Question

Does the hydrogen atom "expand" or "contract" when an electron is excited from the \(n=1\) state to the \(n=3\) state?

Short answer

When an electron in a hydrogen atom is excited from the \(n=1\) state to the \(n=3\) state, the average distance of the electron from the nucleus increases, as calculated using the formula \(r_n = a_0n^2\). Since the average distance is greater in the \(n=3\) state compared to the \(n=1\) state, the hydrogen atom "expands" during this transition.

Step-by-step solution

Recall the Bohr Model for Hydrogen Atom

The Bohr model is a simplified model of a hydrogen atom where the electron orbits around the nucleus in specific energy levels. Each energy level is represented by an integer number called the principal quantum number, denoted by n.

Understand the meaning of "expanding" and "contracting"

As an electron moves from a lower energy level to a higher energy level, its average distance from the nucleus, also known as the radius of the orbital, tends to increase. If the average distance (radius) increases, then the hydrogen atom is said to "expand." Conversely, if the average distance (radius) decreases, the hydrogen atom is said to "contract."

Calculate the average distance of the electron in energy levels n=1 and n=3

The average distance of the electron in the nth orbital can be calculated using the formula: \(r_n = a_0n^2\), where \(a_0\) is the Bohr radius and roughly equals \(5.29 \times 10^{-11}\) meters. For n=1: \(r_1 = a_0(1)^2 = a_0\) For n=3: \(r_3 = a_0(3)^2 = 9a_0\) Now we need to compare these distances.

Compare the average distances to determine if the hydrogen atom expands or contracts

Since \(r_3 = 9a_0\) is larger than \(r_1 = a_0\), it means that the electron is on average farther from the nucleus when it is in the n=3 state than when it is in the n=1 state. Therefore, the hydrogen atom "expands" when an electron is excited from the n=1 state to the n=3 state.

Key concepts

Energy Levels

In the Bohr model of the hydrogen atom, energy levels are depicted as discrete circular orbits around the nucleus where the electron can reside. Each energy level corresponds to a specific amount of energy that the electron has. Think of these levels like rungs on a ladder – the electron can "jump" from one rung to another, but it cannot reside in between these rungs.

The lower the energy level, the less energy is required for the electron to remain there. As the electron moves to higher energy levels, it absorbs energy corresponding to the energy difference between the levels.

• The lowest energy level (closest to the nucleus) is called the ground state and is designated by the principal quantum number, \( n = 1 \).
• Higher energy levels (further from the nucleus) are termed excited states, such as \( n = 2 \), \( n = 3 \), and so on.

When an electron transitions to a higher energy level, it occupies a position further from the nucleus, resulting in an increased orbital radius and, thus, a larger "size" of the atom as seen in the hydrogen atom expansion from \( n=1 \) to \( n=3 \).

Principal Quantum Number

The principal quantum number, symbolized as \( n \), is a fundamental concept in quantum mechanics. It defines the main energy level that an electron occupies in an atom. The principal quantum number is always a positive integer (\( n = 1, 2, 3, \ldots \)), indicating the relative distance of an electron's orbit from the nucleus.

Each principal quantum number correlates with a specific energy level:

• \( n = 1 \) defines the first energy level or ground state.
• \( n = 2 \), \( n = 3 \), etc., correspond to higher, excited states.
• The larger the \( n \), the higher the energy and the farther the electron is from the nucleus.

For instance, when an electron "jumps" from \( n = 1 \) to \( n = 3 \), it moves to a significantly higher energy level, increasing the atomic orbital's average distance from the nucleus. This expansion from the core is due to the increased energy and thus increased principal quantum number.

Hydrogen Atom Expansion

The concept of hydrogen atom expansion comes into play when discussing electrons transitioning between different energy states. In the Bohr model, when an electron absorbs energy, it transitions to a higher energy level, characterized by a larger principal quantum number. This transition leads to an increase in the average orbital radius of the electron.

Mathematically, the average distance from the nucleus, or the radius \( r_n \), can be described by the formula:

• \( r_n = a_0n^2 \)

Where \( a_0 \) is the Bohr radius (approximately \( 5.29 \times 10^{-11} \) meters). Hence, as the principal quantum number \( n \) increases, \( r_n \) grows, indicating that the atom "expands."

For example, the transition of an electron in hydrogen from \( n=1 \) to \( n=3 \) results in the radius growing from \( a_0 \) to \( 9a_0 \), a clear expansion of the hydrogen atom. This larger radius means the electron is now farther from the nucleus, effectively expanding the size of the atom in its excited state.