Question

In the hydrogen atom, what is the physical significance of the state for which \(n=\infty\) and \(E=0\) ?

Short answer

In the hydrogen atom, the physical significance of the state for which \(n=\infty\) and \(E=0\) is that the electron is becoming infinitely far away from the nucleus and is no longer bound to it, turning into a free electron. This state represents the ionization limit, where the hydrogen atom loses its electron and becomes ionized.

Step-by-step solution

Understand the meaning of principal quantum number n

The principal quantum number, n, is a positive integer that describes the electron shell of an electron within an atom. Higher values of n correspond to shells with higher energy and larger radii. In other words, as n increases, the electron is further away from the nucleus and has a higher energy.

Analyze the implications of n approaching infinity

When the principal quantum number n approaches infinity, it means that the electron is moving very far away from the nucleus. In fact, the electron is becoming so far from the nucleus that the attractive force between them becomes weaker. Eventually, the electron is hardly bound to the nucleus at all.

Evaluate the significance of the energy state E being 0

The energy of an electron in a hydrogen atom is given by the formula: \[E = \frac{-13.6 \, eV}{n^2}\] As n approaches infinity, the energy E effectively becomes 0. A 0 energy state means that the electron is no longer bound to the nucleus. It has gone from a bound state within an atom to a free state, where it is considered a free electron.

Conclude the physical significance of the state for which n approaches infinity and E=0

In conclusion, the physical significance of the hydrogen atom state for which n approaches infinity and E=0 is that the electron is becoming infinitely far away from the nucleus and is no longer bound to it, turning into a free electron. This is essentially the ionization limit, where the hydrogen atom loses its electron and becomes ionized.

Key concepts

Principal Quantum Number

The principal quantum number, often denoted by the letter \( n \), is a fundamental concept in quantum physics that pertains to hydrogen atoms and indeed all atoms. It's a positive integer that indicates an electron's energy level and the general size of its orbit around the nucleus.
A helpful way to think about the principal quantum number is as a kind of address, telling us how far the electron's orbit is from the nucleus. The principal quantum number can take on values like 1, 2, 3, and so on.

• When \( n = 1 \), the electron is in the closest orbit to the nucleus, often called the ground state.
• As \( n \) increases to 2, 3, 4, the electron occupies shells further and further from the nucleus.

Each of these shells has a larger radius and corresponds to a higher energy level. Thus, the principal quantum number not only indicates distance but is also intrinsically tied to the energy an electron possesses.

Energy Levels

Energy levels in an atom are precisely what they sound like—they denote the specific energies that electrons can have while orbiting the nucleus. The electron's energy level is synonymous with the shell it resides in, as determined by the principal quantum number \( n \).
The formula to determine the energy level of an electron in a hydrogen atom is given by:
\[ E = \frac{-13.6 \, \text{eV}}{n^2} \]
This equation tells us a crucial fact: as the principal quantum number \( n \) increases, the absolute value of the energy decreases, meaning the electron becomes less and less tightly bound to the nucleus.

• With smaller values of \( n \), electrons are closer to the nucleus and thus, have lower energy levels.
• As \( n \) grows larger, electrons have higher energy levels and are further away from the nucleus.

In the context of the hydrogen atom, when the energy goes to zero, this signifies an electron at the ionization limit, transitioning from a bound state to being free and unbound, something that happens as \( n \) approaches infinity.

Ionization Limit

The ionization limit is a key concept when discussing the hydrogen atom, where it relates to the process of an electron leaving an atom entirely. Ionization occurs when an electron's energy level rises enough such that it is no longer confined to orbiting the nucleus, effectively making it a free electron.
When we say an electron approaches the ionization limit, we mean:

• The principal quantum number \( n \) approaches infinity, implying the electron is exceedingly far from the nucleus.
• The electron's energy level approaches zero, as per the energy formula \( E = \frac{-13.6 \, \text{eV}}{n^2} \).

At this point, the energy is sufficient to overcome the attractive forces of the nucleus entirely. Therefore, the ionization limit represents the threshold where the atom can no longer hold the electron. In practical terms, reaching this limit in a hydrogen atom means the electron becomes free, leading to "ionization"— a state important in fields from astrophysics to modern electronics, where ionized gases or plasmas are commonly encountered.