Question

If the radius of first Bohr orbit is \(x \mathrm{pm}\), then the radius of the third orbit would be (a) \((3 \times x) \mathrm{pm}\) (b) \((6 \times x) \mathrm{pm}\) (c) \(\left(\frac{1}{2} \times x\right) \mathrm{pm}\) (d) \((9 \times x) \mathrm{pm}\)

Short answer

The radius of the third orbit would be (d) (9 \times x) \mathrm{pm}.

Step-by-step solution

Understand the Bohr Model

According to the Bohr model of the hydrogen atom, the radius of the nth orbit is directly proportional to the square of the orbit number (n^2). The radius of the first Bohr orbit is given as x pm.

Apply the Formula for Orbit Radius

Use the formula for the radius of nth orbit, which is given by r_n = n^2 x r_1 where r_1 is the radius of the first orbit and r_n is the radius of nth orbit.

Calculate the Radius of the Third Orbit

To find the radius of the third orbit, plug in n = 3 into the formula: r_3 = 3^2 x (x pm) = 9 x (x pm).

Key concepts

Bohr Orbit Radius

Understanding the Bohr orbit radius is critical when delving into the atomic structure according to the Bohr model. The Bohr model presents a quantized picture of electron positions around a nucleus. In this framework, the term 'Bohr orbit radius' refers to the distance from the nucleus to the electron within a given orbit level in a hydrogen atom.

Electrons can exist only in specific circular paths, or orbits, without radiating energy. These paths are differentiated by their radius. The simplistic beauty of the Bohr model is that it allows us to easily visualize and calculate the position of an electron in one of these orbits. For example, if the radius of the first orbit (closest to the nucleus) of a hydrogen atom is known, the radii of all other orbits can be predicted using Bohr's quantization rule.

This rule states that the radius of an orbit is proportional to the square of the orbit number (also called the principal quantum number). As a result, the radius increases quite rapidly with each subsequent orbit, since each radius is the orbit number squared, multiplied by the first orbit's radius.

Bohr Formula for Orbit Radius

The Bohr formula for orbit radius is fundamental to predicting the size of orbits in the Bohr model. This equation serves as a mathematical expression of the concepts behind Bohr's postulates.

To apply this formula effectively, one must recall the proportionality: the radius of the nth orbit is proportional to the square of the orbit's number times the radius of the first orbit. Mathematically, this is expressed as:\[ r_n = n^2 \times r_1 \]where:\[ r_n \]represents the radius of the nth orbit,\[ n \]is the orbit number, and\[ r_1 \]is the radius of the initial Bohr orbit. For instance, if the first orbit radius is known, you multiply this value by the square of the desired orbit number to find the radius of that orbit.

The Bohr formula for orbit radius connects the macroscopic world we can measure (the radius of an atom's orbit) with the quantized microscopic world, providing a bridge between classical and quantum physics. It offers a convenient way to understand the layout of an electron's energy levels within an atom.

Hydrogen Atom

The hydrogen atom plays a pivotal role in the development and illustration of many atomic models, including the Bohr model. It's the simplest of atoms, consisting of only one electron and one proton. This simplicity makes it an ideal candidate for studying the fundamental principles of atomic structure and behavior.

In the context of the Bohr model, the hydrogen atom is often the focus because the model was initially developed to explain the hydrogen spectrum. According to Bohr, the energy levels of an electron in a hydrogen atom are quantized, which means the electron can occupy only certain well-defined orbits. Transitions between these orbits result in the absorption or emission of light at specific wavelengths, explaining why the hydrogen spectral lines are discrete.

Although the Bohr model is no longer the most accurate depiction of an atom's structure—having been superseded by quantum mechanics—it remains a significant stepping stone that helps students grasp the quantum leap concept and offers a framework for introducing more complex models. The hydrogen atom, consequently, is not just a fundamental part of chemistry and physics curricula but also a bridge to understanding more intricate atomic behavior.