Problem 32 - Master Resource Book in JEE Main Physics

Question

The ionization energy of hydrogen atom is \(13.6 \mathrm{eV}\). Following Bohr's theory, the energy corresponding to a transition between 3 rd and 4 th orbit is (a) \(3.40 \mathrm{eV}\) (b) \(1.51 \mathrm{eV}\) [c) \(0.85 \mathrm{eV}\) (d) \(0.66 \mathrm{eV}\)

Step-by-Step Solution

Answer

The energy corresponding to the transition is \(0.66 \mathrm{eV}\) (Option d).

Step 1: Understand Energy Levels in a Hydrogen Atom

According to Bohr's theory, the energy of an electron in a hydrogen atom at a particular orbit (n) is given by the formula \(E_n = - \frac{13.6 \, \mathrm{eV}}{n^2}\), where \(13.6 \, \mathrm{eV}\) is the ionization energy of the hydrogen atom.

Step 2: Calculate Energy for n=3

For the orbit with \( n = 3 \), the energy is \( E_3 = - \frac{13.6 \, \mathrm{eV}}{3^2} = - \frac{13.6 \, \mathrm{eV}}{9} = -1.51 \, \mathrm{eV} \).

Step 3: Calculate Energy for n=4

For the orbit with \( n = 4 \), the energy is \( E_4 = - \frac{13.6 \, \mathrm{eV}}{4^2} = - \frac{13.6 \, \mathrm{eV}}{16} = -0.85 \, \mathrm{eV} \).

Step 4: Determine the Transition Energy

The energy corresponding to a transition from the 4th orbit to the 3rd orbit is \( \Delta E = E_3 - E_4 = (-1.51 \, \mathrm{eV}) - (-0.85 \, \mathrm{eV}) \).

Step 5: Calculate the Energy Difference

Calculate the difference \( \Delta E = -1.51 \, \mathrm{eV} + 0.85 \, \mathrm{eV} = -0.66 \, \mathrm{eV} \). Thus, the energy released during the transition is \( 0.66 \, \mathrm{eV} \) because it is a transition to a lower energy state (emission).

Key Concepts

Hydrogen Atom Energy Levels

In Bohr's model of the hydrogen atom, the energy level of an electron is quantized, meaning it can only occupy certain defined "rungs" on an energy ladder. Each rung corresponds to a specific orbit labeled by the quantum number \( n \). The energy levels are negative because the electron is bound to the nucleus. The closer an electron is to the nucleus (lower \( n \)), the lower its energy. The energy of an electron in the \( n^{th} \) orbit is described by the formula:

• \( E_n = -\frac{13.6 \, \mathrm{eV}}{n^2} \)

This formula tells us that as \( n \) increases, the absolute value of this negative energy increases, showing that the electron is less tightly bound. In other words, higher energy levels are less stable and the electron requires less energy to transition to a higher state or to escape entirely from the atom.

Electron Transition

An electron transition occurs when an electron moves from one energy level to another, either absorbing or releasing energy in the process. This is crucial in explaining atomic spectra, where such transitions emit or absorb specific wavelengths of light. When an electron drops to a lower energy level, it releases energy, often in the form of a photon.For transitions in the hydrogen atom:

• The energy change \( \Delta E \) associated with a transition between two levels \( n_i \) and \( n_f \) is given by \( \Delta E = E_i -E_f \).
• Here, \( E_i \) and \( E_f \) are the initial and final energy levels computed using the formula \( E_n = -\frac{13.6 \, \mathrm{eV}}{n^2} \).

For example, the transition from the 4th to the 3rd orbit in hydrogen results in energy release of \( 0.66 \, \mathrm{eV} \). This energy difference corresponds to the emission of light with a specific frequency.

Ionization Energy

Ionization energy is an important concept when discussing the binding of electrons in an atom. In the case of hydrogen, the ionization energy is the energy required to remove its single electron from the ground state to infinity, effectively freeing it from the nucleus. For hydrogen, this energy is precisely \( 13.6 \, \mathrm{eV} \), which reflects the energy of the lowest orbit (\( n = 1 \)).Here’s what happens during ionization:

• The electron absorbs exactly \( 13.6 \, \mathrm{eV} \) of energy, increasing its energy to zero as it moves from being bound to free.
• This signifies that the electron has overcome the attractive force of the proton, illustrating the concept that zero energy marks the threshold of being unbound.

Understanding ionization energy helps in comprehending how atoms absorb energy from external sources to trigger electron liberation or excite them to higher energy levels. In any heavier atom, this energy quantification also aids in explaining why multi-electron atoms require additional energy compared to a hydrogen atom for ionization.