Question

If in Bohr’s model for a unielectronic atom the time period of a revolution is represented as Tn,Z where n represents the shell no. and Z represents the atomic number, then what will be the value of T1,2 : T2,1?

Answer 1

Let's derive time period using Bohr's theory,

according to Bohr's theorem,

angular momentum of electron in nth orbit is integral multiple of h/2π.

or, $mvr=\frac{nh}{2\pi}$.....(1)

also electrostatic force is balanced by centripetal force.

kZe²/r² = mv²/r

or, v² = KZe²/mr

putting value of v in equation (1),

then, r = n²h²/4π²mKZe² .....(2)

equation (1),

$mvr=m\omega r^2=\frac{nh}{2\pi}$

or, $\frac{2\pi}{T}=\frac{nh}{2\pi r^2}$

or, $T=\frac{4\pi^2 r^2}{nh}$

from equation (2),

$T=\frac{n^3h^3}{4\pi^2m^2K^2Z^2e^4}$

hence, time period is directly proportional to (n³/Z²)

then, $\frac{T_{(1,2)}}{T_{(2,1)}}=\frac{1\times2^3}{2^2\times1}$

= 2 : 1