Bohr Atom and Specific Heats

Models of the atom #

1. JJ Thompson’s plum pudding model
2. Rutherford’s planetary model. But it came with some issues
   1. Maxwell’s theory of electromagnetism predicted that accelerating charges emit radiation
   2. Using which, even Hertz had generated man-made EM waves (radio waves) in the lab
   3. If the electrons in the model emit radiation, they lose energy and fall into the nucleus destroying the very structure of the atom
3. Bohr’s model
   1. He stated that there are some stable orbits where the electrons are exempt from this Maxwell’s rule of accelerating charges emiting radiation

Stable orbits of Bohr’s model #

Bohr’s quantization condition: Angular momentum=integer multiple of $\hbar$ $$ mvr=n\hbar $$

Radius, Energy Levels in the Bohr’s atom #

$$ \begin{align} mvr=n\hbar\Rightarrow v=\frac{n\hbar}{mr}\\ \\ (1)\\ \frac{mv^2}{r}=\frac{1}{4\pi \epsilon_0}\frac{e^2}{r^2}\Rightarrow v^2=\frac{1}{4\pi \epsilon_0}\frac{e^2}{mr}\\ \\ (2)\\ \frac{n^2\hbar^2}{m^2r^2}=\frac{1}{4\pi \epsilon_0}\frac{e^2}{mr}\\ \\ (1)^2=(2)\\ r_n=(4\pi \epsilon_0)\frac{n^2 \hbar^2}{me^2} \end{align} $$

Using (2) to find K.E. and adding with Potential energy of electron, we find the total energy

$$ \begin{align} E_n=-\frac{1}{4\pi \epsilon_0}\frac{e^2}{2r}\\ Substituting\\ r=r_n\\ \\ E_n=-\frac{-13.6\\ ev}{n^2} \end{align} $$

• Was able to explain Rydberg formula $$ E_{photon} = hv = \frac{hc}{\lambda}=13.6\left(\frac{1}{n^2}-\frac{1}{m^2}\right) $$

Specific Heat #

Of Gases #

• Equipartition theorem: 1/2 kT is the average K.E. at temperature T due to each type of motion
• For a monotonic gas which has 3 degrees of freedom:
  • Total internal energy of molecule : 3/2 k¹T
  • For one mole of such molecules : 3/2 R²T
• For a mole of diatomic molecules
  • We apply bohr’s angular momentum quantization.
  • Very low temperatures: 1.5 RT, from three translational degrees of freedom
  • 2.5 RT at room temperature. An extra 1/2 RT * 2 from two types of rotations that can take place
  • 3.5 RT at high temperatures. A further 1/2 RT from potential energy and 1/2 RT from kinetic energy of vibration along axis of the diatomic molecule. (∴ A vibrational mode contributes 1 RT)

Of Solids #

A solid with N particles has 3N-6=3N³ number of vibrational modes.

Since vibrational modes contribute an internal energy of 1RT, and we have 3N vibrational modes the specific heat of solids should be 3R at all temperatures.

Though, it was shown that the specific heat C goes to 0 at low temperatures

If vibrational modes are quantized, the average energy for model with frequency ν is given by Plank’s Formula for average energy

$$ \begin{align} U=\frac{3Nhv}{e^{\frac{hv}{kT}}-1}\\ for\\ low\\ temperatures\\ U=3Nhve^{-(\frac{hv}{kT})}\\ C=\frac{\partial U}{\partial T}=3R(\frac{hv}{kT})^2e^{-(\frac{hv}{kT})} \end{align} $$

and we can see that C→0 as T→0