Problem 51 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The energy of an electron in the first Bohr orbit of \(\mathrm{H}\) atom is \(-13.6 \mathrm{eV}\). The possible energy value(s) of the excited state(s) for electrons in Bohr orbits of hydrogen is (are) (a) \(-3.4 \mathrm{eV}\) (b) \(-4.2 \mathrm{eV}\) (c) \(-6.8 \mathrm{eV}\) (d) \(-1.5 \mathrm{eV}\)

Step-by-Step Solution

Answer

The possible excited state energies are (a) -3.4 eV and (d) -1.5 eV.

Step 1: Understanding the Energy Levels

The energy levels in the Bohr model of the hydrogen atom can be calculated using the formula: \[ E_n = - \frac{13.6}{n^2} \text{ eV} \] where \( n \) is the principal quantum number (orbit number). The given energy for the first Bohr orbit (\( n = 1 \)) is \(-13.6\) eV. We need to find energies for higher energy levels (\( n = 2, 3, \ldots \)) to determine the possible excited states.

Step 2: Calculating Energy for n=2

Using the formula for \( n = 2 \), we find:\[ E_2 = - \frac{13.6}{2^2} = - \frac{13.6}{4} = -3.4 \text{ eV} \] This energy matches option (a).

Step 3: Checking Other Options

Calculate the energy for \( n = 3 \) using the formula:\[ E_3 = - \frac{13.6}{3^2} = - \frac{13.6}{9} = -1.51 \text{ eV} \] This approximates option (d). Note: Options (b) and (c) do not match any calculated energies for any integer value of \( n \).

Step 4: Conclusion of Excited State Energies

The only possible excited state energies from the given options, based on integer values of \( n \), are \(-3.4\) eV for \( n = 2 \) and approximately \(-1.5\) eV for \( n = 3 \). Therefore, the possible energy values of excited states for electrons in the Bohr orbits of hydrogen are \(-3.4\) eV and \(-1.5\) eV.

Key Concepts

Energy Levels

In the Bohr model of the hydrogen atom, electrons are found in distinct energy levels or orbits around the nucleus. The idea of energy levels helps explain why electrons don't just spiral into the nucleus. Instead, these energy levels are quantized, meaning electrons can only occupy certain allowed orbits.
These orbits correspond to specific energy values, which can be determined using Bohr's energy equation:

• The ground state is the lowest energy level, where an electron naturally resides.
• Excited states are higher energy levels that an electron can move to if it gains enough energy.
• Transitions between these levels can result in absorption or emission spectra.

When an electron moves to a higher energy level, it absorbs a specific amount of energy proportional to the difference between energy levels. When the electron returns to a lower energy state, it releases energy, emitting light or other electromagnetic radiation in usually distinct wavelengths.

Principal Quantum Number

The principal quantum number, represented by the symbol \( n \), is a fundamental concept in quantum mechanics. It's crucial for understanding the structure of an atom and the arrangement of its electrons.
In the context of the Bohr model, \( n \) determines:

• The size of an electron's orbit: Higher principal quantum numbers correspond to orbits further from the nucleus.
• The energy of an electron: The larger the value of \( n \), the higher the energy level of the electron.

The principal quantum number can take on any positive integer value: 1, 2, 3, and so on.
- \( n = 1 \) is the first orbit, with the lowest energy level, known as the ground state.- As \( n \) increases, each orbit represents an excited state with increasing energy levels.
The energy of each level can be calculated using the formula for energy levels, \( E_n = -\frac{13.6}{n^2} \, \text{eV} \). The negative sign shows that the electron is bound to the nucleus, with energy becoming less negative or more positive as \( n \) increases.

Electron Orbitals

While the Bohr model views electrons as particles moving in fixed orbits similar to planets around the sun, it helps simplify the concept of electron orbitals. In a more modern understanding, with quantum mechanics, an electron orbital describes a region in an atom where there is a high probability of finding an electron.

• Each orbital is defined by a set of quantum numbers, including the principal quantum number \( n \).
• Orbitals can have different shapes and sizes, depending on the other quantum numbers, but Bohr's model keeps it straightforward with circular paths.

In Bohr's hydrogen atom, these orbits are like steps up a ladder:
- An electron can only "jump" from one step to another by gaining or losing a fixed amount of energy.- The step or orbit an electron is on is analogous to its energy level, defined by its principal quantum number, \( n \).
Thus, while electron orbitals might be a simplified view in the Bohr model, it captures the essence of quantized electron behavior within atoms. This sets the foundation for the more detailed quantum mechanical models that describe electron positions and energies.