Bohr Hypothesis: Proving Orbital Radius is Quantized

[Jason Gomez]

Homework Statement

Question:
If we assume that an electron is bound to the nucleus (assume a H atom) in a circular orbit, then the Coulomb force is equal to the centripetal force:
mv^2/r= ke^2/r^2
In the Bohr hypothesis, angular momentum, L = mvr is quantized as integer multiples of (h-bar): L = n(h-bar). Show that if this is true, orbital radius is also quantized: r = n^2aB.
aB = (hbar)^2/(ke^2m(electron))

Homework Equations

I have to be honest, I am completely lost. The book doesn't go in any detail and nor did the professor so I am stuck. If I could just get a helpfull hint or advice it would be greatly appreciated

The Attempt at a Solution

 

[kreil]

$$\frac{mv^2}{r} = \frac{ke^2}{r^2} \implies r = \frac{ke^2}{mv^2}$$

$$L = mvr = nh \implies r = \frac{nh}{mv}$$

If you combine these expressions and solve for the quantity (mv), you can then plug this back into solve for r:

$$\frac{ke^2}{mv^2} = \frac{nh}{mv}$$

 

[Jason Gomez]

Oh, let me work with that a little, I will be back on later, thank you

 

[Jason Gomez]

I forgot to come back because I have been busy, but thank you, that did help I figured it out.

 

[paranormal]

The Bohr hypothesis states that the angular momentum of an electron in an orbit around the nucleus of a hydrogen atom is quantized, meaning it can only take on certain discrete values. This is expressed as L = n(h-bar), where n is an integer and h-bar is the reduced Planck's constant. We can use this information to show that the orbital radius, r, is also quantized.

Starting with the equation provided, mv^2/r = ke^2/r^2, we can rearrange it to solve for v:

v = ke^2/(mr)

Now, we can substitute this value for v into the formula for angular momentum, L = mvr:

L = m(ke^2/(mr))r

Simplifying, we get:

L = ke^2/m

Since we know that L = n(h-bar), we can substitute this into the equation:

n(h-bar) = ke^2/m

Rearranging for r, we get:

r = (h-bar)^2/(ken)

Now, we can substitute for the value of k, which is the Coulomb constant, and the mass of the electron, me, to get:

r = (h-bar)^2/(ke^2m(electron)n)

Finally, we can simplify the equation by substituting in the value for aB, which is the Bohr radius, defined as aB = (h-bar)^2/(ke^2m(electron)), giving us the final equation:

r = n^2aB

This shows that the orbital radius, r, is quantized and can only take on certain discrete values, determined by the integer value of n. This supports the Bohr hypothesis and provides evidence for the quantization of angular momentum in the hydrogen atom.