Bohr's model of atom

Question

In our textbook, under developments that lead to Bohr's atomic model, it is stated

1. Dual nature of electromagnetic radiation.

2. atomic spectra which could be explained only by assuming quantized electronic energy levels.

Now Bohr used quantization of energy by $E=hf$ to explain $H$ spectra, but I do not understand how the dual nature of radiation comes into play here? As far as I understand only particle nature has been considered here by assuming energy is released in form of packets when electrons jump to a lower state.

Also, how did Bohr overcome the limitation of Rutherford that electrons should lose energy due to acceleration and spiral into the nucleus? Everywhere I searched it's simply stated that they have "fixed energy orbits". But so what? They are still accelerating either way. I just can't take that in, it sounds too make-believe.

Answer 1

What's the importance of the dual nature of particles in Bohr's model?

Bohr's condition, that the angular momentum is an integer multiple of $\hbar$ was later reinterpreted in $1924$ by de Broglie as a standing wave condition: the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit:

$$n\lambda=2\pi r$$ $$\lambda=\frac{h}{mv} \ \ \text{De Broglie Hypothesis}$$ $$\Rightarrow \frac{nh}{2\pi}=mvr$$ $$l=n\hbar$$ The stationary orbits are attained at distances for which the angular momentum of the revolving electron is an integer multiple of the reduced Planck constant.

How Bohr's model overcomes the problem with the crashing of electrons in the nucleus?

According to Bohr's model:

Certain stationary states exist in atoms, which differ from the classical stable states in that the orbiting electrons do not continuously radiate electromagnetic energy. The stationary states are states of definite total energy.

which was at that time put by the Bohr somewhat arbitrarily. But It was later explained by Quantum theory. Bohr believed that the assumption is self-evident as the electron does not crash to the nucleus and thus must not emit radiation. He did not give any classical reason for such behavior.