Question

An electron is moving in Bohr's fourth orbit, its de-Broglie wavelength is \(X\). What is the circumference of the fourth orbit? (a) \(2 \lambda\) (b) \(2 / \lambda\) (c) \(3 \lambda\) (d) \(4 \lambda\)

Short answer

The circumference of the fourth orbit is \(4 \lambda\) (option d).

Step-by-step solution

Understand the Concept

In Bohr’s model of the hydrogen atom, an electron orbits the nucleus in certain allowed circular paths. The de Broglie hypothesis states that the electron will only occupy orbits where the orbit's circumference is an integral multiple of its de Broglie wavelength.

Identify the Orbit Number

The problem states that the electron is in the fourth orbit. In Bohr's model, the primary condition requires the circumference of the orbit to be an integral multiple of the electron's de Broglie wavelength.

Apply Bohr's Quantization Condition

According to Bohr's quantization condition, the circumference of the nth orbit is given by: \[ 2 \pi r_n = n \lambda \]where \( n \) is the orbit number and \( \lambda \) is the de Broglie wavelength.

Substitute Known Values

For the fourth orbit, substitute \( n = 4 \) into the equation:\[ 2 \pi r_4 = 4 \lambda \].This shows that the circumference of the fourth orbit equals four times the de Broglie wavelength.

Select the Correct Answer

Compare the derived expression with the given options: (a) \(2 \lambda\), (b) \(2 / \lambda\), (c) \(3 \lambda\), (d) \(4 \lambda\). The correct answer is the one that matches our derived circumference: \(4 \lambda\).

Key concepts

de Broglie wavelength

The concept of the de Broglie wavelength emerges from the interesting idea that all particles have wave-like properties. According to de Broglie's hypothesis, particles such as electrons exhibit wave characteristics, and their wavelength, known as the de Broglie wavelength, can be determined by dividing the Planck constant by the momentum of the particle:\[ \lambda = \frac{h}{p} \]where:

• \( \lambda \) is the de Broglie wavelength
• \( h \) is the Planck constant
• \( p \) is the momentum of the particle

This assertion suggests that the wavelength decreases as the particle moves faster and gains more momentum. In the context of Bohr's model, an electron orbiting a nucleus will exhibit these wave properties, affecting the possible orbits it can occupy. Consequently, the electron will only maintain stable orbits where the complete orbit is a whole number multiple of its de Broglie wavelength. Understanding this is crucial to comprehending how electrons form the structure of atoms.

quantization condition

Bohr's model was groundbreaking in introducing the concept of quantization to the orbits of electrons. The quantization condition states that an electron will only occupy certain allowed orbits around the nucleus. These allowed orbits correspond to the integer multiples of the de Broglie wavelength of the electron. The quantization condition is articulated mathematically as:\[ 2 \pi r_n = n \lambda \] where:

• \( r_n \) is the radius of the nth orbit
• \( \lambda \) is the de Broglie wavelength
• \( n \) is an integer representing the orbit number

This equation suggests that as electrons move to higher orbits, the allowed circumference proportionally increases. Consequently, electrons can transition between these quantized orbits by absorbing or emitting energy, a fundamental idea leading to the explanation of atomic spectra. The notion of quantized orbits provides a framework for explaining why electrons don't spiral into the nucleus, thereby contributing to the stability of atoms.

fourth orbit

In Bohr's model, specific orbits serve as defined pathways that electrons must follow around a nucleus. The fourth orbit indicates that the electron is in the fourth allowed pathway from the nucleus in Bohr's atomic model. This configuration has significant implications due to the quantization condition discussed earlier. For Bohr's fourth orbit, where the de Broglie wavelength is given as \( X \), the circumference can be calculated directly using Bohr’s quantization condition:\[ 2 \pi r_4 = 4 \lambda \]Substituting the de Broglie wavelength \( \lambda \) with the value \( X \), the equation becomes:\[ 2 \pi r_4 = 4 X \]This calculation reveals that the entire path length (or circumference) is four times the de Broglie wavelength of the electron. The important takeaway is that each defined orbit number dictates a unique set of permissible distances from the nucleus, underlining the precise nature of atomic structuring in Bohr’s model. Understanding this lays the groundwork for further exploration into quantum mechanics and atomic behavior.