Question

(a) What are the similarities of and differences between the \(1 \mathrm{~s}\) and \(2 s\) orbitals of the hydrogen atom? (b) In what sense does a \(2 p\) orbital have directional character? Compare the "directional" characteristics of the \(p_{x}\) and \(d_{x^{2}-y^{2}}\) orbitals. (That is, in what direction or region of space is the electron density concentrated?) (c) What can you say about the average distance from the nucleus of an electron in a \(2 s\) orbital as compared with a \(3 s\) orbital? (d) For the hydrogen atom, list the following orbitals in order of increasing energy (that is, most stable ones first): \(4 f, 6 s, 3 d, 1 s, 2 p\).

Short answer

(a) Both 1s and 2s orbitals are spherical and have no angular nodes. However, the 2s orbital is larger with one radial node, while the 1s orbital has no radial nodes. (b) A 2p orbital has directional character due to electron density concentration along the axes. The px and dx^2-y^2 orbitals both have electron density along the x and y axes but differ in shape and orientation. (c) The average distance from the nucleus is smaller for electrons in a 2s orbital compared to a 3s orbital. (d) The order of increasing energy for the given orbitals is: 1s, 2p, 3d, 4f, 6s.

Step-by-step solution

(a) Similarities and differences between 1s and 2s orbitals

The main similarities between the 1s and 2s orbitals are that both are spherical regions around the nucleus where there is a high probability of finding an electron, and both have no angular nodes. The differences between 1s and 2s orbitals are: 1. The principal quantum number (n) is different for both orbitals. n=1 for the 1s orbital and n=2 for the 2s orbital. 2. The size of the 2s orbital is larger than the 1s orbital, which means that the electron in the 2s orbital is farther away from the nucleus on average than the electron in the 1s orbital. 3. The 2s orbital has one radial node, while the 1s orbital has no radial nodes.

(b) Directional character of 2p orbital and comparison between px and dx^2-y^2 orbitals

A 2p orbital has directional character because its electron density is concentrated along the x, y, or z-axis. The px, py, and pz orbitals are dumbbell-shaped, with the electron cloud situated along the x, y, and z axes, respectively. The px and dx^2-y^2 orbitals both have electron density concentrated along the x and y axes. However, they differ in the shape and orientation of electron density. The px orbital is a single dumbbell along the x-axis, while the dx^2-y^2 orbital has four lobes arranged in a square planar shape around the x and y axes. The lobes in dx^2-y^2 orbital lie in the xy plane with 45 degrees angle to the axes.

(c) Average distance from the nucleus for electrons in 2s and 3s orbitals

The average distance from the nucleus for an electron in a 2s orbital is smaller than that for an electron in a 3s orbital. This is because the principal quantum number (n) for the 2s orbital is 2 and for the 3s orbital is 3. As the principal quantum number increases, the size of the orbital increases, meaning that the electron is, on average, farther away from the nucleus.

(d) Listing the orbitals in order of increasing energy

We can use the following rules to determine the order of increasing energy for hydrogen atom orbitals: 1. Orbitals with lower n values have lower energy (i.e., are more stable). 2. For orbitals with the same n value, lower l values result in lower energy levels. Following these rules, we can list the given orbitals in order of increasing energy: 1s, 2p, 3d, 4f, 6s.

Key concepts

Quantum Numbers

Quantum numbers are crucial in determining the unique address of an electron within an atom. They describe the properties of atomic orbitals and the electrons in those orbitals.

There are four quantum numbers, each indicating different properties:

• Principal Quantum Number ( n ): This number specifies the energy level and size of the orbital. It can be any positive integer. A higher n value means the electron is further from the nucleus and the orbital is larger.
• Angular Momentum Quantum Number ( l ): This number indicates the shape of the orbital. It ranges from 0 up to n-1 . Different l values correspond to different orbital shapes, such as spherical ( s ), dumbbell-shaped ( p ), or more complex shapes like d or f orbitals.
• Magnetic Quantum Number ( m_l ): This number shows the orientation of the orbital in space. It can range from -l to +l .
• Spin Quantum Number ( m_s ): This number indicates the direction of an electron's spin, and it can be either +1/2 or -1/2. The spin creates a magnetic field that affects how electrons interact with each other.

Knowing these numbers allows scientists to predict not only where electrons are likely to be found, but also how they interact with each other. This understanding is key to chemical reactions and properties.

Electron Density

Electron density is a concept that helps in understanding where electrons are most likely to be found in relation to the nucleus of an atom. It's a probability distribution that can vary based on the type of atomic orbital. Electron density is critical for visualizing the behavior of electrons within an atom or a molecule. For example, the 2p orbital has electron density concentrated along the x, y, or z axes. This directional nature gives these orbitals unique characteristics in chemical bonding.

Key points about electron density:

• Electron density decreases as the distance from the nucleus increases. In hydrogen-like orbitals, such as 1s and 2s, the density is highest closest to the nucleus and decreases outward.
• Nodes are areas where there is zero probability of finding an electron. In a 2s orbital, radial nodes exist, meaning there are spherical shells where electron density is zero.
• Different orbitals ( ext{px} , ext{dx}^2 ext{-y}^2 , etc.) concentrate electron density differently, which affects how electrons from these orbitals interact to form bonds.

These principles of electron density are fundamental for predicting the behavior of atoms and molecules in chemical reactions.

Orbital Energy Levels

Orbital energy levels describe the energy associated with electrons residing within a particular orbital. They help determine the stability of electrons in atoms and their interactions in chemical bonding.

Several factors affect orbital energy levels:

• The principal quantum number ( n ): Higher n values mean higher energy levels, since the electron is farther from the nucleus.
• Subshell type ( l ): Within the same energy level n , subshells with higher l values have higher energy. For example, p orbitals are higher in energy than s orbitals in the same energy level.
• Electron-electron interactions: These interactions can slightly alter energy levels, especially in multi-electron atoms.

For hydrogen, energy levels depend mainly on n , making 1s the lowest (most stable) and 6s one of the higher and less stable energy levels. In multi-electron atoms, the pattern of filling orbitals complicates due to electron-electron repulsion and shielding effects. Recognizing these energy levels helps understand atomic behavior, ionization energy, and the formation of chemical bonds.