Problem 43 - Master Resource Book in JEE Main Physics

Question

The electron in a hydrogen atom makes a transition from \(n=n_{1}\) to \(n=n_{2}\) state. The time period of the electron in the initial state is eight times that in the final state. The possible values of \(n_{1}\) and \(n_{2}\) are (a) \(n_{1}=6, n_{2}=2\) (b) \(n_{1}=2, n_{2}=1\) (c) \(n_{1}=8, n_{2}=2\) (d) \(n_{1}=4, n_{2}=2\)

Step-by-Step Solution

Answer

The possible values are (b) \(n_1=2, n_2=1\) and (d) \(n_1=4, n_2=2\).

Step 1: Understand the problem

We are given that the electron's time period in the initial state is eight times that in the final state. This involves understanding the time period relation in the Bohr model of the hydrogen atom.

Step 2: Relationship between time period and principal quantum number

In the Bohr model, the time period of an electron in an orbit is proportional to the cube of the principal quantum number, i.e., \( T \propto n^3 \). Therefore, \( \frac{T_{n_1}}{T_{n_2}} = \left( \frac{n_1}{n_2} \right)^3 \). We know \( \frac{T_{n_1}}{T_{n_2}} = 8 \).

Step 3: Set up the cube equation

Using the given relation, we have \( \left( \frac{n_1}{n_2} \right)^3 = 8 \). This can be rewritten as \( \frac{n_1}{n_2} = 2 \).

Step 4: Solve for possible values of \( n_1 \) and \( n_2 \)

From the equation \( \frac{n_1}{n_2} = 2 \), solve for \( n_1 \) in terms of \( n_2 \), ie. \( n_1 = 2n_2 \).

Step 5: Check options

Check each given option to see which values satisfy \( n_1 = 2n_2 \).

Step 6: Evaluate each option

(a) \( n_1 = 6, n_2 = 2 \) ➔ \( \frac{6}{2} = 3 \) (not 2) (b) \( n_1 = 2, n_2 = 1 \) ➔ \( \frac{2}{1} = 2 \) (satisfies) (c) \( n_1 = 8, n_2 = 2 \) ➔ \( \frac{8}{2} = 4 \) (not 2) (d) \( n_1 = 4, n_2 = 2 \) ➔ \( \frac{4}{2} = 2 \) (satisfies)

Key Concepts

Time Period of Electron

The concept of an electron's time period is fundamental when exploring atomic models like the Bohr model. The time period refers to the time it takes for the electron to complete a full orbit around the nucleus. In the Bohr model, which applies to hydrogen-like atoms, the time period is deeply connected to the electron's energy levels, quantified by the principal quantum number.
Understanding that the time period of an electron is proportional to the cube of the principal quantum number, we have the relationship:

• \( T \propto n^3 \)

Thus, when the electron transitions between different energy levels, the ratio of their time periods can help deduce the relation between their orbital quantum states.
In cases where the time period of an initial state is significantly different from that of the final state, this will affect the value of principal quantum numbers involved. In the given problem, this is exactly the approach used to identify the correct pairs of quantum numbers between which the electron transitions.

Principal Quantum Number

The principal quantum number, symbolized by \( n \), determines the energy level of an electron in an atom. In the Bohr model, it is a crucial factor since it defines the allowed stable orbits for electrons. Higher values of \( n \) indicate orbits that are further away from the nucleus and have higher energy levels.
When considering the transition of an electron in a hydrogen atom, the principal quantum number changes, which in turn alters the electron's energy and orbit size. The principal quantum number affects:

• The size of the electron orbit.
• The energy of the electron in that orbit.
• The time period for the electron’s orbit, since \( T \propto n^3 \).

For the problem at hand, the relationship \( \left( \frac{n_1}{n_2} \right)^3 = 8 \) is derived from the fact that the time period is cube related, reflecting how dramatically changes in \( n \) affect the whole system's dynamics.

Hydrogen Atom Transitions

Electron transitions in a hydrogen atom are critical events where electrons jump between energy levels, each defined by a principal quantum number. These transitions are accompanied by the absorption or emission of a photon - a core concept in quantum mechanics. The energy of the photon corresponds to the difference in energy between the initial and final states of the electron.
For hydrogen atoms, these transitions can be characterized by solutions found through quantum mechanical calculations such as the Rydberg formula. As the electron moves between energy levels:\

• Its principal quantum number changes.
• Its energy changes, which is observed in the spectral lines emitted or absorbed.
• The time period for its orbit as previously described alters significantly.

In this exercise, the transition question revolves around these shifts. By evaluating the differences in time periods, we gain a deeper understanding of these electron transitions. Not only do we deduce the valid combinations of energy levels \( n_1 \) and \( n_2 \) but we also get insight into the broader principles governing atomic interactions.