How far from the nucleus in angstroms ( 1 angstrom \(=1 \times 10^{-10} \mathrm{m}\) ) is the electron in a hydrogen atom if it has an energy of \(-8.72 \times 10^{-20} \mathrm{J} ?\)

The principal quantum number, denoted as 'n', plays a crucial role in the Bohr model of the hydrogen atom. It is a positive integer that specifies the energy level and size of the electron orbit. The value of 'n' determines how far an electron is from the nucleus, with each increasing number indicating a larger orbit with higher energy.

In the context of the exercise, we identified the energy level of an electron by rearranging the Bohr energy formula and solving for 'n'. Since the principal quantum number must be a whole number (1, 2, 3, ...), and it defines the discrete energy levels of an electron in a hydrogen atom, we rounded the initial calculation to the nearest whole number to determine the appropriate energy level where the electron resides.

In the Bohr model, the energy levels of a hydrogen atom are quantized, meaning an electron can only possess certain discrete energy values. These energy levels are negatively valued and become less negative as you move away from the nucleus (i.e., at higher principal quantum numbers). The formula given by the Bohr model to calculate the energy is: \( E_n = -\frac{13.6 \text{eV}}{n^2} \).

When we know the energy value, such as the \( -8.72 \times 10^{-20} \text{J} \) given in the problem, we can find out which energy level (principal quantum number 'n') the electron occupies. As exemplified, converting the energy to electron volts and inputting it into the Bohr energy formula, we are able to identify that the electron is on the second energy level, corresponding to \( n = 2 \).

The Bohr radius formula provides the means to calculate the radius of an electron's orbit based on its principal quantum number. For a hydrogen atom, this formula is \( r_n = n^2 \times 0.529 \) angstroms, where \( r_n \) is the radius for a given principal quantum number 'n'.

Utilizing the principal quantum number found from the energy level, we apply this formula to determine the exact distance of the electron from the nucleus, which is crucial for visualizing the structure of a hydrogen atom. In the exercise, applying the formula with \( n = 2 \) gives us a radius of \( r_2 = 4.116 \) angstroms. This radius quantifies the space an electron occupies and is pivotal for understanding atomic dimensions and further calculations related to the hydrogen atom.