Problem 45 - Master Resource Book in JEE Main Physics

Question

The electron in a hydrogen atom makes a transition \(n_{1} \rightarrow n_{2}\) where \(n_{1}\) and \(n_{2}\) are the principal quantum numbers of the two states. Assume the Bohr model to be valid. The time period of electron in the initial state is 8 times that in the final state. The possible values of \(n_{1}\) and \(n_{2}\) are (a) \(n_{1}=6, n_{2}=3\) (b) \(n_{1}=8 n_{2}=2\) (c) \(n_{1}=n_{2}=1\) (d) \(n_{1}=8, n_{2}=1\)

Step-by-Step Solution

Answer

The possible values are (a) \(n_1 = 6, n_2 = 3\) and (b) \(n_1 = 8, n_2 = 2\).

Step 1: Understanding the Bohr Model

The Bohr model describes electrons in hydrogen atoms as orbiting the nucleus in quantized energy levels labeled by the principal quantum number, \(n\). The time period of an electron in an orbit is proportional to \(n^3\). Therefore, if \(T_1\) and \(T_2\) are the time periods of the electron in states \(n_1\) and \(n_2\) respectively, then \(T_1 = n_1^3\) and \(T_2 = n_2^3\).

Step 2: Relate the Time Periods

The problem states that the time period in the initial state is 8 times that in the final state. Therefore, we have the equation \(T_1 = 8T_2\). Substituting the time periods in terms of \(n_1\) and \(n_2\), we have \(n_1^3 = 8n_2^3\).

Step 3: Solve for Ratio of Quantum Numbers

From the equation \(n_1^3 = 8n_2^3\), we can simplify to \((\frac{n_1}{n_2})^3 = 8\). Taking the cube root on both sides, we have \(\frac{n_1}{n_2} = 2\). This means \(n_1 = 2n_2\).

Step 4: Determine Valid Options

Using the relation \(n_1 = 2n_2\), we test the options given: (a) \(n_1 = 6, n_2 = 3\): valid as \(\frac{6}{3} = 2\). (b) \(n_1 = 8, n_2 = 2\): valid as \(\frac{8}{2} = 2\). (c) \(n_1 = n_2 = 1\): not valid as \(\frac{1}{1} eq 2\). (d) \(n_1 = 8, n_2 = 1\): not valid as \(\frac{8}{1} eq 2\).

Step 5: Conclude with Solution

Based on the valid options, the possible transitions that satisfy \(n_1 = 2n_2\) are (a) \(n_1 = 6, n_2 = 3\) and (b) \(n_1 = 8, n_2 = 2\).

Key Concepts

Hydrogen Atom

The hydrogen atom is the simplest atom in the universe and is pivotal in understanding atomic structure. It consists of a single proton at its center, known as the nucleus, and one electron orbiting around it. This electron is held in place by the electrostatic force of attraction between the oppositely charged proton and electron.

The study of hydrogen atoms has greatly influenced atomic physics as it was the first system to be successfully modeled accurately using quantum mechanics. This achievement has led to important discoveries about the nature of atoms and quantum behavior. The simplicity of the hydrogen atom makes it an ideal subject for testing theories in quantum mechanics.

Principal Quantum Number

In quantum mechanics, the principal quantum number, denoted by the symbol \( n \), is one of the quantum numbers assigned to an electron in an atom. It is a crucial number because it defines the major energy level or shell that the electron occupies.

The value of \( n \) is always a positive integer \((n = 1, 2, 3, \ldots)\). Each increment in \( n \) corresponds to a higher energy level and typically a larger orbital size. Thus, \( n = 1 \) is the closest orbital to the nucleus and has the lowest energy, while \( n = 2 \), \( n = 3 \), and higher are progressively farther and have increased energy.

In the Bohr model, the principal quantum number determines the radius of the electron's orbit around the nucleus. The higher the value of \( n \), the further away the electron orbits from the nucleus.

Electron Transitions

Electron transitions refer to the movement of an electron from one energy level (or shell) to another within an atom. In the context of a hydrogen atom, this involves moving between the quantized orbits defined by different principal quantum numbers \( n_1 \) and \( n_2 \).

When an electron transitions from a higher energy level to a lower one, it releases energy, often in the form of light. Conversely, moving from a lower to a higher energy level requires the absorption of energy. This process is responsible for the emission and absorption spectra observed in hydrogen and other elements.

• Emission: Electron releases energy as it drops to a lower energy level.
• Absorption: Electron absorbs energy to move to a higher energy level.

Time Period in Quantum States

The time period of an electron in a quantum state, particularly in the Bohr model, refers to the time it takes for an electron to complete one full orbit around the nucleus. In the hydrogen atom, this time period is proportional to the cube of the principal quantum number \( n \), expressed as \[ T \propto n^3 \].

This relationship implies that electrons in higher quantum states (larger \( n \)) have significantly longer orbital periods than those in lower states. For instance, if an electron transitions from an initial state with quantum number \( n_1 \) to a final state with quantum number \( n_2 \), the time period changes according to the ratio \( \frac{n_1^3}{n_2^3} \).

Understanding these periods is essential in calculating the dynamics of electron transitions and predicting the spectral lines in hydrogen's emission spectrum.