Question

According to Bohr's postulates which quantity can take up only discrete values? (1) Kinetic energy (2) angular momentum (3) Momentum (4) Potential energy

Short answer

Angular momentum

Step-by-step solution

- Understand Bohr's Postulates

Bohr's postulates primarily address the behavior of electrons in atoms. According to Bohr, electrons travel in specific orbits without radiating energy, meaning they can only exist at certain discrete energy levels.

- Identify the Key Postulate

One of Bohr's key postulates states that the angular momentum of an electron is quantized. This means the angular momentum can only take on certain discrete values and is given by the formula: \[ L = n \hbar \] where \(n\) is a positive integer (also known as the quantum number), and \( \hbar \) is the reduced Planck's constant.

- Evaluate the Options

Given the options: 1) Kinetic energy 2) Angular momentum 3) Momentum 4) Potential energy According to Bohr's postulates, only the angular momentum is explicitly stated to take discrete values, as per the formula: \[ L = n \hbar \].

- Choose the Correct Answer

From the evaluation and explanation of Bohr's postulates, the quantity that can take up only discrete values is angular momentum.

Key concepts

Quantized Angular Momentum

One of the cornerstones of Bohr's atomic theory is the concept of quantized angular momentum. Bohr proposed that the angular momentum of an electron orbiting the nucleus is not arbitrary but can only take on specific, discrete values.

This quantization is expressed mathematically as: \( L = n \hbar \)
where \( L \) is the angular momentum, \( n \) is a positive integer known as the quantum number, and \( \hbar \) is the reduced Planck's constant.

• The quantum number, \( n \), determines the specific orbit or energy level that an electron can occupy.
• The reduced Planck's constant, \( \hbar \), is a fundamental physical constant that scales the quantized values.

In simple terms, electrons can only orbit the nucleus in specific paths where their angular momentum matches the formula above. They cannot possess angular momentum values in between these quantized levels.
This was a revolutionary idea because it introduced the concept of quantization into atomic theory, reshaping our understanding of atomic structure and electron behavior.

Discrete Energy Levels

Bohr's model also introduced the idea of discrete energy levels for electrons in an atom. According to this postulate, electrons can only exist in certain allowable energy states, not in between. These energy levels are determined by the quantum number, \( n \), and each level corresponds to a specific amount of energy.

• When an electron is in a specific orbit, it has a fixed energy.
• Electrons can move between energy levels by absorbing or emitting a photon with an energy equal to the difference between the levels.

Mathematically, the energy of an electron in a particular orbit is given by:
\[ E_n = - \frac{{13.6 \, \text{eV}}}{{n^2}} \]
where:

• \( E_n \) is the energy of the electron in the \( n \)-th energy level.
• 13.6 eV is the ground state energy of a hydrogen atom.
• \( n \) is the principal quantum number.

Therefore, each energy level is associated with a fixed amount of energy, and the electron must gain or lose that exact amount of energy to jump between levels. This explains the emission and absorption spectra of atoms, which show lines at discrete energy intervals.

Electron Orbitals

In Bohr's model, electrons move in fixed circular paths known as orbitals around the nucleus. Each of these orbitals corresponds to a specific energy level defined by the principal quantum number, \( n \).
These electron orbitals are crucial because they determine the spatial distribution of electrons around the nucleus and their energy levels.

• The lowest energy orbital, closest to the nucleus, is called the ground state.
• Higher energy orbitals, further from the nucleus, are called excited states.

In modern quantum mechanics, the concept of orbitals has evolved from fixed circular paths to probabilistic distributions that describe where an electron is likely to be found. These regions are often depicted as clouds rather than precise paths but still reflect Bohr's initial quantized energy levels.
Understanding electron orbitals helps explain chemical bonding, atomic interactions, and many other fundamental processes in chemistry and physics.