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Chapter 2: Problem 39

The electron density in the \(X Y\) plane in \(3 d_{x^{2}-y^{2}}\) orbital is zero.

Short Answer

Expert verified
The electron density in the XY plane in the \(3d_{x^{2}-y^{2}}\) orbital is zero due to its nodal plane.

Step by step solution

01

Understanding the problem statement

We need to analyze the electron density pattern for a specific orbital denoted as \(3d_{x^{2}-y^{2}}\). The task is to find in which plane the electron density is zero.
02

Understanding the orbital structure

The \(3d_{x^{2}-y^{2}}\) orbital is one of the five \(3d\) orbitals. It has a unique shape characterized by lobes lying along the \(x\) and \(y\) axes, and it shows zero electron density along its nodal plane.
03

Identify nodal planes of the orbital

In the \(3d_{x^{2}-y^{2}}\) orbital, the electron density is concentrated in regions along the \(x\) and \(y\) axes. There are nodal planes where there is no electron density. For \(3d_{x^{2}-y^{2}}\), the nodal planes are the planes where \(z = 0\), along the \(xy\) axis.
04

Analyzing electron density in the XY plane

Since the lobes of the \(3d_{x^{2}-y^{2}}\) orbital lie along the \(x\) and \(y\) axes, there is no electron density exactly on the \(x\) and \(y\) axes but rather between them. Therefore, the plane where the electron density is zero is where the axes intercept, which are specifically not along these axes for the \(x^{2}-y^{2}\) direction as they are the nodal regions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electron Density
In quantum chemistry, electron density is a key concept, referring to how electrons are spatially distributed in an atom or molecule. It essentially gives us a map of where electrons are likely to be found around the nucleus. In atomic orbitals, electron density patterns are crucial for understanding chemical bonding and molecular shapes. These patterns indicate where electrons are most and least likely to be located.
Electron density is a probabilistic field, which means that it doesn't show exact positions of electrons, but rather regions where they have a higher probability of being found. For example, in the case of a hydrogen atom, which has a spherical electron distribution, the electron can be found with high probability anywhere within its sphere.
  • High electron density regions: Areas where electrons are likely to be found.
  • Low electron density or nodal regions: Areas where electrons are unlikely or have zero probability of being found.
It's important to differentiate between areas of high and low electron density, as these are critical when analyzing molecular structures and predicting chemical properties.
3d Orbitals
The 3d orbitals are a set of five orbitals found in the third shell of an atom, characterized by their unique shapes and energy levels. These orbitals include: 3d鈧撫掸, 3d鈧撯倠y鈧�, 3d鈧撯倐鈧媦鈧�, 3d岫宦�, and 3d岫烩倱. Each orbital has a distinct spatial orientation and symmetry, determining how they interact with other atomic orbitals in chemical bonds.
For the 3d orbitals:
  • The 3d鈧撫掸 and 3d鈧撯倐鈧媦鈧� orbitals have lobes oriented between the axes, unlike others which align along axes.
  • The 3d岫宦� orbital has a different shape with a donut-shaped electron cloud around the nucleus along the z-axis.
Among these, the 3d鈧撯倐鈧媦鈧� orbital stands out because of its nodals, areas of zero electron density that manifest closest to the xy-plane. These orbitals become crucial in transition metals where they play essential roles in d-orbital filling used for bonding.
Nodal Planes
Nodal planes are regions within atomic orbitals where the probability of finding an electron is zero. They are intrinsic to understanding the spatial orientation and characteristics of different orbitals. Nodal planes contribute to the unique shapes and energy levels of orbitals due to their definition of regions of zero electron probability.
For instance:
  • In the 3d鈧撯倐鈧媦鈧� orbital, nodal planes are along the xy-plane, meaning electrons cannot be found exactly on this plane.
  • Nodal planes depend on the orbital's geometry; more complex orbitals have multiple nodal planes.
The existence of nodal planes affects orbital energies and bonding characteristics. They limit where electrons can be found, thus influencing how orbitals overlap and bond in molecules, playing a pivotal role in chemical reactions and molecular stability.

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Most popular questions from this chapter

Consider the hydrogen atom to be a proton embedded in a cavity of radius \(a_{0}\) (Bohr radius) whose charge is neutralised by the addition of an electron to the cavity in vacuum, infinitely slowly. Estimate the average total energy of an electron in its ground state in a hydrogen atom as the work done in the above neutralisation process. Also, if the magnitude of the average kinetic energy is half the magnitude of the average potential energy, find the average potential energy.

The shortest wavelength of \(\mathrm{H}\) atom in the Lyman series is \(\lambda_{1}\). The longest wavelength in the Balmer series is \(\mathrm{He}^{+}\)is : (a) \(\frac{36 \lambda_{1}}{5}\) (b) \(\frac{5 \lambda_{1}}{9}\) (c) \(\frac{9 \lambda_{1}}{5}\) (d) \(\frac{27 \lambda_{1}}{5}\)

According to Bohr's theory, the electronic energy of hydrogen atom in the \(n^{\text {ti }}\) Bohr's orbit is given \(\operatorname{by} E_{n}=\frac{-21.76 \times 10^{-19}}{n^{2}} \mathrm{~J}\). Calculate the longest wavelength of light that will be needed to remove an electron from the third Bohr orbit of the \(\mathrm{He}^{+}\)ion.

Many elements have non-integral atomic masses because: (a) they have isotopes (b) their isotopes have non-integral masses (c) their isotopes have different masses (d) the constitutents, neutrons, protons and electrons, combine to give fractional masses

The work function of sodium metal is \(4.41 \times 10^{-19} \mathrm{~J}\). If photons of wavelength \(300 \mathrm{~nm}\) are incident on the metal, the kinetic energy of the ejected electrons will be \(\left(h=6.63 \times 10^{-34} \mathrm{~J} \mathrm{~s} ; c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)\) \(\times 10^{-21} \mathrm{~J}\)
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