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Chapter 2: Problem 36

The angular momentum of an electron in a given stationary state can be expressed as \(m_{e} v r=n \frac{h}{2 \pi}\). Based on this expression an electron can move only in those orbits for which its angular momentum is (a) equal to \(n\) (b) integral multiple of \(\frac{h}{2 \pi}\) (c) multiple of \(n\) (d) equal to \(\frac{h}{2 \pi}\) only.

Short Answer

Expert verified
The correct answer is (b) integral multiple of \(\frac{h}{2 \pi}\).

Step by step solution

01

Analyze the Given Expression

The expression provided for the angular momentum of an electron in a stationary state is given by the equation: \(m_{e} v r = n \frac{h}{2 \pi}\). In this equation, \(m_e\) is the mass of the electron, \(v\) is the velocity of the electron, \(r\) is the radius of the orbit, \(n\) is a quantum number (which can take on any positive integer value), and \(h\) is Planck's constant.
02

Determine the Condition for Angular Momentum

The equation can be rearranged to state that the quantized angular momentum \(L\) is equal to an integer multiple of a reduced Planck constant: \(L = n \frac{h}{2 \pi}\). This means that for an electron to move in a permissible orbit, its angular momentum must be an integer multiple of \(\frac{h}{2 \pi}\).
03

Identify the Correct Options

Based on the derived condition for the angular momentum to be quantized, we can determine the correct options from the ones provided. The revised angular momentum condition does not indicate that the angular momentum is simply equal to \(n\), a multiple of \(n\), or equal to \(\frac{h}{2 \pi}\) only; rather, it must be an integral multiple of \(\frac{h}{2 \pi}\). Therefore, the option that correctly describes the orbits an electron can move in is (b) integral multiple of \(\frac{h}{2 \pi}\).

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Key Concepts

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

Bohr's Model
Niels Bohr's model of the atom was a groundbreaking concept that introduced the idea of quantization into atomic physics. According to Bohr's model, electrons orbit the nucleus of an atom in certain permissible paths known as orbits or shells. Each orbit corresponds to a specific energy level. One of the model's key postulates is that an electron in an atom does not radiate energy as long as it stays in its orbit. This addresses the concept of 'stationary states'.

When an electron transitions between these orbits, it must either absorb or emit energy equal to the difference between the energy levels of these orbits. This energy is transferred in the form of quanta of light, known as photons. The frequencies of the emitted or absorbed light directly correspond to the energy differences and hence can be correlated to the atomic structure.

Improving on the step-by-step solution, it's important to underline that these orbits are not arbitrary but correspond to specific quantized angular momenta, as suggested in the exercise, ensuring that the electron's angular momentum is an integral multiple of \( \frac{h}{2\pi} \) with \( h \) being Planck's constant and \( n \) representing the principal quantum number.
Stationary States
In atomic physics, stationary states refer to the stable energy levels an electron can occupy around a nucleus without emitting radiation. These are the specific conditions under which the electrons do not lose energy, and as such, their orbits can be considered 'stationary'. The concept of stationary states is central to Bohr's model, where an electron in a stationary state has a fixed energy and is not in the process of emitting or absorbing energy.

This concept dismisses classical mechanics which would predict a spiraling-in situation due to electromagnetic radiation emitted by accelerating electrons. Instead, Bohr introduced quantum conditions to constrain the behavior of electrons, further implying that transitions between these states result in the absorption or emission of energy in discrete amounts.

Understanding this concept helps clarify why the angular momentum needs to be quantized, as the specific condition for these stable orbits is a result of quantum mechanics defining that not all values of angular momentum are permissible. The quantized nature of these states defines why only certain orbits are available to the electron, which links directly to the expression given in the exercise.
Planck's Constant
Planck's constant, symbolized by \( h \), is a fundamental constant in quantum mechanics. It was first introduced by Max Planck and represents the proportionality constant between the energy \( E \) of a photon and the frequency \( u \) of its associated electromagnetic wave. The relationship is described by the famous equation \( E = h u \).

Planck's constant has the dimension of action, which is energy multiplied by time. The significance of \( h \) in quantum mechanics cannot be overstated as it sets the scale at which quantum effects become significant. It is the key to understanding the quantization of energy levels and angular momentum as seen in atomic structures.

In the context of the original exercise, Planck's constant assists in defining the quantization condition for the angular momentum of the electron. The constant helps form part of the equation that relates the quantization of angular momentum to the integral multiples of \( \frac{h}{2\pi} \), further illustrating that physical quantities at the quantum level, like angular momentum, can only take on discrete values as opposed to any continuous range.

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Most popular questions from this chapter

The configuration of the valence orbital of an element with atomic number 22 is (a) \(3 d^{5} 4 s^{1}\) (b) \(3 d^{2} 4 s^{2}\) (c) \(4 s^{\prime} 4 p^{1}\) (d) \(3 d^{2} 4 s^{1} 4 p^{1}\)

An electron is in one of the \(3 d\)-orbitals. What are the possible values of \(n, l\) and \(m\) for this electron? (a) \(n=3, l=0, m_{l}=0\) (b) \(n=3, l=1, m_{1}=-1,0,+1\) (c) \(n=3, l=2, m_{l}=-2,-1,0,+1,+2\) (d) \(n=3, l=3, m_{l}=-3,-2,-1,0,+1,+2,+3\)

Few electrons have following quantum numbers, (i) \(n=4, l=1\) (ii) \(n=4, l=0\) (ii) \(n=3, l=2\) (iv) \(n=3, l=1\) Arrange them in the order of increasing energy from lowest to highest. (a) (iv) \(<\) (ii) \(<(\mathrm{iii})<(\mathrm{i})\) (b) (ii) \(<(\mathrm{iv})<(\mathrm{i})<\) (iii) (c) \((\mathrm{i})<(\mathrm{iii})<(\mathrm{ii})<(\mathrm{iv})\) (d) (iii) \(<(\mathrm{i})<(\mathrm{iv})<(\mathrm{ii})\)

Match the values of column II with column \(\mathrm{I}\) and mark the appropriate choice. $$ \begin{array}{|l|l|l|l|} \hline {\text { Column I }} & && {\text { Column II }} \\ \hline \text { (A) } & \begin{array}{l} \text { The shape of rubber heel } \\ \text { changes under stress } \end{array} & \text { (p) } & \begin{array}{l} \text { Young's } \\ \text { modulus of } \\ \text { elasticity is } \\ \text { involved } \end{array} \\ \hline \text { (B) } & \begin{array}{l} \text { In a suspended bridge, } \\ \text { there is a strain in the } \\ \text { ropes by the load of the } \\ \text { bridge } \end{array} & \text { (q) } & \begin{array}{l} \text { Bulk modulus } \\ \text { of elasticity is } \\ \text { involved } \end{array} \\ \hline \text { (C) } & \begin{array}{l} \text { In an automobile tyre, } \\ \text { when air is compressed, } \\ \text { the shape of tyre changes } \end{array} & \text { (r) } & \begin{array}{l} \text { Modulus of } \\ \text { rigidity is } \\ \text { involved } \end{array} \\ \hline \text { (D) } & \begin{array}{l} \text { A solid body is subjected } \\ \text { to a deforming force } \end{array} & \text { (s) } & \begin{array}{l} \text { All the moduli } \\ \text { of elasticity } \\ \text { are involved } \end{array} \\ \hline \end{array} $$ (a) \((A) \rightarrow(i),(B) \rightarrow(i i),(C) \rightarrow(\) iv),\((D) \rightarrow(\) iii) (b) (A) \(\rightarrow\) (iii), (B) \(\rightarrow\) (i), (C) \(\rightarrow\) (ii), (D) \(\rightarrow\) (iv) (c) (A) \(\rightarrow\) (ii), (B) \(\rightarrow\) (iii), (C) \(\rightarrow\) (iv), (D) \(\rightarrow\) (i) (d) \((\mathrm{A}) \rightarrow(\mathrm{i}),(\mathrm{B}) \rightarrow\) (iii), (C) \(\rightarrow\) (ii), (D) \(\rightarrow\) (iv)

Which of the following configurations represents a noble gas? (a) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{10} 4 s^{2} 4 p^{6} 4 d^{10} 5 s^{2}\) (b) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{10} 4 s^{2} 4 f^{14} 5 s^{2}\) (c) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{10} 4 s^{2} 4 p^{6} 4 d^{10} 5 s^{2} 5 p^{6}\) (d) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{10} 4 s^{2} 4 p^{6} 4 d^{10} 5 s^{2} 5 p^{3}\)
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