Problem 22 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is \(\left[a_{0}\right.\) is Bohr radius] : (a) \(\frac{h^{2}}{4 \pi^{2} m a_{0}^{2}}\) (b) \(\frac{h^{2}}{16 \pi^{2} m a_{0}^{2}}\) (c) \(\frac{h^{2}}{32 \pi^{2} m a_{0}^{2}}\) (d) \(\frac{h^{2}}{64 \pi^{2} m a_{0}^{2}}\)

Step-by-Step Solution

Answer

The kinetic energy is given by option (c): \(\frac{h^{2}}{32 \pi^{2} m a_{0}^{2}}\).

Step 1: Identify the Formula for Kinetic Energy in Bohr Model

In the Bohr model, the kinetic energy (KE) of an electron in an orbit is given by \( KE = \frac{1}{2} m v^2 \), but it's also expressed as \( KE = \frac{Z^2 e^2}{8 \pi \varepsilon_0 n^2 a_0} \), where \( Z \) is the atomic number, \( e \) is the charge of an electron, \( \varepsilon_0 \) is the permittivity of free space, \( n \) is the principal quantum number, and \( a_0 \) is the Bohr radius.

Step 2: Substitute the Values for Hydrogen Atom

In the hydrogen atom (\( Z = 1 \)), substituting the known values into the expression for kinetic energy gives the equation: \[ KE = \frac{1^2 e^2}{8 \pi \varepsilon_0 n^2 a_0} \]. Since the electron is in the second Bohr orbit, \( n = 2 \).

Step 3: Express in Terms of Bohr Model Constants

Recognize that in the Bohr model, the term \( \frac{e^2}{4 \pi \varepsilon_0 a_0} = \frac{h^2}{4 \pi^2 m a_0^2} \), which relates to the basic constants of the Bohr model. This allows simplification and understanding of the terms.

Step 4: Simplify the Kinetic Energy Expression

Rearrange the kinetic energy expression for an electron in the second Bohr orbit: \[ KE = \frac{h^2}{8 \pi^2 m a_0^2 n^2} \]. Since \( n = 2 \), substitute it into the equation: \[ KE = \frac{h^2}{8 \pi^2 m a_0^2 \times 4} = \frac{h^2}{32 \pi^2 m a_0^2} \].

Step 5: Match with Given Options

Compare the calculated expression for kinetic energy \( \frac{h^2}{32 \pi^2 m a_0^2} \) with the given options. It matches option (c).

Key Concepts

Kinetic Energy

Kinetic energy (KE) is an important concept in physics, representing the energy that an object possesses due to its motion. In the context of the Bohr model of the atom, the kinetic energy of an electron orbiting a nucleus is particularly significant.
Within this model, KE is expressed by the formula \( KE = \frac{1}{2} m v^2 \), where \( m \) is the mass of the electron, and \( v \) is its velocity. However, for electrons in atoms, there is a more specific formula: \( KE = \frac{Z^2 e^2}{8 \pi \varepsilon_0 n^2 a_0} \).
This refined formula considers the atomic number \( Z \), the elementary electric charge \( e \), the permittivity of free space \( \varepsilon_0 \), the principal quantum number \( n \), and the Bohr radius \( a_0 \). For hydrogen, where \( Z = 1 \), this simplifies the calculations when determining kinetic energy in specific orbits, such as the second Bohr orbit.

Hydrogen Atom

The hydrogen atom is the simplest atom and consists of only one proton and one electron. This simplicity makes it an ideal model for understanding atomic structure and electron behavior.
When studying quantum mechanics and atomic models, hydrogen often serves as the starting point due to its straightforwardness. The Bohr model uses the hydrogen atom to illustrate key concepts, showing that electrons orbit the nucleus much like planets orbit the sun.
Though the hydrogen atom has only a single electron, its various energy levels can still be explored through different values of the principal quantum number \( n \). Each of these levels corresponds to a specific orbit and energy associated with the electron, laying the groundwork for understanding more complex atoms and molecules.

Second Bohr Orbit

The second Bohr orbit specifically refers to the orbit of an electron in a hydrogen atom when the principal quantum number \( n \) equals 2. This orbit is one step further from the nucleus compared to the ground state.
With each increase in \( n \), the electron's orbit gets larger, having more energy and less tightly bound to the nucleus. For \( n = 2 \), this also means that the kinetic energy of the electron changes according to the Bohr model's formula.
For a hydrogen atom, the kinetic energy is calculated using the expression \( KE = \frac{h^2}{8 \pi^2 m a_0^2 n^2} \). In the second Bohr orbit, since \( n = 2 \), this simplifies to \( KE = \frac{h^2}{32 \pi^2 m a_0^2} \). This simple calculation illustrates how fundamental constants and quantum numbers come together to determine the properties of atomic orbits.