Question

The distance between 4th and 3rd Bohr orbits of$H{e^ + }$ is:
A. $2.645 \times {10^{ - 10}}m$
B. $ 1.322 \times {10^{ - 10}}m$
C. $1.851 \times {10^{ - 10}}m$
D. None of the above

Answer 1

Hint: In a hydrogen atom in its ground state, the Bohr radius is the most likely distance between the nucleus and the electron (non-relativistic and with an infinitely heavy proton). It was named after Niels Bohr because of its importance in the Bohr atom model.

Complete answer:
Niels Bohr proposed the Bohr model for atomic structure in 1913, in which electrons circle a central nucleus under electrostatic attraction. The original derivation proposed that electrons had orbital angular momentum in integer multiples of the reduced Planck constant, which correctly predicted discrete energy levels in emission spectra as well as a set radius for each of these levels. A single electron circles the nucleus of the simplest atom, hydrogen, and its smallest feasible orbit, with the lowest energy, has an orbital radius almost equal to the Bohr radius. The electron probability cloud following the Schrodinger equation has supplanted the Bohr model of the atom, which is further complicated by spin and quantum vacuum effects to generate fine and hyperfine structure. Despite this, the Bohr radius formula is still used extensively in atomic physics computations, thanks to its straightforward connection with other basic constants.
Hence radius of ${n^{th}}$ orbit is given by the formula $r = \dfrac{{0.529{n^2}}}{Z}\dot A$
The distance between 4th and 3rd Bohr orbits of $H{e^ + }$ is:
Z = 2
$r = \dfrac{{0.529{{(4)}^2}}}{2} - \dfrac{{0.529{{(3)}^2}}}{2}\dot A$
$r = \dfrac{{0.529[{{(4)}^2} - {{(3)}^2}]}}{2}\dot A$
$r = \dfrac{{0.529[7]}}{2}\dot A$
$r = \dfrac{{3.703}}{2}\dot A$
$ \Rightarrow r = 1.851 \times {10^{ - 10}}m$

Note:
Typically, Bohr model relations for these unusual systems may be easily changed by simply substituting the electron mass with the system's decreased mass. The Bohr radius is the radius with the highest radial probability density, not the anticipated radial distance. Because of the large tail of the radial wave function, the predicted radial distance is 1.5 times the Bohr radius. Another key contrast is that the highest probability density in three-dimensional space occurs at the nucleus, not at the Bohr radius, but the radial probability density peaks at the Bohr radius.