De Broglie's hypothesis and Bohr's atom

Bohr's first postulate

• Recall that Bohr postulated that electrons existed in distinct quantised orbits with an unexplainable stability.
• Typically an electron orbiting in a circle would emit EM radiation as it is accelerating.
• Bohr had no explanation for why this isn't so, but merely postulated that it just didn't.
• De Broglie's new hypothesis accounted for this by suggesting that the electrons exist as standing waves.
• This means they have a closed orbit with no movement of energy.
• Energy is instead constantly transferred between kinetic energy and potential energy, but not lost.
• For this to occur:
(1)
\begin{align} 2 \pi r = n \lambda \end{align}
• Where n is an integer.

Bohr's third postulate

• Bohr postulated that the angular momentum of electrons were quantised.
• He had no basis for this, and it is only true if the equation $mvr = \frac{nh}{2\pi}$ was true.
• De Broglie's hypothesis was able to prove it, as shown below:
(2)
\begin{align} n \lambda =2 \pi r \end{align}

Using matter waves ($\lambda = \frac{h}{mv}$):

(3)
\begin{align} \frac{nh}{mv} =2 \pi r \end{align}

Rearranging:

(4)
\begin{align} mvr = \frac{nh}{2\pi} \end{align}
• Thus de Broglie's hypothesis had been able to explain Bohr's postulates and validify his model of the atom.
page revision: 2, last edited: 03 Nov 2011 07:08