Question

In the Bohr atomic model, what determines what the wavelength of possible electron orbitals are? Is the first orbital defined by experiment and the rest are just 1/n frequencies of the lowest?

Answer 1

Or is it that the lowest frequency electron is 1, so that its energy is E=h(1)?

Answer 2

The allowed orbits are determined by the requirement that the cirumference of the orbit is an integer times the electron's wavelength.

Answer 3

What IS the wavelength?

Answer 4

It's a balance between kinetic energy and potential energy. As the orbit gets smaller, the potential energy falls but the kinetic energy rises. The minimum energy orbit is where the two factors just balance.

As for how you can calculate where that balance lies -- that's what the eigenvalue equation applied to the hydrogen atom Hamiltonian does for you. There are formulas for the kinetic and potential energy, derived from basic laws of physics and principles of quantum mechanics, and then you do the math.

The size of the first orbital depends on two key parameters:

(1) Nuclear charge Z. The higher the charge, the smaller the orbit, because higher Z makes potential energy fall more steeply with smaller size of orbit.

(2) Mass of the electron m. The higher the mass, the smaller the orbit, because (contrary to what you might think), a lighter mass makes the kinetic energy rise faster with decreasing size of orbit. A heavier mass slows down how fast kinetic energy rises with decreasing orbit size.

Aside from these, it also depends on some universal constants, like the speed of light.

Answer 5

The energies in the bohr tom are determined by three conditions

Newton's 2nd Law for Circular orbits
\[F= mv^2/r = -kZe^2/r^2\]

Definition of Mechanical Energy

\[E= 1/2 mv^2 -kZe^2/r\]

and the Bohr quantization condition(this is the new bit)
\[mvr =nh/2 \pi\]

This last is equivalent to

\[\lambda = h/mv =2 \pi r/n\]

Most of this is just slogging through the algebra to solve for E, r, and v,
but once you have them, the wavelength is easy.

Answer 6

Thank you both- @fwizbang, why does\[mv^2=-kZe^2/r\]

Answer 7

for circular motion, a centripetal force is needed which is here the Coulombic force.
\[mv^2/r = kZe^2/r^2\]

Answer 8

Thanks. Damn- the sign that you've rote learned something is where you don't recognise the same thing using different symbols! Thanks again.