# What actually makes a Bohr's radius stable?

I was told that Bohr introduced the concept of Stationary orbits in which electrons were stable in my school but I never got the reason behind this stability.

So can someone explain why is Bohr's radius considered stationary and why is it so stable? How does quantized angular momentum make it stable?

I know that this theory is now invalid and we have quantum model but I want to know why Bohr's model got attention in the first place.

• Feb 8, 2021 at 13:59

I know that this theory is now invalid and we have quantum model but I want to know why Bohr's model got attention in the first place.

This was the first model that made a successful prediction of atomic (hydrogen) emission frequencies (the Rydberg formula). Other than that, Bohr's model doesn't give too much insight into how the atoms work. It just consisted of "strange" postulates, created in order to avoid collapse of the atom.

Just 12 years after its introduction, Heisenberg and Schrödinger made the first steps to create the actual consistent theory that's now called quantum mechanics. It's just a historical curiosity that Bohr's model is still being taught.

Bohr postulated that the angular momentum of such circular orbit would be quantized in multiples of $$\hbar$$, that is

$$L=n\hbar$$

This is a postulate, which means it's a happy idea that we consider true, but he gavee no proof.

Now, if we apply a classical definition, $$\vec{L}=m \vec{r}\times \vec{v}$$, and, for a circular orbit, we have

$$L=mrv$$

So the equation is $$mrv=n\hbar$$. Now, if we consider that the only attraction is the electrostatic one, we apply Newton's 2nd law. The electrostatic force must act as the centripetal force, so

$$\dfrac{1}{4\pi\epsilon_0}\frac{q_e^2}{r^2} = m \frac{v^2}{r}$$

$$\dfrac{1}{4\pi\epsilon_0}\frac{q_e^2}{mr} = v^2$$

Then replace

$$m r v = n \hbar$$

$$m^2 r^2 v^2 = n^2 \hbar^2$$

$$m^2 r^2 \dfrac{1}{4\pi\epsilon_0}\frac{q_e^2}{mr} = n^2 \hbar^2$$

$$m r \dfrac{1}{4\pi\epsilon_0}\frac{q_e^2}{1} = n^2 \hbar^2$$

$$r = n^2 4\pi\epsilon_0 \dfrac{\hbar^2}{mq_e^2} = n^2 \cdot a_0$$

with $$a_0$$ being $$4\pi\epsilon_0 \dfrac{\hbar^2}{mq_e^2}$$

Edit:

As I said, Bohr does not explain why it is like that, it is a postulate.

Then, Louis de Broglie came up with an explanation. The electron is like a stationary wave around the orbit.

The wave must be stationary (otherwise it would emit EM radiation). Such a stationary wave requieres a junction of the begining and the end. It's liek a usual stationary wave in which you join both extremes forming a circumference.

So, the junction of both endings requires the circumference lenght to be a multiple of the wavelenght

$$lenght = n\cdot \lambda$$, or $$2\pi r= n \lambda$$

Then include de Broglie's hypothesis and you have

$$2\pi r = n \frac{h}{mv}$$

which is the same condition from the Bohr's postulate. This was a curious explanation.

However, the real truth behind this solution is that it is solution to the Schrödinger's equation.

• wher is the reason behind stability ? How does quantized angular momentum make it stable ? Feb 2, 2021 at 14:49
• The reason behind stability is beyond Bohr's model, but see the edit Feb 2, 2021 at 15:03

People at the time knew from the sharp lines seen in spectroscopy that an electron in hydrogen (for example) could only have certain energies. If they only have certain energies that meant (for classical miniature-solar-system models) these electrons only had certain speeds, certain radii, certain momenta and so on. There were obviously rules.

But these rules for such allowed quantities are non-obvious and complicated. What Bohr spotted was that if you impose a rule on angular momentum, in the form $$L=n \hbar$$, then the rules for energy and all the rest come out right ('One rule to ring them all'). This rule is simple and it only involves $$\hbar$$ and not $$e$$ or $$\epsilon_0$$: it has nothing to do with the electromagnetic interaction which is binding the electron to the nucleus, but comes from something more basic.

We now understand this (thanks to quantum mechanics) as coming from a basic property of space: if you rotate a system through 360 degrees about any axis you have to get back to the system you started with, so any angular dependence must be some integer fraction of $$1 \over 2 \pi$$, and as angular momentum is the conserved quantity matching angular dependence it all - with a wrinkle to accomodate intrinsic spin - follows neatly.

In classical physics an electron orbiting the nucleus will constantly lose energy by emitting electromagnetic radiation and eventually will collapse to the nucleus. But If you accept the fact, as Bohr did, that angular momentum is quantized then you can tell that a change in orbital radius cannot happen in small steps. Therefore the electron has to lose or gain a lot of energy for this change to happen. This is the stability of Bohr's model.

Historically, in 1913 Bohr proposed a model to explain the transitions spectra of hydrogen atom. The postulates he took contrary to classical mechanics were

1. Discrete set of circular stable orbits
2. Discrete set of orbital angular momentum
3. Discrete set of energy transition

These are semi-classical argument by Bohr as he noticed Max Planck successfully obtained the black-body radiation spectrum by postulating energy is discretised. However, in quantum mechanics, discretised physical quantities emerged naturally from calculation result, instead of postulate.

In short, quantum mechanics is a probabilistic theory which incorporates wave properties and particle properties of matter into one, the mathematical entity to describe the state is called wave function. A striking feature is superposition of state, for instance a cat is simultaneously alive and dead before measurement.

In quantum mechanics we don't speak of orbit as orbit in mental picture means you know a matter position and momentum to definite amount at every time. Yet, Heisenberg uncertainty principle gives $$\Delta x \Delta p \geq \hbar/2$$. From this inequality, if one uncertainty is zero, another uncertainty is infinite. Thus, the mental picture breaks down.

Stationary orbits, means transition from one wave function to another is difficult. After tedious calculation of hydrogen atom, each wave function correspond to different orbital angular momentum and Energy.

Conservation laws are still obey in quantum mechanics, and hydrogen will not under goes spontaneously transition by itself restricted by second law of thermodynamics, different discretised energy and orbital angular momentum require very accurate magnitude of disturbance are required for transitions.

These lead to constant most probable position of particle to be found. In classical terms, which means stationary orbit.

This concept is direct result of the postulate that angular momentum $$L$$ of an electron in an orbit is integral multiple of a given value and is given by: $$L = \frac {nh}{2\pi}$$

At the time it was quite a mystery as to why this postulate should hold, until de Broglie came up with his hypothesis (which was later verified to be true).

de Broglie in his hypothesis, in its essence, stated that all of the matter is dual natured (i.e., a wavevicle), the nature that it shows depends on the type of situation it is in. It behaves like a particle when interacting with other matter to exchange energy and/or momentum whereas it behaves like wave when in motion i.e., without energy and momentum exchange with other particles. The wavelength of such a matter-wave is given by:$$\lambda = \frac hp$$

These electrons in an atom behave like waves. In a hydrogen atom there is a single electron so the calculations for it's different properties become much simpler.

de Broglie reasoned that if an electron revolves around a proton in an orbit of radius $$r$$ then this wave will interfere with itself. He argued that these matter waves will survive only in those orbits where a standing wave forms. This means if this wave had a wavelength of $$\lambda$$ ($$= \frac hp$$) then the orbit radius should be such that:

$$2\pi r = n\frac hp$$

Since $$pr= L$$ this means that:

$$L = \frac {nh}{2\pi}$$

It is worth mentioning here that these matter-waves were later interpreted as probability waves by Max Born.

Note: The method use to analyse the situation is semi-classical in nature hence it doesn't provide complete picture.