Question

Which of the following cannot be circumference of an orbit in H-atom? \(\left(r_{0}=\right.\) radius of the first orbit \()\) (a) \(2 \pi r_{0}\) (b) \(4 \pi r_{0}\) (c) \(8 \pi r_{0}\) (d) \(18 \pi r_{0}\)

Short answer

Option (d) \(18 \text{π}r_0\) cannot be the circumference of an orbit in a hydrogen atom according to the Bohr model.

Step-by-step solution

Understanding the Bohr Model

In the Bohr model of the hydrogen atom, the orbiting electron is assumed to revolve in certain stable orbits without the emission of radiant energy. The circumference of these orbits must be an integral multiple of the de Broglie wavelength of the electron.

de Broglie Wavelength Relation

The de Broglie wavelength (\( \text{λ} \)) of an electron is given by \( \text{λ} = \frac{h}{m_ev} \), where \(h\) is Planck's constant, \(m_e\) is the mass of the electron, and \(v\) is the velocity of the electron. The circumference of an allowed orbit can be written as \( n\text{λ} \) where \(n\) is a positive integer (quantum number).

Circumference of Allowed Orbit

The circumference of the nth orbit in the hydrogen atom can be described by \( 2\text{π}r_n = n\text{λ} \). For the first orbit \(n = 1\), the circumference is \( 2\text{π}r_0 \).

Evaluating Options

Since the circumference of allowed orbit must be an integral multiple of the de Broglie wavelength, only option (d) \(18 \text{π}r_0\) does not fit the form of \(n\text{λ}\), where \(n\) should be a positive integer. The numbers 2, 4, and 8 are all integers whereas 18 is not an allowed quantum number in the simple Bohr model.

Key concepts

de Broglie Wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics, bridging the gap between particle and wave behavior. Named after the physicist Louis de Broglie, it represents the wavelength exhibited by particles with mass under certain conditions. According to de Broglie's hypothesis, every particle with momentum has an associated wavelength, given by the equation:
\[ \text{λ} = \frac{h}{p} \]
where \( \text{λ} \) is the wavelength, \( h \) is Planck's constant, and \( p \) is the momentum of the particle. In the context of an electron in a hydrogen atom, its momentum \( p \) is the product of its mass \( m_e \) and velocity \( v \). This reveals the electron’s wave-like properties on a nanoscopic scale, and in the Bohr model, it determines the allowed orbits for the electron.

Allowed Orbit Circumference

In the Bohr model of the hydrogen atom, the allowed orbit circumference is a fundamental concept that states the electron can only occupy certain orbits around the nucleus. For the electron to be in a stable orbit, without radiating energy, the circumference of its orbit must be an integral multiple of its de Broglie wavelength. This condition is expressed mathematically as:
\[ 2\text{π}r_n = n\text{λ} \]
where \( r_n \) is the radius of the orbit, \( n \) is a positive integer representing the principal quantum number, and \( \text{λ} \) is the de Broglie wavelength of the electron. Therefore, the circumference of each allowed orbit is quantized, meaning it can only take on certain discrete values. This quantization leads to the concept of energy levels within an atom and is pivotal for understanding the emission and absorption spectra of elements.

Quantum Numbers

Quantum numbers are integral numbers that describe the discrete energy levels and properties of an electron within an atom. They are the outcome of the solutions to the Schrödinger equation for the hydrogen atom and arise naturally from the constraints of the Bohr model. There are four quantum numbers: the principal quantum number (\( n \)), the angular momentum quantum number (\( l \)), the magnetic quantum number (\( m_l \)), and the spin quantum number (\( s \)).

• The principal quantum number, \( n \), defines the size of the electron's orbit and its energy level. It can take any positive integer value, starting from 1.
• The angular momentum quantum number, \( l \), relates to the shape of the electron's orbital and can take on any integer value from 0 to \( n-1 \).
• The magnetic quantum number, \( m_l \), describes the orientation of the orbital in space. It can range from \( -l \) to \( l \), including zero.
• The spin quantum number, \( s \), relates to the intrinsic angular momentum or ‘spin’ of the electron. It can be either +1/2 or -1/2.

In the problem given, when discussing the allowed circumference of an electron's orbit, we are primarily concerned with the principal quantum number, which dictates that orbits be quantized into distinct levels, reflecting the 'stepped' nature of electron energy within an atom.