Question
How did de Broglie hypothesis lead to Bohr’s quantum condition of atomic orbits?
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Answer
According to Bohr’s quantum condition “Only those atomic orbits are allowed as stationary orbits in which angular momentum of an electron is the integral multiple of $\frac{\text{h}}{2\pi}."$ If m is the mass, v velocity and r radius of orbit, then angular momentum of electron L = mvr. According to Bohr’s quantum condition, $\text{mvr}=\text{n}\frac{\text{h}}{2\pi}\dots(\text{i})$ According to de Broglie quantum condition only those atomic orbits are allowed as stationary orbits in which circumference of electron-orbit is the integral multiple of de Broglie wavelength associated with electron, i.e., $2\pi\text{r}=\text{n}\lambda\dots(\text{ii})$ According to de Broglie hypothesis, $\lambda=\frac{\text{h}}{\text{mv}}\dots(\text{iii})$ Substituting this value in (ii), we get $2\pi\text{r}=\text{n}\Big(\frac{\text{h}}{\text{mv}}\Big)\Rightarrow\ \text{mvr}=\text{n}\frac{\text{h}}{2\pi}$This is Bohr's quantum condition.
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