Problem 110 - A Complete Resource Book in Chemistry for JEE Main

Question

The ratio of the difference between 2 nd and 3 rd Bohr's orbit energy to that between 3 rd and 4 th orbit energy is (a) \(7 / 20\) (b) \(20 / 7\) (c) \(27 / 9\) (d) \(9 / 27\)

Step-by-Step Solution

Answer

The correct ratio is \( \frac{20}{7} \), corresponding to option (b).

Step 1: Understand Bohr's Energy Formula

Bohr's model states that the energy of an electron in the nth orbit of a hydrogen atom is given by \( E_n = - \frac{13.6}{n^2} \text{ eV} \) where \( n \) is the principal quantum number. We'll use this formula to compute the energies for the specified orbits.

Step 2: Compute Energy for the 2nd, 3rd, and 4th Orbits

Calculate energies for \( n=2, 3, \text{and} 4 \):\( E_2 = - \frac{13.6}{4} \), \( E_3 = - \frac{13.6}{9} \), \( E_4 = - \frac{13.6}{16} \).

Step 3: Calculate Energy Differences

Calculate the differences for the given transitions: \( E_2 - E_3 = \left(- \frac{13.6}{4} \right) - \left(- \frac{13.6}{9} \right) \) and \( E_3 - E_4 = \left(- \frac{13.6}{9} \right) - \left(- \frac{13.6}{16} \right) \).

Step 4: Simplify Energy Differences

For \( E_2 - E_3 \), solve: \( - \frac{13.6}{4} + \frac{13.6}{9} = 13.6 \left( -\frac{1}{4} + \frac{1}{9}\right) = 13.6 \left(\frac{-9 + 4}{36} \right) = 13.6 \left( \frac{-5}{36} \right) = -1.889 \text{ eV} \). Similarly, for \( E_3 - E_4 \), solve: \( 13.6 \left( \frac{-16 + 9}{144} \right) = 13.6 \left( \frac{-7}{144} \right) = -0.661 \text{ eV} \).

Step 5: Compute the Ratio of Differences

Calculate: \( \text{Ratio} = \frac{E_2 - E_3}{E_3 - E_4} = \frac{-1.889}{-0.661} \approx 2.86 \). This value corresponds to a ratio of \( \frac{20}{7} \).

Step 6: Choose the Correct Option

From our calculation, the ratio \( \frac{20}{7} \) matches option (b).

Key Concepts

Energy Levels

In Bohr's model of the hydrogen atom, energy levels are like the floors of a building that electrons inhabit. Each of these floors represents a specific amount of energy. The higher the floor, the more energy an electron has.

When an electron is in the lowest energy level or 'ground state,' it is most stable. To move to a higher energy level, it would need to gain more energy, just like you would need more energy to climb stairs or use an elevator to reach a higher floor. It's important to remember that energy levels are quantized, meaning electrons can only exist at specific energies, not between them.

• Electrons closest to the nucleus (lower energy levels) have less energy.
• As electrons move to higher energy levels, their energy increases.

Principal Quantum Number

The principal quantum number, often symbolized as 'n,' is like a code that tells us which energy level an electron occupies. The larger the value of 'n,' the higher the energy level and the farther from the nucleus that electron will be.

Bohr's formula for the energy of an electron in a hydrogen atom is expressed as \( E_n = - \frac{13.6}{n^2} \text{ eV}\). Here, 'n' plays a crucial role:

• 'n' determines the size of the orbit of the electron.
• Lower values of 'n' correspond to lower energy states (closer to the nucleus).
• Higher values of 'n' mean higher energy states and orbits further away from the nucleus.

Understanding the principal quantum number helps us understand how electrons are arranged in atoms, which is essential for predicting chemical reactions and properties.

Electron Transitions

Electron transitions occur when electrons jump from one energy level to another. These jumps can result in the absorption or emission of energy, usually in the form of photons (light).

When an electron absorbs energy, it jumps to a higher energy level (farther from the nucleus). Conversely, when it falls back to a lower energy level, it releases energy by emitting a photon.

• Energy is absorbed when an electron moves up to a higher energy level.
• Energy is emitted when an electron falls to a lower energy level.
• The difference in energy between the two levels is what determines the energy (or wavelength) of the emitted or absorbed photon.

These transitions explain phenomena like the emission spectrum of hydrogen, where each line corresponds to a specific transition between energy levels. Understanding these transitions helps in fields like spectroscopy and astrophysics.