The radius of the $n$th Bohr orbit in a hydrogen-like atom of nuclear charge $Z$ scales as
A$r_n \propto \tfrac{n}{Z^2}$
B$r_n \propto nZ$
C$r_n \propto \tfrac{n^2}{Z}$
D$r_n \propto \tfrac{Z}{n^2}$
Answer & Solution
Correct answer: C. $r_n \propto \tfrac{n^2}{Z}$
Equating the Coulomb force $\tfrac{kZe^2}{r^2}$ with the centripetal demand and substituting Bohr's quantisation $mvr = n\hbar$ gives $r_n = \tfrac{n^2 \hbar^2}{m k Z e^2}$, i.e. $r_n \propto \tfrac{n^2}{Z}$. Energy scales the opposite way: $E_n \propto -\tfrac{Z^2}{n^2}$.
Related questions
The ionisation energy of the hydrogen atom is about:The lowest energy state of an atom is called the:The energy of the emitted photon equals the difference between the:By Bohr's third postulate, a transition to lower energy emits a:Allowed angular momenta are integral multiples of h divided by:Bohr's second postulate quantises which quantity?Bohr called the stable non-radiating orbits the:By Bohr's first postulate, an electron in a stable orbit emits: