Question

a) How many different 5\(g\) states does hydrogen have? (b) Which of the states in part (a) has the largest angle between \(\vec L\) and the z-axis, and what is that angle? (c) Which of the states in part (a) has the smallest angle between \(\vec L\) and the z-axis, and what is that angle?

Short answer

There are 9 different 5g states. The largest angle is 90° and the smallest is approximately 26.57°.

Step-by-step solution

Understand the Quantum Numbers

For a hydrogen atom, the principal quantum number is denoted by \(n\), and the azimuthal or angular momentum quantum number is \(l\). The magnetic quantum number \(m_l\) ranges from \(-l\) to \(+l\). A 5\(g\) state indicates \(n = 5\) and \(l = 4\), since \(g\) corresponds to \(l = 4\).

Determine the Number of 5g States

The number of different magnetic states for a given \(l\) is determined by the number of possible \(m_l\) values, which are \(2l +1\). For \(l = 4\), the \(m_l\) values range from \(-4\) to \(+4\), giving us \(9\) possible states: \(-4, -3, -2, -1, 0, 1, 2, 3, 4\).

Use Vector Model of Angular Momentum

The angle \(\theta\) between \(\vec{L}\) (the angular momentum vector) and the z-axis is given by \(\cos{\theta} = \frac{m_l}{\sqrt{l(l+1)}}\). We will use this formula to find the angles for different \(m_l\) values.

Largest Angle Calculation

The largest angle occurs when \(|m_l|\) is smallest because the cosine value is smallest. For \(m_l = 0\), the angle \(\theta\) is \(\cos^{-1}(0) = \frac{\pi}{2} \text{ radians} = 90^{\circ}\).

Smallest Angle Calculation

The smallest angle occurs when \(|m_l|\) is largest, \(m_l = \pm4\). The angle is \(\theta = \cos^{-1}\left(\frac{4}{\sqrt{4(4+1)}}\right) = \cos^{-1}\left(\frac{4}{\sqrt{20}}\right) = \cos^{-1}\left(\frac{4}{2\sqrt{5}}\right)\). Simplifying gives \(\cos^{-1}\left(\frac{2}{\sqrt{5}}\right)\).

Calculate Smallest Angle Numerically

Compute \(\cos^{-1}(\frac{2}{\sqrt{5}})\): Use a calculator to find \(\cos^{-1}(0.894)\), which is approximately \(26.57^{\circ}\).

Key concepts

Angular Momentum

In quantum mechanics, angular momentum is an intrinsic property of particles, such as electrons in an atom. It is analogous to the classical concept of angular momentum which describes the rotation of objects. However, in quantum systems, angular momentum is quantized, meaning it only takes on specific discrete values.

For atoms, the angular momentum of electrons is described using quantum numbers. These quantum numbers allow us to determine the energy level and shape of an electron's orbit. Angular momentum is an essential factor in understanding atomic structures and electron configurations.

• The total angular momentum \(\vec L\) is a vector quantity combining both magnitude and direction.
• Magnitude is calculated as \(|\vec L| = \sqrt{l(l+1)}\hbar\), where \(\hbar\) is the reduced Planck's constant.
• Direction is often described with respect to a chosen axis, typically the z-axis in spherical coordinates.

By applying these principles to a hydrogen atom, we can analyze the behavior and orientation of electron pathways, leading to a deeper comprehension of chemical bonding and material properties.

Magnetic Quantum Number

The magnetic quantum number, denoted by \(m_l\), is one of the quantum numbers that describe an electron's state in an atom. It specifically pertains to the orientation of the angular momentum vector in the presence of an external magnetic field.

This quantum number can take values ranging from \(-l\) to \(+l\), where \(l\) is the azimuthal (or orbital angular momentum) quantum number. Each of these values corresponds to a different possible orientation of the electron cloud around the nucleus.

• For \(l = 4\) as in the 5\(g\) state, \(m_l\) values are \(-4, -3, -2, -1, 0, 1, 2, 3, 4\).
• The number of different states is given by \(2l + 1\).

The magnetic quantum number plays a crucial role in understanding the splitting of spectral lines under a magnetic field, a phenomenon known as the Zeeman effect. It provides insight into how electrons are arranged around the nucleus under different conditions.

Hydrogen Atom

The hydrogen atom is the simplest atom and serves as a fundamental model in quantum mechanics. It comprises one electron and one proton, which makes it ideal for studying the principles of atomic structure and the quantum mechanics governing electron behavior.

In the hydrogen atom, the electron moves under the influence of the electrostatic force exerted by the proton. The distinct energy levels and orbitals of the electron are described using various quantum numbers, which emerge from solving the Schrödinger equation for this system.

• It has a principal quantum number (\(n\)), which determines the main energy level of the electron.
• A hydrogen atom exhibits a spherical symmetry, allowing the usage of spherical coordinates \(r, \theta, \phi\) to describe electron position.

Studying the hydrogen atom allows us to grasp the basics of more complex atoms and explore phenomena such as spectral lines and atomic transitions that are foundational in both physics and chemistry.

Azimuthal Quantum Number

The azimuthal quantum number, also known as the angular momentum quantum number and symbolized by \(l\), defines the shape of an electron's orbital. It is crucial in determining the spatial distribution of the electron cloud around the nucleus.

The values of \(l\) range from 0 to \(n-1\), where \(n\) is the principal quantum number. Each value of \(l\) corresponds to a different type of orbital, designated by letters (s, p, d, f, g, etc.). For a given principal quantum number, multiple possible orbital shapes are defined by the azimuthal quantum number.

• \(l = 0\) corresponds to s-orbitals, \(l = 1\) to p-orbitals, and so on.
• For \(l = 4\), it is associated with the g-orbital, as seen in the 5\(g\) state.

The azimuthal quantum number also contributes to calculating the angular momentum of the electron as \(\sqrt{l(l+1)}\hbar\). Understanding this quantum number is essential for analyzing electron configurations and predicting atomic behavior in reaction mechanisms.