1.3 Rutherford, Planck, and Bohr

Rutherford's Model (1910)

Provided experimental evidence that an atom has a dense, positively charged nucleus.
The nucleus accounts for only a small portion of the atom's volume.

Planck's Quantum Theory

Developed the first quantum theory in 1900s.
Energy emitted as electromagnetic radiation comes in discrete bundles called quanta.
Planck Relation: $E = hf$
- $E$ : Energy of a quantum.
- $h$ : Planck's constant ( $6.626 \times 10^{-34} \text{ J s}$ ).
- $f$ : Frequency of the radiation (sometimes denoted by the Greek letter $\nu$ ).
Relationship between speed of light, frequency, and wavelength:
- $c = f\lambda$
- $c$ : Speed of light.
- $f$ : Frequency.
- $\lambda$ : Wavelength.

Bohr Model (1913)

Danish physicist Niels Bohr used Rutherford's and Planck's work to develop his model of the electronic structure of the hydrogen atom.
Assumptions:
- Hydrogen atom consists of a central proton.
- An electron travels in a circular orbit around the proton.
- Centripetal force on the electron is provided by the electrostatic force between the proton and electron.
Bohr used Planck's quantum theory to correct classical physics assumptions about electron pathways.
Classical mechanics postulates that an electron revolving may assume an infinite number of values for radius and velocity, but Bohr placed restrictions on angular momentum values.
Quantized Angular Momentum:
- $L = \frac{nh}{2\pi}$
  - $L$ : Angular momentum.
  - $n$ : Principal quantum number (any positive integer).
  - $h$ : Planck's constant.
The angular momentum of an electron changes only in discrete amounts with respect to the principal quantum number.
Energy of the Electron:
- $E = -\frac{R_H}{n^2}$
  - $R_H$ : Rydberg unit of energy (experimentally determined).
  - $n$ : Principal quantum number.
A value of zero energy was assigned to the state in which the proton and electron are separated completely, meaning that there is no attractive force between them. Therefore, the electron in any of its quantized states in the atom will have an attractive force toward the proton, this is represented by the negative sign in equation 1.3.
The energy of the electron increases (becomes less negative) the farther it is from the nucleus (increasing $n$ ).
Analogy: Quantized energy is like ascending or descending a staircase, allowing only certain discrete changes of potential energy.
Bohr described the hydrogen atom as a nucleus with one proton around which a single electron revolved in a defined pathway (orbit) at a discrete energy value.
Ground State: Orbit with the smallest, lowest energy radius ( $n = 1$ ).
Excited State: When an electron is promoted to an orbit with a larger radius (higher energy).
Atoms of any element will generally exist in the ground state unless subjected to extremely high temperatures or irradiation.

Applications of the Bohr Model

Useful for explaining atomic emission and absorption spectra of one-electron systems (e.g., H atom).

Mnemonic

As electrons go from a lower energy level to a higher energy level, they Absorb light (Higher potential Excited Distant from the nucleus

Atomic Emission Spectra

Electrons can be excited to higher energy levels by heat or other forms of energy.
Excited states are brief, and electrons rapidly return to the ground state, emitting photons.
Energy of a Photon:
- $E = hf = \frac{hc}{\lambda}$
  - $h$ : Planck's constant.
  - $c$ : Speed of light.
  - $\lambda$ : Wavelength of the radiation.
Energy transitions are quantized, resulting in a line spectrum.
Each element has a unique atomic emission spectrum, acting as a "fingerprint."
Application: Analysis of stars and planets by matching light wavelengths to known line spectra.
*Real world emissions from electrons dropping from an excited state to a ground state give rise to fluorescence. What we see is the color of the emitted light.
The Bowen model of the hydrogen atom explain the atomic emission spectrum of hydrogen, which is the simplest emission spectrum among all the elements.
Hydrogen Emission Series:
- Lyman series: transitions from energy levels n ≥ 2 to n = 1 (UV region, shorter wavelengths, larger energy transitions).
- Balmer series: transitions from energy levels n ≥ 3 to n = 2(includes four wavelengths in the visible region).
- Paschen series: transitions from N ≥ 4 to N = 3.
Energy Associated with Change in Quantum Number:
- E+hc/wavelngth =R $E=\frac{hc}{\lambda}=RH=\left\lbrace\frac{1}{ni^2}-\frac{1}{nF^2}\right\rbrace$
  - $n_i$ : Initial principal quantum number.
  - $n_f$ : Final principal quantum number.
- If an atom emits a photon, the equation gives a negative value for energy, indicating a decrease.

Atomic Absorption Spectra

Electrons absorb energy to move to a higher energy level.
The wavelengths of absorption correspond exactly to the wavelengths of emission.
Identification of elements in the gas phase requires absorption spectra.
$ΔE$ is the same for absorption or emission between any two energy levels.
Each element has a characteristic set of energy levels.
*Real world Absorption is the basis for the color of compounds. We see the color of the light that is not absorbed by the compound.

MCAT Concept Check 1.3

The valence electron in a lithium atom jumps from energy level n equals two to n equals four. What is the energy of this transition in joules? In eve.
If an electron emits three eve of energy, what is the corresponding wavelength of the emitted photon?
- Note. One F equals 1.6 X ten nineteen J, H equals 6.626 X ten thirty four J S.
Calculate the energy of a photon of wavelength 662 nm. Note.
- H equals 6.626 x ten thirty four j s.