Bohr Model of Atom: Stability Condition Explained

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The discussion centers on the stability condition of the Bohr model of the atom, specifically the equation k Ze²/r² = mv²/r. A participant questions why the atomic number Z is not squared in the equation, suggesting that with Z electrons and Z protons, the interaction should yield a term of -Z²e². It is clarified that the equation applies to a one-electron atom, such as hydrogen, and Z is left open to accommodate other ions like He+. Additionally, the equation presented is identified as the classical force equation for circular orbits, while the actual stability condition is Bohr's condition mvr = nħ. Understanding these distinctions is crucial for grasping the Bohr model's framework.
connor415
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Eisberg and Resnick says that the condition for stability of the atom is:

k Ze2/r2=mv2/r

What I fail to understand is why the Z isn't square too. I mean surely you have Z electrons interacting with Z protons which which would surely give q1q2=(Ze)(-Ze)=-Z2e2?

What am I missing?
 
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No. They are dealing with a one electron atom. You might say: then it's hydrogen, so why not put Z = 1? They leave Z open so the same treatment will do for, an He+ ion and so on.

Incidentally the equation you give isn't really the condition for stability. It is the classical F = ma equation for circular orbits. The stability condition is Bohr's condition: mvr = n\hbar.
 
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