15.2 The Bohr Model of Atomic Structure

The Bohr model revolutionized our understanding of atomic structure. It explained the discrete energy states of electrons in atoms, particularly for hydrogen, by combining classical physics with quantum concepts.

This model introduced the idea of quantized electron orbits and energy levels. It laid the groundwork for modern quantum mechanics, despite its limitations in accurately describing more complex atoms beyond hydrogen.

Properties of Atoms

Internal Structure of Atoms

Atoms are the fundamental building blocks of matter, with a complex internal architecture that determines their physical and chemical properties. 🔍

• The atom consists of a dense, positively charged nucleus at the center surrounded by a cloud of negatively charged electrons
• The nucleus contains protons (positively charged) and neutrons (electrically neutral)
• Nuclear notation provides a compact way to represent the composition of an atom:
  • Written as $^A_Z X$ where:
  • X is the chemical symbol
  • Z is the atomic number (number of protons)
  • A is the mass number (protons + neutrons)
  • Example: $^{12}_{\;6}\text{C}$ represents carbon with 6 protons and 6 neutrons

Ions form when atoms gain or lose electrons while maintaining the same number of protons:

• Positive ions (cations) form when atoms lose electrons, creating a net positive charge
• Negative ions (anions) form when atoms gain electrons, creating a net negative charge
• The charge of an ion equals the difference between the number of protons and electrons

Unique Proton Numbers

The number of protons in an atom's nucleus serves as its defining characteristic and determines which element it is.

• The atomic number (Z) uniquely identifies each element on the periodic table
• Elements with the same atomic number but different neutron counts are called isotopes
  • Example: Carbon-12 ($^{12}_{\;6}\text{C}$) and Carbon-14 ($^{14}_{\;6}\text{C}$) both have 6 protons but differ in neutron count
  • Isotopes have identical chemical properties but different physical properties (like radioactivity)

Electron configuration plays a crucial role in determining chemical behavior:

• Valence electrons (those in the outermost energy level) primarily determine chemical reactivity
• Elements in the same column of the periodic table have similar chemical properties due to similar valence electron configurations

The mass of an atom is concentrated in its nucleus. 🪨

• Protons and neutrons each have approximately the same mass, about 1,836 times greater than an electron
• The atomic mass unit (amu) is defined as 1/12 the mass of a carbon-12 atom
• Electrons contribute negligibly to the total mass of an atom (less than 0.05%)

Bohr Model of Atoms

The Bohr model represents a pivotal moment in our understanding of atomic structure, providing the first quantum mechanical description of the atom. 🪐

Niels Bohr proposed this model in 1913 by applying both classical physics and early quantum theory:

• Electrons orbit the nucleus in circular paths similar to planets orbiting the sun
• The electric force between the positively charged nucleus and negatively charged electron follows Coulomb's law:
  • $F_{e}=k \frac{q_{1} q_{2}}{r^{2}}$ where k is Coulomb's constant, q₁ and q₂ are charges, and r is the distance
• This electric force provides the centripetal force needed for circular motion:
  • $F_{\text{net}}=m \frac{v^{2}}{r}$ where m is electron mass, v is velocity, and r is orbital radius

The revolutionary aspect of Bohr's model was the quantization of electron orbits:

• Electrons can only exist in specific allowed energy levels, not in between
• The energy of an electron in the nth energy level of hydrogen is given by:
  • $E_n = -\frac{13.6 \text{ eV}}{n^2}$ where n is a positive integer (1, 2, 3, etc.)
• When electrons transition between energy levels, they emit or absorb photons with energy:
  • $E_{photon} = |E_f - E_i| = hf$ where h is Planck's constant and f is frequency

The standing wave concept explains why only certain orbits are allowed:

• An electron's orbit must contain a whole number of de Broglie wavelengths
• This creates a standing wave pattern where the electron wave reinforces itself
• Mathematically: $n\lambda = 2\pi r$ where n is an integer, λ is wavelength, and r is orbit radius

Practice Problem 1: Bohr Model Energy Levels

Solution

To solve this problem, we need to find the energy difference between the initial and final states:

1. Calculate the energy at n=3: $E_3 = -\frac{13.6 \text{ eV}}{3^2} = -\frac{13.6 \text{ eV}}{9} = -1.51 \text{ eV}$

2. Calculate the energy at n=1: $E_1 = -\frac{13.6 \text{ eV}}{1^2} = -13.6 \text{ eV}$

3. Find the energy of the emitted photon: $E_{photon} = |E_f - E_i| = |E_1 - E_3| = |-13.6 \text{ eV} - (-1.51 \text{ eV})| = |{-13.6 \text{ eV} + 1.51 \text{ eV}}| = 12.09 \text{ eV}$

Therefore, the photon emitted has an energy of 12.09 eV.

Practice Problem 2: Atomic Structure

Solution

Let's analyze this atom step by step:

1. (a) The element is determined by the number of protons (atomic number):

   • 17 protons means this is Chlorine (Cl)

2. (b) The mass number is the sum of protons and neutrons:

   • Mass number = 17 protons + 18 neutrons = 35

3. (c) The nuclear notation is written as $^A_Z X$ where A is the mass number, Z is the atomic number, and X is the chemical symbol:

   • Nuclear notation: $^{35}_{17}\text{Cl}$

4. (d) The net charge is determined by the difference between protons and electrons:

   • Net charge = 17 protons - 18 electrons = -1
   • This atom has one more electron than protons, making it a negative ion (anion)
   • The ion would be written as Cl⁻

Therefore, this atom is a chloride ion (Cl⁻) with mass number 35.