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Why do the \(3 s, 3 p,\) and \(3 d\) orbitals have the same energy in a hydrogen atom but different energies in a many-electron atom?

Short Answer

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In a hydrogen atom which is a one-electron system, the energy of electron depends purely on the principal quantum number (n). Therefore, 3s, 3p, and 3d orbits have identical energy as they share the same n. In contrast, a many-electron system presents an additional complexity in which electron-electron interactions and differences in penetration and shielding come into play. As such, in these atoms, the 3s, 3p and 3d orbitals have different energy levels.

Step by step solution

01

Understanding atomic orbitals in hydrogen

Hydrogen is known as a one-electron atom which means it consists of one electron orbiting a single proton. In this case, the energy levels of the orbitals only depends on the principal quantum number, n. Therefore, all orbitals having the same value of n have the same energy. In hydrogen atom, orbitals such as 3s, 3p and 3d all have the same principal quantum number, 3. Thus, in hydrogen atom, these orbitals possess the same energy.
02

Understanding atomic orbitals in many-electron atoms

In many-electron atoms, the energy of electrons does not solely depend on the principal quantum number. It is also influenced by electron-electron interactions and differences in penetration and shielding. According to the Pauli's exclusion principle, each electron in an atom is described by a unique set of quantum numbers. The principal quantum number defines the energy level of the electron and the possible values of the other three quantum numbers divide this energy level into sublevels. The shape of the orbital is determined by the azimuthal quantum number (l), which also affects the energy of the atom. For a many-electron atom the energies of 3s, 3p and 3d orbitals are different from each other due to different electron-electron interactions and differences in penetration and shielding. s orbitals are generally lower in energy than p orbitals, and p orbitals are lower in energy than d orbitals, because s orbitals have a greater ability to penetrate to the nucleus of the atom.

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Most popular questions from this chapter

How does de Broglie's hypothesis account for the fact that the energies of the electron in a hydrogen atom are quantized?

Which of the four quantum numbers \(\left(n, \ell, m_{\ell}, m_{s}\right)\) determine (a) the energy of an electron in a hydrogen atom and in a many- electron atom, (b) the size of an orbital, (c) the shape of an orbital, (d) the orientation of an orbital in space?

When an electron makes a transition between energy levels of a hydrogen atom, there are no restrictions on the initial and final values of the principal quantum number \(n\). However, there is a quantum mechanical rule that restricts the initial and final values of the orbital angular momentum \(\ell\). This is the selection rule, which states that \(\Delta \ell=\pm 1,\) that is, in a transition, the value of \(\ell\) can only increase or decrease by one. According to this rule, which of the following transitions are allowed: (a) \(1 s \longrightarrow 2 s\), (b) \(2 p \longrightarrow 1 s\) (c) \(1 s \longrightarrow 3 d\) (d) \(3 d \longrightarrow 4 f\), (e) \(4 d \longrightarrow 3 s ?\)

Thermal neutrons are neutrons that move at speeds comparable to those of air molecules at room temperature. These neutrons are most effective in initiating a nuclear chain reaction among \({ }^{235} \mathrm{U}\) isotopes. Calculate the wavelength (in \(\mathrm{nm}\) ) associated with a beam of neutrons moving at \(7.00 \times 10^{2} \mathrm{~m} / \mathrm{s}\). (Mass of a neutron \(\left.=1.675 \times 10^{-27} \mathrm{~kg} .\right)\)

When copper is bombarded with high-energy electrons, X rays are emitted. Calculate the energy (in joules) associated with the photons if the wavelength of the X rays is \(0.154 \mathrm{nm}\).

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