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Question: Calculate the radius of \( {1^{(st)}},{2^{(nd)}},{3^{(rd)}},{4^{(th)}} \) Bohr's Orbit of hydrogen....

Calculate the radius of 1(st),2(nd),3(rd),4(th){1^{(st)}},{2^{(nd)}},{3^{(rd)}},{4^{(th)}} Bohr's Orbit of hydrogen.

Explanation

Solution

Hint : 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 possessed 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.

Complete Step By Step Answer:
Using fundamental physics, the planetary model of the atom, and some very significant new ideas, Bohr was able to establish the formula for the hydrogen spectrum. His initial idea is that only certain orbits are permitted: we refer to electron orbits in atoms as quantized orbits. Each orbit has a distinct energy, and electrons can absorb energy to travel to a higher orbit or release energy to fall to a lower orbit. The quantity of energy received or released is quantized when the orbits are quantized, resulting in discrete spectra. The major mechanisms of moving energy into and out of atoms are photon absorption and emission.
We know that Radius of Bohr's orbit r=0.529×n2Zr = 0.529 \times \dfrac{{{n^2}}}{Z}
(a) Radius of 1st{1^{st}} orbit:
r=0.529×121=0.529A∘r = 0.529 \times \dfrac{{{1^2}}}{1} = 0.529{\mathop A \limits^\circ}
(b) Radius of 2nd {2^{{\text{nd }}}} orbit:
r=0.529×221=0.529×4=2.116A∘r = 0.529 \times \dfrac{{{2^2}}}{1} = 0.529 \times 4 = 2.116\mathop {\mathop A \limits^\circ}
(c) Radius of 3rd {3^{{\text{rd }}}} orbit:
r=0.529×321=0.529×9=4.761A∘r = 0.529 \times \dfrac{{{3^2}}}{1} = 0.529 \times 9 = 4.761{\mathop A \limits^\circ}
(d) Radius of 4th {4^{{\text{th }}}} orbit:
r=0.529×421=0.529×16=8.464A∘r = 0.529 \times \dfrac{{{4^2}}}{1} = 0.529 \times 16 = 8.464{\mathop A \limits^\circ} .

Note :
No one had ever been able to do what Bohr had. He not only explained the hydrogen spectrum, but he also properly computed the atom's size using simple physics. Some of his concepts may be applied to a wide range of situations. All atoms and molecules have quantized electron orbital energies. The quantity of angular momentum is quantized. The electrons do not spiral toward the nucleus as conventionally predicted (accelerated charges radiate, causing electron orbits to decay fast and the electrons to sit on the nucleus, causing matter to collapse).