Rutherford's Atomic Model and Geiger-Marsden Experiment

• Leucippus and Democritus first proposed matter is made of atoms in the 5th century BC

Dalton's Atomic Theory (early 19th century)

• Matter is made of indestructible particles
• Atoms of a given element are identical
• Atoms combine to form new substances
• J.J. Thomson's experiments disproved the indestructibility of atoms, discovering electrons in 1897

Atomic Structure

• Atoms contain a tiny nucleus (radius ~100,000 times smaller than the atom)
• Nucleus contains all the positive charge and 99.9% of the atom's mass

Thomson's Atomic Model (1903)

• Atom is a sphere with uniform positive charge, with electrons embedded inside
• Also known as the "Plum-pudding model"
• Total positive charge equals total negative charge so the atom is electrically neutral
• Explained ion and ionic compound formation
• Further experiments revealed a different charge distribution

Geiger-Marsden Experiment (1908-1913)

• Rutherford suggested an experiment for understanding atomic structure
• Involved scattering alpha particles (helium nuclei, +2 charge) by atoms
• Most alpha particles passed straight through a gold foil
• A few were deflected at various angles
• Only ~0.14% of alpha particles scattered at angles > 0.1°
• About 1 in 8000 alpha particles deflected at angles > 90°

Rutherford's Atomic Model

• Geiger-Marsden experiment results couldn't be explained by Thomson's model
• Rutherford proposed positive charge is in a small, massive particle (nucleus) containing most of the atom's mass
• Nucleus size was found to be ~10 fm (femtometre, 10⁻¹⁵ m), which is about 10⁻⁵ times the size of the atom
• Nucleus volume is ~10⁻¹⁵ times the atom's volume
• Electrons revolve around the nucleus in circular orbits, like planets around the Sun
• Electrons' revolution prevents them from falling into the positively charged nucleus, which would cause the atom to collapse
• Most alpha particles pass through the empty space undeflected, only getting repelled if in direct line with the nucleus
• This model explained why atoms don't emit positively charged particles but do emit negatively charged electrons, due to the nucleus’ large mass
• This model, in its basic form, is still accepted

Difficulties with Rutherford's Model

• Maxwell's equations state that accelerated charges emit electromagnetic radiation
• Electrons moving in circular orbits are accelerating
• The electron should emit electromagnetic radiation continuously, lose energy, reduce orbit radius and spiral into the nucleus
• Atoms are stable; they don't constantly emit electromagnetic radiation

Atomic Spectra

• Heating a metallic object emits radiation of different wavelengths, producing a continuous spectrum when passed through a prism

• Heating hydrogen gas emits radiation of a few selected wavelengths, producing a line spectrum

• Hydrogen emits radiation at wavelengths 410, 434, 486, and 656 nm

• Lines seen in the spectrum are called emission lines

• Hydrogen emits radiation in the ultraviolet (UV) and infrared (IR) ranges

• Spectral lines divided into series (Lyman, Balmer, Paschen, Brackett, Pfund) based on discoverers

• Separation between successive lines decreases as wavelength shortens, reaching a limiting value

• Observed wavelengths of emission lines obey the equation 1/λ = R (1/n² - 1/m²)

• λ is the wavelength of a line; R is a constant

• n and m are integers (n = 1, 2, 3,... for Lyman, Balmer, Paschen... series)

• m takes all integral values greater than n for a given series

• Wavelength decreases as m increases

• Atoms of other elements also emit line spectra, unique for each element

• Rutherford's model could not explain atomic spectra.

Bohr's Atomic Model

• Niels Bohr modified Rutherford's model using quantum physics ideas
• Postulates
  • Electrons revolve around the nucleus in circular orbits
  • Radius of an electron's orbit can only take discrete fixed values, that is the angular momentum is a multiple of h/2π, where h is the Planck's constant
  • When an electron transitions from one orbit to another it will release a photon equal to the difference in it's energy

Bohr's Model Equations

• The positive integer n is called the principle quantum number, while the radii of the orbits are

  • r = (n²h²ε₀) / (πmZe²) which can be simplified into r = a₀n²

• Energies of the Orbits are

  • v = (Ze²) / (2ε₀hn) which can be simplified into E = -(13.6 Z²) / n² eV

• Ground state of an atom is the first orbit with minimal energy, with subsequent orbits called excited states

Limitations of Bohr's Model

• It could not explain the line spectra of atoms other than hydrogen
• Even for hydrogen, more accurate study of the observed spectra showed multiple components in some lines which could not be explained on the basis of this model.
• The intensities of the emission lines seemed to differ from line to line and Bohr's model had no explanation for that.
• On theoretical side also the model was not entirely satisfactory as it arbitrarily assumed orbits following a particular condition to be stable. There was no theoretical basis for that assumption.

De Broglie's Explanation

• Instead of considering the orbiting electrons inside atoms, it is better to view them as sanding waves
• Linear Momentum P = h / λ

Atomic Nucleus

• Atomic nucleus is made up of subatomic particles called protrons and neutrons, which together, is known as nucleons

• Protons are 1836 times the mass of an electron, which neutrons are roughly the same size as

• Number of protons is the atomic number, and is designated Z

Masses of subatomic particles breakdown:

• Electron: = 9.109383 × 10-31 kg
• Proton: = 1.672623 × 10-27 kg
• Neutron: = 1.674927 × 10-27 kg

Sizes of Nuclei

• The number of nucleons it's atomic number A decides the sizes of nuclei - R=R₀A^(⅓) where R₀ is the radius which equals R₀ = 1.2 x 10-15 m

Nuclear Forces

• It has a strength of 50-60 times larger then electrostatic force
• The nuclear force is independent of the charge of the nucleons, i.e., the nuclear force between two neutrons with a given separation is the same as that between two protons or between a neutron and a proton at the same separation.

Nuclear Binding Energy

• binding energy is the minimum amount of energy required to be given to an electron in the ground state of that atom to set the electron free

• ΔM( mass defect of the nucleus) = Zmp + Nm』-M is smaller than

• • ∆M c² = (Zmp + Nm』-M)c² ---(BInding energy) can be used to find atomic mass where m₁ is the mass of a hydrogen atom and M is the atomic mass of the element being considered. We will be using atomic masses in what follows, unless otherwise specified.

• (Eb/A = binding energy of a nucleon, allows for the comparison between the relative strength to whcih nucleons are bound in a nuclues for different species

Radioactive Decays

• Radioactive isotopes: isotopes that are unstable and spontaneously emit particles and/or energy
• Parent nucleus: the original unstable nucleus Daughter nucleus: the nucleus resulting from radioactive decay
• These decays are of three types as described below

Types of Decay

• Alpha Decay: Releases two protons, two neutrons, and is expressed as X→Y+a; Total mass of the products of an alpha decay is always less than the mass of the parent atom Q(value) = [mx−my−mHe]c² which is an amount of energy
• Beta Decay: emits an electron produced by converting a neutron in the nucleus into a proton. n → p + e + antineutrino and is written as X→Y+e + antineutrino
• Gamma Decay: releases gamma rays emitted by the parent nucleus, where A nucleon can make a transition from a higher energy level to a lower energy level, emitting a photon in the process, and is written as X→Y+Ƴ

Law of Radioactive Decay

• Materials with apha, beta or gamma decays are called radioactive materials Activity is proportional to, and can be written as dN = −λN(t)dt

Other Laws And Formulas

• Decay law of radioactivity : N(t) = Noe^-λt where λ= the disintegration constant
• T(1/2)= (0.693 / λ) - Half - life (Time take for half of the material in a radioactive sample to disintegrate)
• Average life is related to decay constant (τ) = 1 / λ

Nuclear Energy / Reaction

• It can be through the prosses of Nuclear Fissioin in which a heavy nuclues is broken into 2 smaller nuclei
• Nuclear Fusion, whee 2 nuclei combine into 1

Isotope Summary Guide

• Isotopes: Same number of protons but different number of neutrons; same atomic number, A; different mass number, Z.
• Isobars: refers to a set of nuclides that have the same mass number, A but different atomic numbers, Z

This lesson explores Rutherford's atomic model, which emerged from the Geiger-Marsden experiment. It covers the purpose of collimating alpha particles, the concentration of atomic mass, and the key observations that led to the concept of a tiny, positively charged nucleus. Also, it includes the implications for atomic radius and the limitations of the plum pudding model.