Problem 38 - Master Resource Book in JEE Main Physics

Question

When a hydrogen atom is bombared, the atom is excited to then \(n=4\) state. The energy released, when the atom goes from \(n=4\) state to the ground state is (a) \(1.75 \mathrm{eV}\) (b) \(12.75 \mathrm{eV}\) (c) \(5 \mathrm{eV}\) (d) \(8 \mathrm{eV}\)

Step-by-Step Solution

Answer

The energy released is 12.75 eV, so the answer is (b) \(12.75\, \mathrm{eV}\).

Step 1: Understanding the Energy Levels

The energy levels of a hydrogen atom are given by the formula \( E_n = -13.6 \frac{1}{n^2} \) eV, where \( n \) is the principal quantum number of the energy level. The ground state energy (\( n=1 \)) is \( -13.6 \) eV.

Step 2: Calculating Energy for n=4

Calculate the energy of the hydrogen atom when it is in the \( n=4 \) state using the formula: \( E_4 = -13.6 \frac{1}{4^2} = -13.6 \frac{1}{16} = -0.85 \) eV.

Step 3: Calculating the Energy Difference

The energy released when an excited atom transitions from a higher energy level \( n=4 \) to a lower energy level \( n=1 \) (ground state) is the difference in energy levels: \( E = E_1 - E_4 \). Substitute the values to get \( E = -13.6 - (-0.85) = -13.6 + 0.85 = -12.75 \) eV.

Step 4: Converting Negative Energy

The negative sign in the calculated energy indicates the energy is released. We calculate the absolute value of energy released: \( |12.75| = 12.75 \) eV.

Step 5: Selecting the Correct Answer

The amount of energy released in eV is 12.75. Thus, the correct choice is option (b) \( 12.75 \) eV.

Key Concepts

Quantum Numbers

Quantum numbers are fundamental in understanding the hydrogen atom's behavior. They describe the specific characteristics of electrons within an atom. There are four types of quantum numbers:

• Principal quantum number ( ): It signifies the energy level of an electron. In the given problem, the hydrogen atom transitions from a higher energy level ( =4) to the ground state ( =1).
• Azimuthal quantum number ( abla): It defines the shape of an orbital.
• Magnetic quantum number ( abla_m): It specifies the orientation of the orbital.
• Spin quantum number ( abla_s): It represents the spin direction of the electron.

In the context of energy levels, the principal quantum number is crucial. It influences the energy associated with the electron's state. Larger quantum numbers correspond to higher energy states and positions further from the nucleus.

Energy Transition

Energy transition in atomic physics refers to the change in energy levels of an atom's electron. This phenomenon is observable in spectroscopy, where emitted or absorbed light corresponds to these transitions.

In our case, we have a hydrogen atom initially excited to a higher energy state (=4). When this electron returns to the ground state (=1), it releases energy. This energy is calculated using the formula for energy levels: \[ E_n = -13.6 \times \frac{1}{n^2} \text{ eV} \]
Calculating for =4 and =1, we find:

• For =4, the energy is \(-0.85\) eV.
• For =1, the energy is \(-13.6\) eV.

The transition energy released as light is calculated by subtracting these values: \(-13.6 - (-0.85) = -12.75\) eV. The absolute value of 12.75 eV represents the released energy. These energy transitions are the basis for many optical technologies and help in understanding atomic spectra.

Atomic Physics

Atomic physics is the study of atoms, focusing on their components and interactions. It plays a crucial role in explaining phenomena like energy transitions in hydrogen.

Key elements of atomic physics include:

• Electrons: Orbit around the nucleus in specific energy levels.
• Protons and Neutrons: Compose the nucleus itself.

The energy levels of electrons are quantized, meaning they can only possess specific energy values. When energy is absorbed by an atom, electrons can "jump" to higher levels (excitation), as seen when our hydrogen atom reaches =4. Conversely, as electrons return to lower energy states, energy is released, usually in the form of a photon.

Atomic physics not only helps in understanding these microscopic interactions but also explains larger phenomena like emission lines in atomic spectra. For hydrogen, these transitions and energy level calculations serve fundamental roles in disciplines ranging from quantum mechanics to astrophysics.