Question

What is the ionization energy of a hydrogen atom excited to the \(n=2\) state?

Short answer

Answer: The ionization energy of a hydrogen atom excited to the n=2 state is 3.4 eV.

Step-by-step solution

Recall the formula for the energy levels of the hydrogen atom

The formula for the energy levels of the hydrogen atom is given by: E_n = - \frac{13.6 eV}{n^2}, where n is the principal quantum number corresponding to different energy levels.

Calculate the energy of the electron in n=2 state

We will use the formula we recalled in step 1 to calculate the energy of the electron in n=2 state: E_2 = -\frac{13.6 eV}{(2)^2} = -\frac{13.6 eV}{4} = -3.4 eV

Calculate the energy of the electron in the unbound state (n=infinity)

As the electron moves to an infinitely large distance from the nucleus (n=infinity), the energy of the electron goes to zero. So, E_\infty = 0

Calculate the ionization energy

The ionization energy (IE) is the difference in energy between the unbound state where the electron has been removed from the atom and the initial excited state (n=2). So, we will calculate the difference between the energies of n=infinity and n=2 states: IE = E_\infty - E_2 = 0 - (-3.4 eV) = 3.4 eV Thus, the ionization energy of the hydrogen atom excited to the n=2 state is 3.4 eV.

Key concepts

Energy Levels of Hydrogen Atom

The hydrogen atom, the simplest atom with only one electron, exhibits quantized energy levels. This quantization is a fundamental tenet of quantum mechanics; it denotes that the electron can only inhabit certain well-defined energy levels.

An electron in the hydrogen atom is bound to its nucleus by electromagnetic forces. Its energy is negative, reflecting the fact that work must be done to free the electron from the atom. The energy level formula for the hydrogen atom is \(E_n = -\frac{13.6 \, \text{eV}}{n^2}\) where \(E_n\) is the energy of the electron at a given level and \(n\) is the principal quantum number designating that level.

The value 13.6 eV, known as the Rydberg unit of energy, represents the photon energy needed to ionize a ground state electron completely (where \(n=1\)). When an electron transitions between these levels, it must absorb or emit energy equal to the difference between these levels, in discrete packets called photons.

Visualizing the Energy Levels

One can visualize these levels as concentric circles around the nucleus, with increasing radius. The ground state, \(n=1\), is the smallest circle right around the nucleus. Each subsequent level, with increasing \(n\), is larger and indicates a higher, less negative energy state.

Principal Quantum Number

The principal quantum number, symbolized by \(n\), is one of four quantum numbers that describe the unique quantum state of an electron. It essentially determines the size and energy of the orbital where the electron resides. The lowest value for \(n\) is 1, which corresponds to the closest orbital to the nucleus, the ground state, where the electron has the lowest energy.

As \(n\) increases, the electron orbitals become larger, encompassing more area around the nucleus, and the energy increases (becomes less negative). This is because the electron is less tightly bound to the nucleus and thus has a higher potential energy.

Significance of the Principal Quantum Number

It is significant because it can help us calculate the energy levels of hydrogen with the aforementioned formula and is an integral part in understanding the electron's position and energy. When calculating ionization energy or energy emitted/absorbed during transitions, knowing the principal quantum number is crucial.

Excited State Ionization

Ionization is the process by which an atom loses an electron, gaining a positive charge to become an ion. When we discuss the ionization from an excited state, we are referring to an electron being removed from an atom that is not in its lowest energy level (ground state), but rather in one of the higher 'excited' states.

An electron can be excited to a higher energy state when it absorbs the right amount of energy. Once in this higher energy state, it is closer to the ionization energy: the energy required to completely remove the electron from the influence of the nucleus.

The exercise we've explored gives us the ionization energy for an electron in the \(n=2\) state. This logical progression of calculations underpins the critical concept that the ionization energy depends on the initial state of the electron. In other words, it requires less energy to ionize an electron from, say, the \(n=2\) level compared to the \(n=1\) level. This is because the electron is already further away from the nucleus and in a higher (less bound) energy state.

In summary, understanding the excited state ionization is key for grasping how atoms interact with light and with each other, leading to phenomena such as spectral lines, lasers, and even the physical properties of materials.