The electron energy in hydrogen atom is given by \(E n=\) \(\left(-2.18 \times 10^{-18}\right) / n^{2} \mathrm{~J}\). Calculate the energy required to remove an electron completely from the \(n=2\) orbit. What is the longest wavelength of light in \(\mathrm{cm}\) that can be used to cause this transition?

Ionization energy is a crucial concept in the study of atomic structure, particularly the hydrogen atom. It represents the amount of energy necessary to completely remove an electron from an atom or ion.

To determine the ionization energy for a hydrogen atom, we start by understanding that the energy levels of an electron in a hydrogen atom can be described using a simple formula: \( E_n = \frac{-2.18 \times 10^{-18}}{n^2} \) joules, where \(n\) is the principal quantum number corresponding to the electron's orbit.

For the electron in the second orbit, when \(n=2\), we simply plug the value into the equation to get \(E_2\). To calculate the energy required to remove this electron (the ionization energy), we consider the energy at infinity (which is zero, because it represents a point where the electron is no longer bound to the nucleus) and subtract \(E_2\) from it. This provides us with the ionization energy, indicating that the electron has been completely removed from the atom.

The value obtained not only represents the ionization threshold but also aids in understanding the strength of the electron's attachment within the atom. The higher the ionization energy, the more tightly bound the electron is to the nucleus.

The relationship between energy and wavelength is one of the cornerstones of quantum mechanics and is described by the equation \(E = h\frac{c}{\lambda}\). Here, \(E\) signifies the photon energy, \(\lambda\) is the wavelength, \(h\) denotes Planck's constant, and \(c\) stands for the speed of light.

The equation shows that energy and wavelength are inversely proportional— as the energy goes up, the wavelength decreases, and vice versa. When dealing with ionization, where photons must supply energy to remove an electron, knowing the longest wavelength of light that can cause this transition is vital.

This wavelength corresponds to the minimum energy needed to ionize the atom. By rearranging the energy-wavelength equation to solve for \(\lambda\), we calculate the maximum wavelength of light that can bring about the electron's ejection from the atom. This not only applies to fundamental scientific research but also has practical implications in technologies such as lasers and spectroscopy. Converting this wavelength into different units, such as centimeters, can further assist with experimental design and comparison.

Planck's constant, symbolized by \(h\), is a vital element of quantum mechanics, integral to the energy-wavelength relationship. It represents the fundamental limits of action in the universe and quantifies the discrete nature of energy exchange in quantum phenomena.

With a value of approximately \(6.626 \times 10^{-34} \) joule seconds, Planck's constant serves as the bridge between the macroscopic and quantum worlds. Understanding Planck's constant is not only important for calculations involving the energy and frequency of photons, but it also underpins the principles of energy quantization in atomic and subatomic systems.

Named after Max Planck, who proposed its existence, this constant allows scientists to calculate the energy of photons from their frequency or wavelength. Essentially, it's Planck's constant that gives us the precise scale to observe and calculate the effects of quantum mechanics, making it a linchpin quantity that students must become comfortable with when delving into the world of atomic physics and beyond.