Question

The circumference of the third orbit of \(\mathrm{He}^{+}\) ion is \(x \mathrm{~m} .\) The de-Broglie wavelength of electron revolving in this orbit will be (a) \(\frac{x}{3} \mathrm{~m}\) (b) \(3 x \mathrm{~m}\) (c) \(\frac{x}{9} \mathrm{~m}\) (d) \(9 x \mathrm{~m}\)

Short answer

The de-Broglie wavelength of the electron is \(\frac{x}{3} \text{~m}\).

Step-by-step solution

Understanding the concept of de-Broglie wavelength

The de-Broglie wavelength \(\lambda\) is associated with a moving particle and is given by the equation \(\lambda = \frac{h}{p}\), where \(h\) is Planck's constant and \(p\) is the momentum of the particle. For an electron in orbit around a nucleus, its momentum is quantized and the circumference of the orbit must be an integer multiple of the de-Broglie wavelength. That is, \(n\lambda = 2\pi r\), where \(n\) is a positive integer (the principal quantum number), \(\lambda\) is the wavelength, \(r\) is the radius of the orbit, and \(2\pi r\) is the circumference of the orbit.

Applying the relationship for the third orbit

For the third orbit of an \(\mathrm{He}^{+}\) ion, we have \(n = 3\). This implies the circumference of the third orbit \(x\) is three times the de-Broglie wavelength. Therefore, the relationship between the circumference and the de-Broglie wavelength is \(3\lambda = x\). Solving for \(\lambda\), the de-Broglie wavelength of the electron in the third orbit, gives us \(\lambda = \frac{x}{3}\).

Selecting the correct answer

Based on Step 2, the de-Broglie wavelength of the electron in the third orbit is \(\frac{x}{3}\). Therefore, the correct answer is (a) \(\frac{x}{3} \text{m}\).

Key concepts

Quantum Mechanics

The beauty of quantum mechanics lies in its ability to describe the behavior of particles on the smallest scales, such as atoms and subatomic particles like electrons. Unlike classical mechanics, which applies to the macroscopic world we experience daily, quantum mechanics introduces a level of uncertainty and the concepts of wave-particle duality.

At its core, this theory suggests that particles can exhibit both wave-like and particle-like properties. The de-Broglie wavelength, which is central to the problem at hand, is a perfect example of this duality, as it describes the wavelength associated with a moving particle. The concept that particles could be described in terms of waves was a revolutionary step in developing quantum mechanics and allowed for the prediction of electron behavior in a given situation, such as the circumference of an electron's orbit.

Electron Orbitals

Electron orbitals are regions around an atom's nucleus where electrons are most likely to be found. These are not fixed paths but areas defined by complex mathematical functions derived from quantum mechanics. Each orbital correlates to a specific energy level and shape, which determines the distribution of electrons around the nucleus.

The concept of quantized orbits comes into play when we discuss the restrictions on the arrangement and energy of these electrons. In the case of the exercise, the electron traveling in the third orbit of a hydrogen-like ion must fit an integer number of de-Broglie wavelengths into the circumference of its orbit. This quantization results from the wave-like properties of electrons, reinforcing the idea that quantum mechanics rules their motion and positioning.

Planck's Constant

Planck's constant (\(h\)) is a fundamental quantity in quantum mechanics. It sets the scale of quantum effects and relates the energy carried by a photon to its frequency. Introduced by Max Planck in 1900, this constant laid the groundwork for quantum theory.

The precise value of Planck's constant is critical in equations that govern quantum physics, including the calculation of the de-Broglie wavelength (\(\lambda = \frac{h}{p}\)). Here, \(p\) stands for the momentum of the particle, with Planck's constant serving as the bridge between the macroscopic parameters (like momentum) and the quantum phenomena (like wave behavior of particles). Understanding Planck's constant is essential for solving problems related to the quantum scale and interpreting the nature of the subatomic world.