CHEM-327 Lecture 12

• When considering an electron within a carbon nanotube:

  • Compare the minimum uncertainty in the electron's momentum to momentum values from its wave function.

  • This involves applying the Heisenberg Uncertainty Principle from quantum mechanics.

  • The goal is to evaluate the range of possible momentum values for the confined electron and relate them to quantized momentum states predicted by the wave function.

Organic Dye Molecules

• In organic dye molecules (crucial for OLEDs):

  • The electron is spatially confined due to a network of delocalized double bonds ( $\pi$-bonds).

  • These $\pi$-electron systems create a potential well, effectively trapping the electron.

• Key characteristic:

  • Larger dye molecules emit at longer wavelengths.

  • This is explained by the particle-in-a-box model:

  • As the size of the conjugated $\pi$-system (the 'box') increases, the energy spacing between quantized electron energy levels decreases.

  • Smaller energy gaps correspond to lower energy photons, which have longer wavelengths ($E = \frac{hc}{\lambda}$).

Hydrogen Atom Approximations

• An atom possesses discrete quantum states, each with a specific energy level.

  • Photon Absorption:

  • When an atom absorbs a photon, its energy ($h\nu$) must precisely match the energy difference between two quantum states.

  • This allows the electron to transition to a higher energy state.

  • The system's energy increases from $E$ to $E + h\nu$.

  • Photon Emission:

  • When an electron transitions from a higher energy state to a lower one, the atom emits a photon.

  • The photon's energy ($h\nu$) is equal to the energy difference between these two states.

  • The system's energy decreases from $E$ to $E - h\nu$.

Particle-in-a-Box Model for Hydrogen

• To illustrate quantized energy, a simplified model for the hydrogen atom can be used:

  • Assume the electron is confined within a one-dimensional box.

  • Length of the box: $L = 10^{-10} \, m$ (approximately the Bohr radius).

  • This allows an estimation of the wavelength of light absorbed or emitted during a transition.

• The energy levels for a particle in a 1D box are given by:
  $E_n = \frac{n^2h^2}{8mL^2}$
  where:

  • $n$ is the principal quantum number $(1, 2, 3, \ldots)$

  • $h$ is Planck's constant $(6.626 \times 10^{-34} \, Js)$

  • $m$ is the mass of the electron $(9.109 \times 10^{-31} \, kg)$

  • $L$ is the length of the box $(10^{-10} \, m)$

• For a transition between $n=1$ and $n=2$ energy levels:

  • The energy difference is:
    $\Delta E = E_2 - E_1 = \frac{2^2h^2}{8mL^2} - \frac{1^2h^2}{8mL^2} = \frac{3h^2}{8mL^2}$

  • This energy difference corresponds to the energy of an emitted or absorbed photon:
    $\Delta E = h\nu = \frac{hc}{\lambda}$

  • Therefore, the wavelength ($\lambda$) can be calculated as:
    $\lambda = \frac{hc}{\Delta E} = \frac{hc}{(\frac{3h^2}{8mL^2})} = \frac{8mL^2c}{3h}$

• Plugging in the values:
  $\lambda = \frac{8 \times (9.109 \times 10^{-31} \, kg) \times (10^{-10} \, m)^2 \times (2.998 \times 10^8 \, m/s)}{3 \times (6.626 \times 10^{-34} \, Js)}$
  $\lambda \approx 1.10 \times 10^{-7} \, m \, \text{or} \, 110 \, nm$

• Comparison to actual values:

  • The calculated wavelength of $110 \, nm$ is reasonably close but not exact.

  • Actual hydrogen absorption wavelengths are in the visible and UV range (e.g., Lyman series $91 \, nm$ to $121 \, nm$, Balmer series $365 \, nm$ to $656 \, nm$).

  • For the $n=1$ to $n=2$ transition, the actual wavelength is $121.6 \, nm$ (Lyman-$\alpha$ line).

  • The model is not perfectly accurate because the potential energy in the hydrogen atom is not an ideal infinite square well.