expecting them to deflect only slightly by the gold atoms’ subatomic particles.
But he got very unexpected results from this experiment.
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As Per His Observations
A larger chunk of fast-moving alpha particles passed straight through the gold foil without any deflection.
There were deflections of small angles for some of the alpha particles.
The most surprising discovery was the complete rebound of a few alpha particles. At least 1 out of every 12,000 particles rebounded.
What Rutherford Concluded From This Experiment
Since most ɑ particles passed through the gold foil, most of the atom is an empty space.
Since the number of particles that got slightly deflected was very few, it was concluded that the charge of the atom occupied a tiny space.
Since there was a complete bouncing off of a few particles from the gold foil, the atom of the gold foil's positive charge was concentrated in a small volume within the atom.
Rutherford Proposed The Following Nuclear Model of An Atom Based on His Experiment
The centre of an atom is positively charged, and almost all of an atom’s mass is contained in the central part called the nucleus.
Electrons have well-defined orbits around the nucleus.
The size of atomic nucleus is quite small in comparison to the atom’s size.
What is the Size of Nucleus?
A nucleus size is much smaller than the atom’s size. The size of a hydrogen atom is 145,000 times its nucleus. The hydrogen nucleus is the smallest (1.75 * 10-15 m) since it has a single proton. The size of nucleus is of the order of 1.2 * 10-15 m, and the nuclear radii range from 1 - 10 * 10-15 m. Some nucleus is spherical while some are flattened and have deformed shapes. The formula to measure the size of nucleus is:
R = R0A1/3
Where R0 = 1.2 * 10-15 m
Density of Nuclear Matter
The density of a nucleus (ρ) is its mass divided by the total volume. The number of protons and neutrons are called nucleons, and the mass of a nucleus is A time the mass of the nucleon (A is the number of nucleons in the atom).
mnucleon ~ ⅙ * 10-27 kg
Volume = 4/3 * π * R3, Here R = R0A1/3
Density of nuclear matter ρ = mnucleon/Volume ~ 2 * 1017 kg/m3
FAQs on Size of the Nucleus and Nuclear Radius in Chemistry
1. What is the size of the nucleus in an atom?
The size of the atomic nucleus is typically about 10-15 m (1 femtometre), which is extremely small compared to the whole atom. For comparison:
- The radius of an atom is about 10-10 m.
- The nucleus is about 100,000 times smaller than the atom.
- Almost all the mass of the atom is concentrated in this tiny nucleus.
2. Why is the nucleus so small compared to the atom?
The nucleus is so small because it contains only protons and neutrons, which are tightly packed together by the strong nuclear force. In contrast:
- Electrons occupy large regions of space called electron clouds or orbitals.
- The attractive electrostatic force between nucleus and electrons spreads electrons over a much larger volume.
- Thus, the atomic radius is much larger than the nuclear radius.
3. How is the radius of a nucleus calculated?
The nuclear radius is calculated using the formula R = R0A1/3, where R0 ≈ 1.2 × 10-15 m and A is the mass number. In this formula:
- R = radius of the nucleus
- R0 = constant (about 1.2 fm)
- A = total number of protons and neutrons
4. Does the size of the nucleus depend on atomic number or mass number?
The size of the nucleus depends on the mass number (A), not directly on the atomic number (Z). This is because:
- The nucleus contains both protons and neutrons.
- The total number of nucleons (A = Z + N) determines its size.
- The radius follows R ∝ A1/3.
5. How does nuclear size change with increasing mass number?
The nuclear radius increases as the cube root of the mass number, meaning R ∝ A1/3. This implies:
- If A increases eight times, the radius only doubles.
- The increase is gradual, not proportional to A directly.
- This relationship indicates nearly constant nuclear density for all nuclei.
6. What is the approximate density of the nucleus?
The density of the atomic nucleus is about 2.3 × 1017 kg m-3, which is extremely high. This high density arises because:
- Almost all atomic mass is concentrated in a very small volume.
- Protons and neutrons are tightly packed together.
- Nuclear density remains nearly constant for all elements.
7. What experiment helped determine the size of the nucleus?
The Rutherford gold foil experiment helped determine the size of the nucleus. In this experiment:
- α-particles were directed at thin gold foil.
- Most particles passed straight through.
- A few were deflected at large angles.
8. How does nuclear size compare with atomic size?
The nuclear radius (~10-15 m) is about 100,000 times smaller than the atomic radius (~10-10 m). This means:
- The nucleus occupies a tiny fraction of the atom’s volume.
- Most of the atom consists of empty space.
- Electrons occupy regions far from the nucleus.
9. Are all nuclei the same size?
No, nuclei are not the same size; their size increases with mass number according to R = R0A1/3. For example:
- Light nuclei like hydrogen have very small radii.
- Heavy nuclei like uranium have larger radii.
- However, all nuclei have nearly constant density.
10. Why is understanding the size of the nucleus important in chemistry?
Understanding the size of the nucleus is important because it explains atomic structure, nuclear reactions, and stability. Specifically:
- It supports the modern nuclear model of the atom.
- It helps explain radioactivity and nuclear fission and fusion.
- It clarifies why chemical reactions involve electrons, not the nucleus.
