Problem 15 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Find the velocity \(\left(\mathrm{ms}^{-1}\right)\) of electron in first Bohr's orbit of radius \(a_{0}\). Also find the de Broglie's wavelength (in \(\mathrm{m}\) ). Find the orbital angular momentum of \(2 p\) orbital of hydrogen atom in units of \(h / 2 \pi\).

Step-by-Step Solution

Answer

The velocity is approximately \(2.18 \times 10^6\, \mathrm{ms}^{-1}\), the de Broglie wavelength is \(3.32 \times 10^{-10}\, \mathrm{m}\), and the angular momentum is \(\frac{h}{\sqrt{2}\pi}\).

Step 1: Understanding Bohr's Model

In Bohr's model of the hydrogen atom, electrons travel in circular orbits around the nucleus. The velocity of the electron in the nth orbit is given by the formula \( v_n = \frac{kZe^2}{nh} \), where \(k\) is Coulomb's constant, \(Z\) is the atomic number, \(e\) is the charge of an electron, \(h\) is Planck's constant, and \(n\) is the principal quantum number. For the first orbit, \(n = 1\) and \(Z = 1\).

Step 2: Calculating Velocity in First Bohr's Orbit

In the first Bohr orbit of hydrogen, we use the formula for velocity: \( v_1 = \frac{k \cdot e^2}{h} \). Substituting known constant values (where \(k = 8.9875 × 10^9 \mathrm{Nm}^2/\mathrm{C}^2\), \(e = 1.6 × 10^{-19} \mathrm{C}\), and \(h = 6.626 × 10^{-34} \mathrm{Js}\)), calculate \(v_1\).

Step 3: Applying de Broglie's Wavelength Formula

The de Broglie wavelength \(\lambda\) is given by \( \lambda = \frac{h}{mv} \). Here, \(m\) is the electron's mass \(9.109 × 10^{-31}\) kg, and \(v\) is the velocity just calculated. Substitute these values to determine \(\lambda\).

Step 4: Finding Orbital Angular Momentum

For the 2p orbital, the orbital angular momentum \(L\) is given by \(L = \sqrt{l(l+1)} \cdot \frac{h}{2\pi}\), where \(l\) is the azimuthal quantum number. In the 2p orbital, \(l = 1\). Substituting into the formula gives \(L = \sqrt{2} \cdot \frac{h}{2\pi}\).

Step 5: Perform Calculations

1. Substitute into the velocity equation to find \(v_1\).2. Substitute into the wavelength equation for \(\lambda\).3. Calculate \(L\) using the angular momentum formula.

Key Concepts

Electron Velocity

In Bohr's model of the atom, the velocity of an electron is essential to understand its dynamics around the nucleus. The velocity, denoted as \( v_n \), in the \( n \)-th orbit, can be calculated using:
\[v_n = \frac{kZe^2}{nh}\]
Here:

• \( k \) represents Coulomb's constant \( (8.9875 \times 10^9 \mathrm{Nm}^2/\mathrm{C}^2) \).
• \( Z \) is the atomic number, which is 1 for hydrogen.
• \( e \) is the electron charge \((1.6 \times 10^{-19} \mathrm{C})\).
• \( h \) is Planck's constant \((6.626 \times 10^{-34} \mathrm{Js})\).
• \( n \) is the principal quantum number, being 1 in the first orbit.

Substituting these values for the first Bohr orbit of hydrogen gives you the detailed velocity of the electron. Understanding this characteristic helps in determining interactions and energy levels in atoms.

de Broglie Wavelength

The de Broglie wavelength is a concept that combines particle and wave behavior, suggesting that all matter has a wavelength associated with its motion. The wavelength \( \lambda \) of an electron is given by:
\[\lambda = \frac{h}{mv}\]
where:

• \( h \) is Planck’s constant \((6.626 \times 10^{-34} \mathrm{Js})\).
• \( m \) is the mass of the electron \((9.109 \times 10^{-31} \mathrm{kg})\).
• \( v \) is the electron velocity, previously calculated using the Bohr model.

This wavelength highlights the wave-like nature of particles on the atomic scale. It provides instrumental insight into quantum mechanics, showing that particles like electrons can exhibit wave characteristics.

Orbital Angular Momentum

Orbital angular momentum is a crucial aspect of quantum mechanics and atomic structure. This concept describes the momentum of electrons due to their orbit around the nucleus. For an electron in a 2p orbital, the angular momentum \( L \) is calculated by:
\[L = \sqrt{l(l+1)} \cdot \frac{h}{2\pi}\]
with:

• \( l \) as the azimuthal quantum number, which is 1 for a 2p orbital.
• \( h \) as Planck's constant.

Substituting \( l = 1 \), the expression becomes simple and shows the quantized nature of angular momentum. This formula is fundamental to understanding electron configurations and transitions in atoms.

Quantum Mechanics

Quantum mechanics is the science that explains the behavior of matter and its interactions at the microscopic scale. It underlies the principles and calculations involving electron velocity, de Broglie wavelength, and angular momentum. Through quantum mechanics:

• We comprehend how particles like electrons display both wave and particle properties.
• It guides the prediction of atomic behaviors and electron arrangements.
• The framework is crucial for explaining chemical bonding, atomic transitions, and spectral lines.

Understanding these principles helps bridge classical mechanics and the phenomena observed in tiny particles, depicting them as both particles and waves.