Problem 12 - University Physics with Modern Physics

Question

(a) Make a chart showing all the possible sets of quantum numbers \(l\) and \(m_{l}\) for the states of the electron in the hydrogen atom when \(n=5 .\) How many combinations are there? (b) What are the energies of these states?

Step-by-Step Solution

Answer

25 quantum states with energy \(-0.544 \, \text{eV}\).

Step 1: Determine the possible values of l

For a given principal quantum number \(n\), the azimuthal quantum number \(l\) can range from 0 to \(n-1\). Since \(n=5\), \(l\) can take the values 0, 1, 2, 3, and 4.

Step 2: Identify possible ml values for each l

The magnetic quantum number \(m_l\) can take values ranging from \(-l\) to \(+l\). For each \(l\), list the possible \(m_l\) values. - For \(l=0\), \(m_l=0\)- For \(l=1\), \(m_l=-1, 0, 1\)- For \(l=2\), \(m_l=-2, -1, 0, 1, 2\)- For \(l=3\), \(m_l=-3, -2, -1, 0, 1, 2, 3\)- For \(l=4\), \(m_l=-4, -3, -2, -1, 0, 1, 2, 3, 4\)

Step 3: Count the number of quantum number combinations

For each \(l\), count the number of \(m_l\) values:- For \(l=0\), there is 1 value.- For \(l=1\), there are 3 values.- For \(l=2\), there are 5 values.- For \(l=3\), there are 7 values.- For \(l=4\), there are 9 values.Total combinations = \(1 + 3 + 5 + 7 + 9 = 25\).

Step 4: Determine the energy levels of the states

In a hydrogen atom, the energy of an electron state depends only on the principal quantum number \(n\) and is given by the formula: \[ E_n = - \frac{13.6 \, \text{eV}}{n^2} \]For \(n=5\), the energy \(E_5\) is: \( E_5 = - \frac{13.6}{5^2} = -0.544 \, \text{eV} \).

Step 5: Summarize the findings

We have 25 different sets of \((l, m_l)\) quantum numbers when \(n=5\), reflecting the possible states of the electron. Regardless of the values of \(l\) and \(m_l\), all these states have the same energy of \(-0.544 \, \text{eV}\).

Key Concepts

hydrogen atom energy levels

Energy levels in a hydrogen atom are fundamental in understanding atomic structure. These energy states depend significantly on a principal quantum number, denoted as \( n \). For the hydrogen atom, the energy of an electron is principally determined by this number, irrespective of other quantum numbers. All states at a given \( n \) share the same energy level.

The formula used to calculate the energy of these states is given by \[ E_n = - \frac{13.6 \, \text{eV}}{n^2} \]This equation highlights that the energy is negative, indicating bound states, and decreases in magnitude with increasing \( n \). Therefore, as \( n \) increases, energy levels become less negative, meaning they are closer to zero or a free electron state. For example, when \( n=5 \), which represents one of the higher energy levels, the calculated energy is \(-0.544 \, \text{eV}\).

Understanding these energy quantizations offers insights into electron transitions, absorption, and emission of light, crucial for spectroscopic analyses.

principal quantum number

The principal quantum number, represented as \( n \), is one of the four quantum numbers that describe the state of an electron in an atom. It is a critical component determining the energy level and size of an electron's orbit in an atom.

\( n \) can take positive integer values (\( n = 1, 2, 3, \dots \)). Each value corresponds to a specific electron shell or energy level. As \( n \) increases:

• The electron's orbit becomes larger.
• The energy of the electron increases (less negative).
• The electron is further from the nucleus, reducing its binding energy.

In our problem, when \( n = 5 \), it means we're considering the 5th energy level of the hydrogen atom, offering multiple possible sublevels and configurations for electrons, represented by other quantum numbers. Each level can hold a maximum of \( 2n^2 \) electrons, indicating the capacity of different orbitals filled by the electrons.

azimuthal quantum number

The azimuthal quantum number, symbolized as \( l \), defines the shape of an atom's electron subshell and is integral to understanding the atom's structure. It is also influenced by the principal quantum number \( n \).

Given a particular \( n \), \( l \) can hold any integer value ranging from 0 to \( n-1 \). Each \( l \) value corresponds to a specific subshell labeled as follows:

• \( l = 0 \): s subshell
• \( l = 1 \): p subshell
• \( l = 2 \): d subshell
• \( l = 3 \): f subshell
• \( l = 4 \): g subshell

These subshells describe the shapes of the electron cloud around the nucleus and where electrons are likely to be found. In the example of \( n = 5 \), \( l \) can take values from 0 to 4, resulting in subshells s, p, d, f, and g, offering varied orbitals for electron occupancy. The number and type of subshells dictate the geometry and chemical behavior of atoms in reactions.

magnetic quantum number

The magnetic quantum number, denoted as \( m_l \), provides insight into the orientation of electron orbitals within a subshell. This is essential for understanding how orbitals occupy space in atoms.

\( m_l \) is tied directly to the azimuthal quantum number (\( l \)), and for each \( l \), \( m_l \) can take values ranging from \(-l\) to \(+l\). This means each \( l \) has \( 2l + 1 \) possible \( m_l \) values. These numbers specify orbital labeling by demonstrating different spatial orientations:

• For \( l = 0 \), \( m_l = 0 \) (one orientation)
• For \( l = 1 \), \( m_l = -1, 0, 1 \) (three orientations)
• For \( l = 2 \), \( m_l = -2, -1, 0, 1, 2 \) (five orientations)

In a scenario where \( n = 5 \), and thus \( l \) ranges from 0 to 4, \( m_l \) accounts for the spatial alignment of 25 individual orbitals. Despite all orientations having the same energy in hydrogen, \( m_l \) impacts how orbitals interact under external influences like magnetic fields, crucial in complex atomic structures.