Calculate the radius of Bohr's fifth orbit for hydrogen atom.

The Bohr model is a key concept in understanding how atoms are structured, particularly the hydrogen atom. It was proposed by Niels Bohr in 1913 to explain how electrons can have stable orbits around the nucleus without radiating energy, which classical physics could not explain. According to Bohr, electrons revolve around the nucleus in specific paths or 'orbits' that have quantized angular momenta. These orbits correspond to different energy levels, and the electron can only exist in these defined orbits, neither radiating nor absorbing energy as it does so.

Every orbit, according to Bohr's theory, is associated with a certain energy level, and to jump from one level to another, electrons must absorb or emit energy equal to the difference between the two levels. This energy is often in the form of a photon. In the case of a hydrogen atom, which has only one electron, these transitions give rise to the emission or absorption spectra characteristic of hydrogen.

The Bohr model was revolutionary because it incorporated quantum concepts into the classical understanding of atomic structure, laying the groundwork for modern quantum mechanics. Despite its limitations, such as not accounting for the behavior of atoms with more than one electron, it remains a fundamental concept in atomic physics and chemistry.

The principal quantum number, denoted as 'n,' is a fundamental concept in quantum mechanics and a crucial part of the Bohr model. It is an integer value that determines the electron's energy level and its orbit within an atom. Within Bohr's hydrogen atom model, the principal quantum number corresponds to the size and energy of an electron orbit.

Values of 'n' start from 1, which is the ground state, or the closest orbit to the nucleus where an electron can be found. As 'n' increases, the electron's energy and the orbit's radius increase, placing the electron further from the nucleus. The significance of the principal quantum number is that it quantizes the electron orbits, meaning the orbits are not arbitrary but rather fixed and only increase in discrete increments as n increases. This quantization is aligned with the observed spectra of atoms, where only certain energy levels are present, validating the existence of these discrete orbits.

Understanding the principal quantum number is vital for comprehending many atomic processes, such as the emission and absorption of light, chemical bonding, and the electron configurations of elements.

Calculating the radius of an atomic orbit within the Bohr model is a mathematical exercise that provides insight into the atomic structure according to early quantum theory. According to Bohr's model, the radius of the nth orbit in a hydrogen atom is given by the formula:
\( r_n = n^2 \times r_0 \),
where \( r_0 \) is the Bohr radius, approximately \( 0.529 \times 10^{-10} \text{m} \), and the value of \( n \) is the principal quantum number for the desired orbit. The Bohr radius represents the size of the smallest orbit, which is the ground state orbit for hydrogen.

To find the radius of any orbit, simply square the principal quantum number for that orbit and multiply it by the Bohr radius. This calculation stems from the quantized nature of electron orbits in the Bohr model. It is particularly useful in visualizing the relative sizes of electron orbits and, by extension, understanding their energy levels.

For example, to calculate the fifth orbit's radius, we use \( n = 5 \). Substituting it into the equation gives \( r_5 = 5^2 \times r_0 \), which simplifies to \( r_5 = 25 \times 0.529 \times 10^{-10} \text{m} \), resulting in \( r_5 = 1.3225 \times 10^{-9} \text{m} \). This gives a clear illustration of how electron orbits in a hydrogen atom expand with increasing principal quantum number.