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Chapter 4: Problem 124

For radial probability distribution curves, which of the following is /are correct? (a) the number of spherical or radial nodes is equal to \((\mathrm{n}-l-1)\) (b) the number of maxima in \(2 \mathrm{~s}\) orbital is two (c) the number of angular nodes is ' \(l\). (d) \(3 \mathrm{~d}_{z^{2}}\) has two spherical nodes.

Short Answer

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(a) and (c) are correct statements; (b) and (d) are incorrect.

Step by step solution

01

Understanding Radial Nodes

To figure out the number of radial (spherical) nodes, use the formula \( n - l - 1 \), where \( n \) is the principal quantum number and \( l \) is the azimuthal quantum number. Now, apply this to each orbital mentioned in the options to check.
02

Counting Maxima in 2s Orbital

Examine the radial probability distribution for the 2s orbital to count the number of maxima (peaks). You will find only one maximum because the node occurs at a minimum, not a maximum.
03

Calculating Angular Nodes

The number of angular nodes is given by the quantum number \( l \). For a given value of \( l \), e.g., \( p=1 \), \( d=2 \), calculate the angular nodes.
04

Assessing Spherical Nodes for 3d Orbitals

For the \( 3d_{z^2} \) orbital, determine the radial nodes using \( n - l - 1 \). Given \( n = 3 \) and \( l = 2 \), the number of radial nodes will be \( 3 - 2 - 1 = 0 \), indicating there are no spherical nodes.

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Key Concepts

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

Quantum Numbers
Quantum numbers are essential in describing the properties and behavior of electrons in atoms. They provide a way to specify the position and energy of an electron within an atom. Four quantum numbers are typically used:
  • Principal Quantum Number ( ) - This indicates the principal energy level that an electron is in, typically any positive integer. It also indicates the relative size of the orbital and its energy level.
  • Azimuthal Quantum Number ( or ) - This defines the shape of the orbital, ranging from 0 to -1. The values of l correspond to different types of orbitals: = 0 (s orbital), = 1 (p orbital), = 2 (d orbital), and so on.
  • Magnetic Quantum Number ( m) - This determines the orientation of the orbital in space. It can take on any integer value between l and l.
  • Spin Quantum Number ( s) - This describes the spin of the electron and can be either +1/2 or -1/2. It accounts for the two possible spin states of an electron within a single orbital.
Understanding these quantum numbers allows us to determine the arrangement of electrons within an atom and predict the atom's chemical properties.
Radial Nodes
A radial node is a spherical region around the nucleus of an atom where there is zero probability of finding an electron.
  • Radial nodes come into play when examining the radial probability distribution curves, which represent the likelihood of finding an electron at various distances from the nucleus.
  • To calculate the number of radial nodes for an orbital, you can use the formula n - l - 1, where n is the principal quantum number and l is the azimuthal quantum number.
  • This formula helps us determine regions in multi-electron atoms where the probability of finding an electron is zero.
For example, the 2s orbital has one radial node, since = 2 and l = 0, resulting in - l - 1 = 2 - 0 - 1 = 1 radial node. Understanding radial nodes is essential for examining electron densities and interpreting electron distributions in atomic orbitals.
Angular Nodes
Angular nodes are areas or planes, excluding the center of the nucleus, where the chances of locating an electron are zero. Unlike radial nodes that refer to spherical shells, angular nodes are associated with the shape or geometry of the orbital.
  • The number of angular nodes in an orbital is determined by the azimuthal quantum number ( l), which defines the orbital shape.
  • If l = 0, like in s orbitals, there are no angular nodes.
  • For p orbitals, where l = 1, there is one plane as an angular node.
  • For d orbitals, with l = 2, two angular nodes exist.
Understanding angular nodes helps us visualize the unique shapes of atomic orbitals, such as the three-dimensional lobes seen in p orbitals or the complex shapes of d and f orbitals. Their presence impacts how electrons are distributed around a nucleus.
Maxima in Orbitals
Maxima in orbitals refer to points within the radial probability distribution curve where the probability is highest for finding an electron. Calculating the maxima can give insights into how electrons organize within different orbitals.
  • When examining a 2s orbital's radial probability distribution, only one maximum is observed. This is because the single radial node corresponds to a point of minimum probability, not maximum.
  • Understanding where these maxima occur enables chemists to understand better the behavior of electrons in atomic and molecular interactions.
Maxima are important for chemists discerning atomic characteristics, as they relate to various physical and chemical properties. Tracking these peaks lets scientists ascertain how electrons bond and interact at atomic levels, vital for predicting reactivity and stability in different chemical environments.

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Most popular questions from this chapter

When potassium metal is exposed to violet light (a) there is no effect (b) ejection of electron takes place (c) the absorption of electrons takes place (d) ejection of some potassium atoms occurs

The quantum number \(+1 / 2\) and \(-1 / 2\) for the electron spin represent (a) rotation of the electron in clockwise and anticlockwise direction respectively. (b) rotation of the electron in anti clockwise and clockwise direction respectively. (c) magnetic moment of the electron pointing up and down respectively. (d) two quantum mechanical spin states which have no classical analogues.

The size of a microscopic particle is one micron and its mass is \(6 \times 10^{-13} \mathrm{gm}\). If its position may be measured to within \(0.1 \%\) of its size, the uncertainty in velocity, in \(\mathrm{cm} \mathrm{s}^{-1}\), is approximately (a) \(10^{-6} / 3 \pi\) (b) \(10^{-7} / 2 \pi\) (c) \(10^{-5} / 4 \pi\) (d) \(10^{-7} / 4 \pi\)

Electron energy of a photon is given as: \(\Delta \mathrm{E} /\) atom \(=3.03 \times 10^{-19} \mathrm{~J}\) atom \(^{-1}\) then, the wavelength of the photon is (a) \(6.56 \mathrm{~nm}\) (b) \(65.6 \mathrm{~nm}\) (c) \(656 \mathrm{~nm}\) (d) \(0.656 \mathrm{~nm}\) Given, h (Planck constant) \(=6.63 \times 10^{-34} \mathrm{Js} \mathrm{c}\) (velocity of light \()=3.00 \times 10^{8} \mathrm{~ms}^{-1}\)

Quantum numbers of an atom can be defined on the basis of (a) Aufbau's principle (b) Heisenberg's uncertainity principle (c) Hund's rule (d) Pauli's exclusion principle
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