Problem 100 - A Complete Resource Book in Chemistry for JEE Main

Question

The wavelength of the de Broglie wave of the electron revolving in the fifth orbit of the hydrogen atom is \(\left(\mathrm{r}_{0}\right.\) is the Bohr's radius \(=0.529 \mathrm{~A}\) ) (a) \(20 \mathrm{r}_{0}\) (b) \((10 \pi) \mathrm{r}_{0}\) (c) \(5 \pi \mathrm{r}_{0}\) (d) \(15 \pi \mathrm{r}_{\mathrm{d}}\)

Step-by-Step Solution

Answer

The de Broglie wavelength is (10 \pi) r_0.

Step 1: Understanding de Broglie Wavelength

The de Broglie wavelength \( \lambda \) of a particle like an electron is given by \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant, and \( p \) is the momentum of the electron. In the case of an electron in the nth orbit of a hydrogen atom, \( p = m v \), and is related by de Broglie's relation \( \lambda = \frac{h}{m v} \).

Step 2: Using the Bohr Model

According to the Bohr model, the angular momentum of the electron in the nth orbit is quantized and given by \( m v r = n \frac{h}{2\pi} \). Thus, \( v = \frac{n h}{2\pi m r} \).

Step 3: Relating Radius to Wavelength

For the n-th orbit, radius \( r = n^2 r_0 \). From de Broglie and Bohr's equations, \( \lambda = \frac{h}{m v} = \frac{2\pi r}{n} = \frac{2\pi n^2 r_0 }{n} = 2 \pi n r_0 \).

Step 4: Finding the Wavelength for the Fifth Orbit

Substituting \( n = 5 \) into the equation for \( \lambda \), we get \( \lambda = 2 \pi \times 5 \times r_0 = 10 \pi r_0 \).

Step 5: Selecting the Correct Answer

Comparing \( 10 \pi r_0 \) with the given options, option (b) \( (10 \pi) r_0 \) is the correct choice.

Key Concepts

Bohr Model

The Bohr model is a fundamental concept in understanding atomic structure, especially for hydrogen atoms. It was introduced by Niels Bohr in 1913 and is an important step in atomic theory. Bohr's model describes electrons as particles that revolve around the nucleus in fixed orbits. These orbits are like paths that electrons can follow without losing or gaining energy. Each orbit corresponds to a specific energy level.

• Electrons can jump from one orbit to another by absorbing or emitting energy.
• The energy levels are quantized, meaning electrons can only exist in specific, discrete levels.
• The Bohr model helps to explain why atoms emit or absorb light at specific wavelengths.

The model provides a useful way to predict the behavior of electrons in simpler atoms like hydrogen, forming the basis for further theories in quantum mechanics.

Hydrogen Atom

The hydrogen atom is the simplest atom in the universe and serves as a perfect example for illustrating atomic theories like the Bohr model. It consists of just one proton and one electron. This simplicity makes it an excellent subject for studying basic atomic principles.

• The electron in a hydrogen atom orbits the nucleus, which contains the single proton.
• In the Bohr model, these orbits are at set distances from the nucleus, called energy levels.
• When the electron moves between these levels, it causes the emission or absorption of a photon.

Understanding the hydrogen atom is crucial for grasping more complex atomic structures, as it lays the groundwork for understanding electron transitions and spectral lines.

Quantized Angular Momentum

Quantized angular momentum is a key concept introduced by the Bohr model. Bohr proposed that the angular momentum of electrons in an atom is quantized, meaning it can only take certain values. This was a groundbreaking shift from classical physics.

• The angular momentum of an electron in an orbit is given by the formula: \( mvr = n\frac{h}{2\pi} \), where \( m \) is the electron's mass, \( v \) is its velocity, \( r \) is the orbit radius, and \( n \) is the principal quantum number (an integer).
• Quantization restricts electrons to certain orbits and makes the energy levels discrete.
• This concept helps in explaining why electrons do not spiral into the nucleus, making atoms stable.

By quantizing angular momentum, Bohr was able to account for the stability of atoms and phenomena such as spectral lines.

Electron Orbits

In the Bohr model, electron orbits are critical to understanding atomic structure. These orbits represent the paths that electrons follow around the nucleus of an atom. However, unlike the elliptical orbits in celestial mechanics, electron orbits are circular.

• Each orbit has a fixed radius, determined by the principal quantum number \( n \). The radii increase with \( n^2 \).
• As electrons move to higher orbits (higher \( n \)), they are at higher energy levels and further from the nucleus.
• These orbits are quantized, meaning electrons jump between them without being in-between states.

Electron orbits are essential for understanding phenomena such as the de Broglie wavelength, ionization energy, and the emission or absorption spectra of atoms.