Problem 70 - A Complete Resource Book in Chemistry for JEE Main

Question

The energy of second Bohr orbit of the hydrogen atom is \(-328 \mathrm{~kJ} \mathrm{~mol}^{-1}\), hence the energy of fourth bohr orbit would be (a) \(-164 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (b) \(-41 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (c) \(-82 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (d) \(-1312 \mathrm{~kJ} \mathrm{~mol}^{-1}\)

Step-by-Step Solution

Answer

The energy of the fourth Bohr orbit is \(-82 \text{ kJ/mol}\).

Step 1: Understand the Bohr's energy equation

Bohr's model of the hydrogen atom expresses the energy of an electron in a given orbit as: \(E_n = \frac{-1312}{n^2} \text{ kJ/mol}\), where \(n\) is the principal quantum number (orbit number).

Step 2: Calculate energy of second orbit

According to the given information, the energy of the second orbit (\(n=2\)) is \(-328 \text{ kJ/mol}\). Check consistency with Bohr's formula: \(E_2 = \frac{-1312}{2^2} = \frac{-1312}{4} = -328 \text{ kJ/mol}\). The given energy for the second orbit is correct.

Step 3: Calculate energy of the fourth orbit

Use Bohr's formula to find the energy of the fourth orbit (\(n=4\)): \(E_4 = \frac{-1312}{4^2} = \frac{-1312}{16}\). Calculate this to find the energy in the fourth orbit.

Step 4: Compute the Result

Perform the division: \(E_4 = \frac{-1312}{16} = -82 \text{ kJ/mol}\). Thus, the energy of the fourth Bohr orbit is \(-82 \text{ kJ/mol}\).

Key Concepts

Principal Quantum Number

The principal quantum number, denoted by \(n\), plays a crucial role in defining the structure and energy levels of an atom. It is an integer that determines the size and energy of an atom's orbitals. In the Bohr model of the hydrogen atom, the principal quantum number represents the electron's orbit, or shell, around the nucleus. These orbits are labeled as \(n=1, 2, 3,\) and so on.

• The value of \(n\) increases with the distance from the nucleus, meaning higher \(n\) values represent orbits further out.
• The smallest \(n\) value, \(n=1\), is called the ground state, while larger values are referred to as excited states.
• Each orbit corresponds to a specific energy level, with energy increasing as \(n\) increases.

Lastly, this number is fundamental in using Bohr's formula, \(E_n = \frac{-1312}{n^2} \text{ kJ/mol}\), to calculate the energy of particular electron orbits.

Energy Levels

Energy levels within an atom correlate directly with the principal quantum number \(n\). Each energy level is quantized, meaning it has a specific value and electrons can only exist in one of these distinct levels. In the Bohr model, energy levels are often visualized as rungs on a ladder. Electrons moving between these levels emit or absorb energy as light, a concept known as the quantum leap.

• The lowest energy level (ground state) is the most stable and occurs at \(n=1\).
• As \(n\) increases, energy levels become closer to one another.
• Energy required to remove an electron from an atom, ionization energy, is an important aspect of these levels.

The relationship \(E_n = \frac{-1312}{n^2} \text{ kJ/mol}\) indicates that the energy associated with each level is inversely proportional to the square of the principal quantum number, leading to decreasing energy magnitudes as \(n\) increases.

Hydrogen Atom

The hydrogen atom, the simplest atom consisting of just one proton and a single electron, is foundational in the study of atomic structure and quantum mechanics. Niels Bohr's model gave insight into how energy levels work within such atoms. His model proposed that electrons move in specific circular orbits, or shells, around the nucleus.

• The model successfully explains the spectral lines of hydrogen seen in its emission spectrum.
• The hydrogen atom's energy levels can be calculated using Bohr's energy equation: \(E_n = \frac{-1312}{n^2} \text{ kJ/mol}\).
• Despite its limitations, Bohr's model was pivotal in advancing atomic theory and laid the groundwork for modern quantum mechanics.

Studying the hydrogen atom helps us understand larger and more complex atoms, as it simplifies concepts like electron transitions, which result in the emission or absorption spectra.