Question

Using the Bohr model, determine the values of the radii of the second and third orbits of the hydrogen atom.

Short answer

The radii of the second and third orbits are 2.116 Å and 4.761 Å, respectively.

Step-by-step solution

Understanding the Bohr Radius Formula

The Bohr model describes the radius of the electron orbits in a hydrogen atom using the formula \( r_n = n^2 \times a_0 \), where \( n \) is the principal quantum number of the electron's orbit and \( a_0 \) is the Bohr radius, approximately equal to 0.529 Å.

Applying Formula for Second Orbit

For the second orbit, the principal quantum number \( n \) is 2. Substitute \( n = 2 \) into the formula: \( r_2 = 2^2 \times a_0 = 4 \times 0.529 \text{ Å} \). Calculate to find \( r_2 \).

Calculating Radius of Second Orbit

Perform the calculation: \( r_2 = 4 \times 0.529 = 2.116 \text{ Å} \). Thus, the radius of the second orbit is 2.116 Å.

Applying Formula for Third Orbit

For the third orbit, the principal quantum number \( n \) is 3. Substitute into the formula: \( r_3 = 3^2 \times a_0 = 9 \times 0.529 \text{ Å} \). Calculate to find \( r_3 \).

Calculating Radius of Third Orbit

Perform the calculation: \( r_3 = 9 \times 0.529 = 4.761 \text{ Å} \). Thus, the radius of the third orbit is 4.761 Å.

Key concepts

Bohr Radius

The Bohr radius is a fundamental constant in the Bohr model of the atom, specifically designed to describe the size of the electron orbits in a hydrogen atom. The standard value of the Bohr radius, denoted as \( a_0 \), is approximately 0.529 Ångströms (Å). This small unit of distance is crucial for understanding atomic-scale structures.

In the Bohr model, electron orbits are quantized, meaning that they can only exist at certain discrete distances from the nucleus. The radius of any given orbit is determined by the formula: \( r_n = n^2 \times a_0 \), where \( n \) is an integer known as the principal quantum number, which denotes the orbit's level.

• For the first orbit, \( n = 1 \), so the radius is simply the Bohr radius \( a_0 \).
• For the second orbit, \( n = 2 \), the radius expands to four times \( a_0 \).
• Each subsequent orbit's radius increases by the square of its principal quantum number.

Thus, the Bohr radius provides a foundational concept for calculating the precise distances of electron orbits in simple atomic models like that of hydrogen.

Hydrogen Atom

The hydrogen atom is the simplest and most fundamental atom in the universe, consisting of only one proton and one electron. Despite its simplicity, it plays a vital role in understanding atomic structure and is often used as a model for studying more complex atoms.

In the Bohr model, the hydrogen atom is described in terms of quantized electron orbits around the nucleus. This model was groundbreaking as it introduced the idea that electron paths are not arbitrary but occur at specific, allowable distances.

• The single electron in a hydrogen atom is held in orbit by the electromagnetic attraction between the negatively charged electron and the positively charged proton.
• The Bohr model is particularly useful for understanding the spectral lines of hydrogen, which result from electron transitions between different orbits.
• Transitional movements of the electron contribute to energy absorptions or emissions, observable as distinct spectral lines.

This simplicity allows the hydrogen atom to serve as a building block for understanding more complex atomic interactions in quantum mechanics.

Quantum Mechanics

Quantum mechanics is the branch of physics that delves into the behavior of matter and energy on very small scales, such as atoms and subatomic particles. It contrasts with classical mechanics by incorporating the principle of quantization, uncertainty, and wave-particle duality.

The development of quantum mechanics revolutionized our understanding of the microscopic world, providing insights into the workings of atoms and molecules. Key concepts relevant to student understanding include:

• Quantization: Energy levels of electrons in an atom are quantized, meaning electrons can only occupy certain discrete energy levels. The Bohr model's quantized orbits are an early representation of this concept.
• Uncertainty Principle: Coined by Heisenberg, this principle explains that one cannot simultaneously know the exact position and momentum of a particle. This limits the precision with which predictions can be made about atomic properties.
• Wave-Particle Duality: Particles, like electrons, exhibit both wave-like and particle-like properties. This dual nature is fundamental to understanding how particles behave in quantum systems.

Quantum mechanics provides the mathematical framework and theoretical understanding needed to grasp phenomena that classical physics cannot explain, such as the peculiar behavior and interaction of electrons within atoms.