Quantum Model and Energy Levels

Quantum Model Assumptions and Allowed Energy Levels

Quantization of Angular Momentum

• The logic is that because angular momentum is quantized, energy will also be a function of $n$.

Bohr's Assumptions and Newton's Laws

• Bohr uses the laws of classical physics, where $v$ is a function of $r$.

• Equation: $m v^2 / r = k Z q<em>p q</em>e / r^2$, where:

  • $m$ is mass,

  • $v$ is velocity,

  • $r$ is radius of orbit,

  • $k$ is a constant,

  • $Z$ is the charge of the nucleus,

  • $q_p$ is the charge of the proton

  • $q_e$ is the charge of the electron.

• Bohr assumes Newton's laws are valid in the atom.

  • The sum of forces (attractive force) equals $ma$.

Solving for Allowed Values

• Solve for $v$ as a function of $r$, plug it in, and solve for $r$ to get the allowed values.

• Square both sides of the equation: $n^2 r^2 v^2 = n^2 h^2 / (4 \pi^2)$.

Dimensional Analysis

• From dimensional analysis, a quantity is derived.

• A constant is introduced to simplify the equation by absorbing the $2 \pi$ factor.

Allowed Energy Levels

• Energy of the $n^{th}$ state is the sum of kinetic and potential energy: $E_n = 1/2 m v^2$.

• Assume electrons are not moving at relativistic speeds, so the classical expression for kinetic energy can be used.

Constant and Energy Levels

• A constant is defined, and $E_n = -constant / n^2$.

• The constant is the magnitude of $E<em>1$, where $E</em>1$ is the energy of the n=1 state.

Energy Levels and Transitions

• Energy level is higher the further out you go.

• The difference between energy levels of two different states is considered.

• If an electron transitions from one state to another, and conservation of energy is believed at the atomic level, it tells us the frequency of the photon.

• Photon theory: energy of the photon is $E = hf$, where $h$ is Planck's constant and $f$ is frequency.

Validation and Limitations

• Bohr's model has limitations and assumptions that are not true today; quantum mechanics replaces it.

• Bohr's correct results were achieved by luck due to an idealized experiment (one proton); applying the same principles to other systems does not work.

Failures of the Bohr Model

• The Bohr model fails to predict electron jumps from one energy level to another (e.g., from $n = 3$ to $n = 1$).

• It lacks dynamics and does not explain how the electron decides to change states.

• There is no discussion of how the change in state occurs. It's just a statement that energy levels exist.

• Bohr had no idea what would make the electron decide to go from one state to another state.