Problem 26 - Master Resource Book in JEE Main Physics

Question

Let the PE of hydrogen atom in the ground state be zero. Then its total energy in the first excited state will be (a) \(27.2 \mathrm{eV}\) (b) \(23.8 \mathrm{eV}\) (c) \(12.6 \mathrm{eV}\) (d) \(10.2 \mathrm{eV}\)

Step-by-Step Solution

Answer

The total energy in the first excited state is \(-3.4 \text{ eV}\), thus standard interpretation of levels backward from zero to absolute means answer choice (d) effectively fits, but the adapted calculation in context traditionally resolves \(10.2\text{eV}\) that alone is traditional rachieved thus undefined here.

Step 1: Understand the Concept

The total energy of an electron in a hydrogen atom in an excited state can be calculated using the Bohr model formula for energy levels: \[E_n = - \frac{13.6}{n^2} \text{ eV} \] where \(n\) represents the principal quantum number or energy level of the electron. This formula calculates the total energy and assumes the potential energy is zero in the ground state.

Step 2: Identify the Energy Level

The first excited state of the hydrogen atom corresponds to \(n = 2\), because the ground state is \(n = 1\) and moving to \(n = 2\) is an excitation to the next available level.

Step 3: Calculate the Total Energy

Applying the formula \[E_n = - \frac{13.6}{2^2} \text{ eV} = - \frac{13.6}{4} \text{ eV} = -3.4 \text{ eV} \] This negative sign indicates that the energy is relative to the zero potential energy of the ground state.

Step 4: Interpret the Result

The total energy in the first excited state compared to the potential energy being zero in the ground state means that it has a lower (more negative) value than zero. The absolute value without polarity is dealt when we discuss energy difference or potential compared to a more traditional 'zero point' energy that wasn't altered.

Key Concepts

Hydrogen Atom Energy Levels

Understanding hydrogen atom energy levels is crucial to grasping atomic physics. In the Bohr model, the energy levels of a hydrogen atom are determined by the allowed orbits that an electron can occupy around the nucleus. Each orbit corresponds to a different energy level, quantized, meaning the electron cannot have energies between these levels.

The energy at each level within a hydrogen atom is quantifiable using the formula:

• \[E_n = - \frac{13.6}{n^2} \text{ eV} \]

where \(E_n\) is the energy of the \(n\)-th level, and \(n\) is the principal quantum number of the orbit.

For a hydrogen atom, the lower the energy level, the more negative the energy. This negative value indicates that energy is required to remove the electron entirely from the atom, which is why the energy decreases as electrons move to higher levels because they get closer to being free of the nucleus. This foundational framework makes it easier to calculate transitions, like from the ground state to excited states.

Principal Quantum Number

The principal quantum number, denoted as \(n\), is an essential concept in understanding atomic structure. It signifies the main energy level occupied by an electron in an atom.

For hydrogen, it determines:

• The size of the electron's orbit
• Its energy level
• Which orbit is being occupied by the electron

The principal quantum number is always a positive integer (1, 2, 3,...).

The ground state of a hydrogen atom is \(n = 1\). In this state, the electron is in its lowest energy orbit, closest to the nucleus. The first excited state is \(n = 2\), indicating the electron has absorbed energy and moved to a higher energy orbit. As \(n\) increases, the orbits become larger, and the electron's energy becomes less negative until it eventually escapes the atom when \(n\) approaches infinity.

Excited States of Hydrogen

Excited states of hydrogen are achieved when an electron absorbs energy, transitioning from a lower energy level to a higher one. Such states are a critical aspect of atomic excitation and deexcitation processes.

• When an electron moves from the ground state \((n = 1)\) to the first excited state \((n = 2)\), it absorbs a specific quantum of energy.
• This absorption results in a change in the energy level, making the electron more active or 'excited.'
• Subsequent excited states, like \(n = 3\), \(n = 4\), and so on, require even more energy.

Each transition up involves absorbing energy, while returning to a lower energy state (like emitting light) releases energy.

The concept of excited states is pivotal because it underpins phenomena like spectral lines, where each transition between levels corresponds to a specific wavelength of emitted or absorbed light. These insights are instrumental for understanding atomic spectra and the workings of lasers, among other applications.