Question

(a) Make a chart showing all possible sets of quantum numbers \(l\) and \(m$$_l\) for the states of the electron in the hydrogen atom when n = 4. How many combinations are there? (b) What are the energies of these states?

Short answer

There are 16 combinations, each has an energy of \(-0.85 \text{ eV}\).

Step-by-step solution

Understanding Quantum Numbers

Quantum numbers describe the properties of electrons in an atom. For any electron state, it is defined by a set of four quantum numbers: principal quantum number \(n\), azimuthal quantum number \(l\), magnetic quantum number \(m_l\), and spin quantum number \(m_s\). Here, \(n = 4\) is given, and we need to find possible values of \(l\) and \(m_l\).

Finding Possible Values of \(l\)

The azimuthal quantum number \(l\) can have integer values ranging from 0 to \(n-1\). Since \(n = 4\), the possible values of \(l\) are 0, 1, 2, and 3.

Determining Possible Values of \(m_l\) for Each \(l\)

The magnetic quantum number \(m_l\) ranges from \(-l\) to \(+l\). Thus, 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 \(l = 3\), \(m_l = -3, -2, -1, 0, 1, 2, 3\).

Compiling the Chart of Quantum Number Sets

Compile a chart with quantum numbers \(l\) and \(m_l\) for each possible value of \(l\): \[(l=0, m_l=0), (l=1, m_l=-1, 0, 1), (l=2, m_l=-2, -1, 0, 1, 2), (l=3, m_l=-3, -2, -1, 0, 1, 2, 3)\].

Counting the Total Combinations

Count the total combinations by adding the possible values of \(m_l\) for each \(l\): 1 (for \(l=0\)) + 3 (for \(l=1\)) + 5 (for \(l=2\)) + 7 (for \(l=3\)) = 16 combinations.

Calculating the Energy of Each State

The energy of an electron in the hydrogen atom depends only on the principal quantum number \(n\) and is given by \(E_n = -\frac{13.6}{n^2} \text{ eV}\). For \(n = 4\), the energy \(E_4 = -\frac{13.6}{4^2} = -0.85 \text{ eV}\). This means all these states have the energy of \(-0.85\text{ eV}\).

Key concepts

Hydrogen Atom

The hydrogen atom is the simplest atom in the universe, consisting of only one proton and one electron. This simplicity uniquely positions it as an excellent model for understanding quantum mechanics and electron behavior. In quantum mechanics, atoms are described in terms of quantum states rather than classical orbits. For hydrogen, which has only one electron, these quantum states are clearly discerned through quantum numbers that we will explore further. Understanding the hydrogen atom is crucial in physics because it provides basic insights into how atomic structures work. Exploring the electron's behavior within a hydrogen atom reveals foundational concepts about energy quantization and the nature of electron configurations. The hydrogen atom's solitary electron makes it ideal for study since it avoids the complexities of electron-electron interactions present in larger atoms.

Electron States

Electron states are specific conditions or configurations that an electron can occupy in an atom. These states are defined by quantum numbers, which describe unique energies and spatial distributions of electrons. For the hydrogen atom, electrons exist in discrete energy levels, visualized as orbits or electron clouds around the nucleus. Each of these discrete levels corresponds to a specific electron state. In simpler terms, think of electron states as different rooms that an electron can occupy within the atom's structure. The electron doesn't just move randomly between these rooms; its position and energy level are quantized, meaning they take on specific values. Since the hydrogen atom only contains a single electron, understanding its electron states is more straightforward, allowing us to develop intuitive models of atomic behavior.

Principal Quantum Number

The principal quantum number, denoted by the symbol \(n\), is a quantum number that describes the energy level or shell the electron occupies. It plays a crucial role in determining the size and energy of orbitals within an atom. The principal quantum number can take positive integer values such as 1, 2, 3, etc. The value of \(n\) determines the main energy level of the electron, with larger values indicating higher energy levels and larger orbital sizes. For hydrogen, the energy of an electron is given by the expression \(E_n = -\frac{13.6}{n^2} \text{ eV}\), showing how energy decreases as \(n\) increases. Understanding \(n\) is essential for comprehending energy levels in atoms. As \(n\) increases, the electron's energy becomes less negative, suggesting it is further from the nucleus and more capable of transitioning between states. This concept is vital for understanding phenomena like atomic spectra and chemical bonding.

Energy Levels

Energy levels in atoms denote the discrete layers or shells where electrons exist. These levels are defined by various quantum numbers, but primarily by the principal quantum number \(n\). Each energy level corresponds to a set energy value, which is quantized, meaning electrons can only occupy specific levels rather than a continuous range. In the hydrogen atom, the simplicity allows us to see how energy levels are ordered and integrated. All electron states with the same \(n\) value share the same energy level, regardless of their sub-level differences in quantum numbers. For \(n=4\), the energy is \(-0.85 \text{ eV}\), illustrating that electrons in higher shells have less negative energies and are more distant from the nucleus than those in lower shells. Energy levels explain how electrons absorb and emit radiation. Transitions between different levels involve absorbing or releasing energy, which is foundational for atomic spectroscopy and the understanding of atomic interactions in chemistry and physics.