Problem 57 - Master Resource Book in JEE Main Physics

Question

Let \(E_{n}=\frac{-1}{8 \varepsilon_{0}^{2} n^{2} h^{2}}\) be the energy of the \(n\)th level of H-atom. If all the H-atoms are in the ground state and radiation of frequency \(\left(E_{2}-E_{1}\right) / \mathrm{h}\) falls on it. [NCERT Exemplar] (a) it will not be absorbed at all (b) some of atoms will move to the first excited state (c) all atoms will be excited to the \(n=2\) state. (d) no atoms will make a transition to the \(n=3\) state

Step-by-Step Solution

Answer

(b) Some of atoms will move to the first excited state.

Step 1: Identify Given Quantum States

We are given the energy levels of a hydrogen atom described by the equation: \[ E_n = \frac{-1}{8 \varepsilon_0^2 n^2 h^2} \]where \( n \) is the principal quantum number of the level. We need to consider two specific energy levels, the ground state \((n=1)\) and the first excited state \((n=2)\).

Step 2: Calculate Energies of Ground and First Excited States

Using the formula, calculate the energy of the ground state \((E_1)\) and the first excited state \((E_2)\):- Ground state \((n=1)\): \[ E_1 = \frac{-1}{8 \varepsilon_0^2 \cdot 1^2 \cdot h^2} = \frac{-1}{8 \varepsilon_0^2 h^2} \]- First excited state \((n=2)\):\[ E_2 = \frac{-1}{8 \varepsilon_0^2 \cdot 2^2 \cdot h^2} = \frac{-1}{32 \varepsilon_0^2 h^2} \]

Step 3: Determine Energy Difference Between First Two Levels

Find the difference in energy between the first excited state and the ground state:\[ \Delta E = E_2 - E_1 = \left(\frac{-1}{32 \varepsilon_0^2 h^2} - \frac{-1}{8 \varepsilon_0^2 h^2}\right) = \frac{-1 + 4}{32 \varepsilon_0^2 h^2} = \frac{3}{32 \varepsilon_0^2 h^2} \]

Step 4: Analyze Radiation Frequency Match

The frequency \( u \) of the radiation is given by \( u = \frac{E_2 - E_1}{h} \). Therefore:\[ u = \frac{3}{32 \varepsilon_0^2 h^3} \] Since this frequency corresponds exactly to the energy difference \( \Delta E \) in Step 3, the atoms can absorb it to move from the ground state \( n=1 \) to the first excited state \( n=2 \).

Step 5: Evaluate Given Options

Based on the calculations, evaluate each given choice:- (a) "It will not be absorbed at all" is incorrect because the frequency matches \( E_2 - E_1 \).- (b) "Some of atoms will move to the first excited state" is correct as the radiation can be absorbed.- (c) "All atoms will be excited to the \( n=2 \) state" is incorrect without further information about the density of the radiation.- (d) "No atoms will make a transition to the \( n=3 \) state" is correct since the frequency does not match the transition energy to \( n=3 \).

Key Concepts

Quantum states

In a hydrogen atom, electrons exist in distinct energy levels known as quantum states. These states are quantized, meaning electrons can inhabit only specific energy levels. The quantum state of an electron is determined by its principal quantum number, denoted by \( n \). This integer indicates the specific energy level occupied by the electron, such as \( n=1 \) for the ground state, and \( n=2 \) for the first excited state.
Each state has a unique energy value, calculated by using the formula \( E_n = \frac{-1}{8 \varepsilon_0^2 n^2 h^2} \). This equation gives the energy of the electron for a given quantum state, taking into account constants like the permittivity of free space \( \varepsilon_0 \) and Planck's constant \( h \).
Understanding quantum states helps in predicting the behavior of electrons, especially during energy transitions, as they can move to higher or lower energy states.

Energy transition

Energy transitions in atoms occur when electrons move between quantum states. These transitions involve a gain or loss of energy corresponding to the difference in energy levels. When an electron moves from a lower to a higher energy state, it absorbs energy; conversely, moving from a higher to a lower state releases energy.
For a hydrogen atom, such transitions are crucial to understanding phenomena like spectral lines. For example, an electron moving from the ground state \( n=1 \) to the first excited state \( n=2 \) absorbs a specific amount of energy that corresponds to the difference between these energy levels. This difference is calculated as

• \( \Delta E = E_2 - E_1 = \frac{3}{32 \varepsilon_0^2 h^2} \)

where \( E_1 \) and \( E_2 \) are the energies of the respective states. Recognizing the concept of energy transition is essential, as it's the mechanism by which atoms interact with radiation.

Radiation absorption

Radiation absorption is a key process in which atoms absorb photons to enable electronic transitions between quantum states. This occurs when the energy of the incoming radiation matches the energy required to shift an electron to a higher energy state. The frequency of radiation necessary for absorption is directly related to the energy difference between two states.
For example, in the hydrogen atom, when radiation of frequency \( \left(\frac{E_2 - E_1}{h}\right) \) is applied, it exactly matches the energy needed for an electron in the ground state \( n=1 \) to jump to the first excited state \( n=2 \). This makes option (b) correct as some atoms move to the excited state, given sufficient radiation.
This selective absorption leads to distinct lines in an atom's absorption spectrum, each corresponding to a specific transition.

Principal quantum number

The principal quantum number, symbolized by \( n \), is a fundamental concept in quantum mechanics which determines the electron's energy level in an atom. It is an integer value that starts from \( 1 \) and increases to higher numbers for higher energy levels.
The principal quantum number not only defines the energy of the electron in its orbit but also affects the size of the electron's orbit; higher \( n \) values correspond to higher energy levels and further distances from the nucleus.
In our scenario, the principal quantum numbers \( n=1 \) and \( n=2 \) pertain to the ground and first excited states of the electron in the hydrogen atom. The energy associated with these states can be calculated using the energy level formula and it succinctly reveals how energy and electron orbit dimensions evolve with varying \( n \). Understanding the principal quantum number is pivotal for dissecting many aspects of atomic structure and electron behavior.