On the basis of Bohr's model, the radius of the 3rd orbit is (a) Equal to the radius of first orbit (b) Three times the radius of first orbit (c) Five times the radius of first orbit (d) Nine times the radius of first orbit

Bohr's atomic theory revolutionized the understanding of the atomic structure in the early 20th century. Proposed by Niels Bohr in 1913, this theory combines classical mechanics with early quantum concepts. According to Bohr, an atom consists of a dense nucleus surrounded by electrons in discrete energy levels, or orbits.

Key to this theory is the idea that electrons cannot spiral into the nucleus, despite electromagnetic attraction, because they are restricted to certain orbits with fixed energy levels. When an electron jumps between these energy levels, it does so without crossing the intervening space—an early nod to quantum leaps.

One of the main tenets of Bohr's atomic theory is that the electrons in atoms orbit the nucleus in specific allowable paths. As long as an electron remains in a given path, it does not radiate energy. Radiation occurs only when an electron jumps from one allowed orbit to another. These transitions are responsible for the atomic spectral lines observed in emission and absorption spectra.

The radius of an electron orbit in Bohr's model is one of the critical factors determining the atom's energy levels. The radius increases with the square of the principal quantum number, which indicates the electron's distance from the nucleus. This relationship is described by the formula:

\( r_n = r_1 n^2 \)
where \( r_n \) is the radius of the nth orbit, \( r_1 \) is the radius of the first orbit, and \( n \) is the principal quantum number. The larger the value of \( n \), the larger the orbit and the higher the energy level.

In terms of size, going from the first to the second orbit already increases the radius by a factor of four, since \( 2^2 = 4 \). This exponential scaling demonstrates that electron orbits become substantially larger as one moves farther away from the nucleus.

One of the groundbreaking aspects of Bohr's model was its suggestion that angular momentum should be quantized. In classical mechanics, angular momentum can take on any value. Bohr, however, proposed that the angular momentum of an electron in orbit around a nucleus can only possess certain discrete values. These allowed values are integral multiples of a reduced Planck’s constant divided by \(2\pi\):

\( L = n\frac{h}{2\pi} \)
where \( L \) represents the angular momentum, \( n \) is the principal quantum number, and \( h \) is Planck's constant. This quantization of angular momentum explains why electrons remain in stable orbits and do not radiate energy continuously, as classical physics would predict.

The principal quantum number, often designated as \( n \), plays a significant role in the Bohr model of the atom. It is a positive integer (\( n=1, 2, 3, ... \)) that determines the size and energy level of an electron's orbit, with \( n=1 \) being the closest orbit to the nucleus and the one with the lowest energy.

As the value of \( n \) increases, so does the radius of the electron's orbit and the energy level. An increase in the principal quantum number also corresponds to higher potential energy for the electron, as it is found further from the nucleus. The concept of energy levels associated with the principal quantum number helps explain why electrons occupy particular orbits and how they transition between them, emitting or absorbing specific amounts of energy, which relates to the characteristic spectra of elements.