Question

(a) What are the similarities and differences between the \(1 s\) 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 \(2 \mathrm{~s}\) 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) Similarities: Both 1s and 2s are s orbitals with a spherical shape and no angular nodes. Differences: 1s has a lower principal quantum number (n=1) and is closer to the nucleus with lower energy, whereas 2s (n=2) has a radial node. (b) 2p orbitals have directional character due to their orientation along Cartesian axes. px has electron density concentrated along the x-axis, while dx^2-y^2 has electron density concentrated in regions along both the positive and negative x and y axes. (c) The average distance from the nucleus for an electron in a 3s orbital (proportional to \(9a_{0}\)) is more than twice the distance compared to an electron in a 2s orbital (proportional to \(4a_{0}\)). (d) The orbitals in order of increasing energy are: 1s, 2p, 3d, 4f, and 6s.

Step-by-step solution

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

Both 1s and 2s orbitals are part of the s-type orbitals, which are characterized by their spherical symmetry around the nucleus. Here are their similarities and differences: Similarities: - Both are s orbitals and have a spherical shape. - They have no angular nodes. Differences: - 1s orbital has a lower principal quantum number (n=1) compared to the 2s orbital (n=2). - 1s orbital is closer to the nucleus and has a lower energy, making it more stable than the 2s orbital. - 2s orbital has a radial node, which means there is a region with zero electron probability density between the maximum probability densities of the 1s orbital and the 2s outer lobe.

b) Directional character of 2p orbitals and comparison with px and dx^2-y^2 orbitals

2p orbitals have directional character because they have a distinct orientation along one of the Cartesian axes (x, y, or z). The electron density in a 2p orbital is concentrated along the corresponding axis, resulting in a dumbbell shape. Comparing px and dx^2-y^2 orbitals: - In the px orbital, electron density is concentrated along the x-axis and has a node along the y and z axes. - In the dx^2-y^2 orbital, electron density is concentrated between the x and y axes in four lobes, forming a square planar shape around the nucleus. This means that the electron density is concentrated in the regions along both the positive and negative x and y axes.

c) Comparison of average distance from the nucleus for an electron in 2s and 3s orbitals

The average distance from the nucleus for an electron in an orbital is given by the expectation value of the radial distance, r, which is proportional to \(n^{2}a_{0}\), where n is the principal quantum number and \(a_{0}\) is the Bohr radius. For a 2s orbital (n=2), the average distance is proportional to \((2)^{2}a_{0} = 4a_{0}\). For a 3s orbital (n=3), the average distance is proportional to \((3)^{2}a_{0} = 9a_{0}\). The average distance from the nucleus of an electron in a 3s orbital is more than twice the distance compared to an electron in a 2s orbital.

d) Ordering the given orbitals by increasing energy

For a hydrogen atom, the energy of an orbital depends only on the principal quantum number (n). A higher n value indicates higher energy. Therefore, we can list the orbitals in order of increasing energy based on their n values. 1s (n=1) < 2p (n=2) < 3d (n=3) < 4f (n=4) < 6s (n=6) The orbitals in order of increasing energy are: 1s, 2p, 3d, 4f, and 6s.

Key concepts

s and p orbitals

Understanding the different types of orbitals in a hydrogen atom is crucial for grasping how electrons are organized around the nucleus. The simplest types of orbitals are 's' and 'p' orbitals. The s orbitals are spherically symmetrical, with the probability of finding an electron being equal in all directions at a given distance from the nucleus. Every principal energy level, denoted by the principal quantum number 'n', has one s orbital.

For example, a 1s orbital, found in the first energy level (n=1), is a single spherical region of electron density. As we move to the second energy level, we encounter the 2s orbital, which is similar in shape but larger, indicating the electron has a higher average distance from the nucleus. This orbital also contains a radial node—a region with zero probability of finding an electron—unlike the 1s orbital.

On the other hand, p orbitals have a dumbbell shape and are oriented along the Cartesian axes. In a hydrogen atom, p orbitals start at the second energy level (n=2). With three p orbitals available (px, py, and pz), each is aligned along a different axis, and this directional orientation gives rise to their characteristic shapes. The px orbital, for instance, has electron density concentrated along the x-axis, while the py and pz orbitals are aligned along the y-axis and z-axis, respectively. This gives p orbitals a directional character within the atom's electron cloud.

orbital angular nodes

Orbital angular nodes are regions within an atom's orbital where the probability of finding an electron is zero. These nodes play a crucial role in defining the shape and size of an orbital, as well as its energy.

The number of angular nodes is determined by the angular momentum quantum number, 'l', which can range from 0 to n-1, where 'n' is the principal quantum number. S orbitals, with an 'l' value of 0, have no angular nodes and therefore are spherically shaped. P orbitals, on the contrary, have an 'l' value of 1 and hence have one angular node. This node is a plane that passes through the nucleus and is perpendicular to the direction of the orbital's orientation (for px, the node is the yz-plane, for py it would be the xz-plane, and for pz, the xy-plane).

As the value of 'l' increases with the orbitals like d and f, the number of angular nodes increases correspondingly, which leads to more complex shapes for these orbitals. Angular nodes are responsible for the unique shapes of p, d, and f orbitals, which affect the chemical bonding and properties of elements.

orbital energy levels

The orbital energy levels in an atom dictate the order in which electrons fill up the orbitals, generally following the Aufbau principle. These energy levels are central to understanding an atom's electron configuration and chemical behavior. In a hydrogen atom, which contains only one electron, the energy of an orbital is determined by the principal quantum number 'n' alone due to the lack of electron-electron interactions that appear in multi-electron atoms.

In this simplistic system, the energy levels increase with increasing n. Therefore, the 1s orbital, having the smallest value of n (n=1), is the lowest in energy and consequently the most stable. As evidenced in the textbook exercise, the order of orbitals from lowest to highest energy level for hydrogen is: 1s, 2p, 3d, 4f, and finally 6s, based on their n values. This sequence showcases the increasing distance from the nucleus with larger 'n' values, and thus the increasing energy and decreasing stability.

These differences in energy levels are crucial not only for atomic structure but also for the spectral characteristics of elements, as transitions of electrons between different energy levels give rise to the emission or absorption of photons with very specific energies, which can be observed as spectral lines.