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 (c) -82 kJ mol^{-1}.

Step 1: Understanding Bohr's Energy Formula

The energy of an electron in the nth orbit of a hydrogen atom is given by the formula \(E_n = \frac{-13.6 \, ext{eV}}{n^2}\), where \(n\) is the orbit number. We need to convert the given energy value to eV since this formula uses electron volts.

Step 2: Converting Energy Units

Given energy of second Bohr orbit, \(-328 \, ext{ kJ mol}^{-1}\). We convert this to \( ext{eV} \) by using the conversion factor \(1 \, ext{eV} = 96.485 \, ext{kJ mol}^{-1}\). Thus, \[-328 \, \text{kJ mol}^{-1} \rightarrow \frac{-328}{96.485} \, ext{eV} \approx -3.4 \, ext{eV}.\] This value matches the theoretical energy for the 2nd orbit.

Step 3: Energy Formula Application

Now we use the formula to find the energy at the fourth orbit, \[E_4 = \frac{-13.6 \, ext{eV}}{4^2} = \frac{-13.6}{16} \, ext{eV} = -0.85 \, ext{eV}.\] Next, convert this energy back to kJ/mol using the conversion factor.

Step 4: Final Energy Conversion

Convert the energy from eV back to \( ext{kJ mol}^{-1} \) using the prior conversion factor: \[-0.85 \, \text{eV} \rightarrow -0.85 \times 96.485 \, ext{kJ mol}^{-1} \approx -82 \, \text{kJ mol}^{-1}.\]

Step 5: Choosing the Correct Answer

The calculated energy for the fourth orbit is \(-82 \, ext{kJ mol}^{-1}\), which matches option (c) in the provided list of answers.

Key Concepts

Energy Levels in the Bohr Model

The Bohr model of the hydrogen atom introduces the concept of energy levels as distinct orbits where electrons can reside. These orbits, or shells, represent specific energy states. Each orbit is characterized by a principal quantum number denoted by \( n \), which determines the energy associated with that orbit.
In the Bohr model for a hydrogen atom, the energy of an electron in the \( n \)-th orbit can be calculated using the formula \( E_n = \frac{-13.6 \, \text{eV}}{n^2} \). The negative sign indicates that the energy is lower than that of an unbound electron, reflecting the attractive force between the positively charged nucleus and the negatively charged electron.
As the principal quantum number \( n \) increases, the energy level becomes less negative, indicating that electrons further from the nucleus have higher energy and are less tightly bound. This means they require less energy to be removed from the atom, and they are also less stable. This principle explains why the energy at the fourth orbit (\( n = 4 \)) is higher (less negative) than that of the second orbit (\( n = 2 \)). It highlights a central feature of quantized energy levels in atomic physics.

Understanding the Hydrogen Atom

The hydrogen atom, being the simplest atom, consists of just one proton in its nucleus and one electron orbiting around it. The Bohr model, which was an early attempt to describe the quantum nature of the atom, specifically applies well to hydrogen.
In the Bohr model, the electron moves in circular orbits around the nucleus, with each orbit corresponding to a certain energy level. The difference in energy between these levels results in the emission or absorption of a photon, giving rise to discrete spectral lines.
Specifically, when an electron transitions from a higher energy orbit to a lower one, it releases energy in the form of electromagnetic radiation, corresponding to the energy difference between these orbits. Conversely, energy must be absorbed for an electron to move to a higher orbit.
The accuracy of Bohr’s model largely results from its application to hydrogen, as hydrogen's single-electron character makes it an ideal candidate for demonstrating quantized orbits. Additional complexities arise in multi-electron atoms, where electron-electron interactions become relevant.

Unit Conversion in Bohr Model Calculations

Converting units is a crucial step in many physics and chemistry problems, especially when dealing with formulas that use specific units. In the context of the Bohr model, energy values are typically calculated in electron volts (eV) but may need to be converted to kilojoules per mole (kJ/mol) depending on the context.
One electron volt is defined as the amount of kinetic energy gained or lost by an electron when it is accelerated through an electric potential difference of one volt. However, in chemistry, kilojoules per mole is often used, making unit conversions essential.
To convert from eV to kJ/mol, we use the conversion factor: \( 1 \, \text{eV} = 96.485 \, \text{kJ mol}^{-1} \). This is because 1 eV per particle equates to approximately 96.485 kJ/mol, considering Avogadro’s number, the number of particles in a mole.
Mastery of unit conversions can significantly ease the solution of numeric problems in physics and chemistry, where cross-verifying units can help in checking the correctness of calculations.