Problem 40 - Master Resource Book in JEE Main Physics

Question

The energy of an electron in \(n\)th orbit of the hydrogen atom is given by \(E_{n}=\frac{-13.6}{n^{2}} \mathrm{eV}\). The energy required to raise an electron from, the first orbit to the second orbit will be (a) \(10.2 \mathrm{eV}\) (b) \(12.1 \mathrm{eV}\) (c) \(13.6 \mathrm{eV}\) (d) \(3.4 \mathrm{eV}\)

Step-by-Step Solution

Answer

The energy required is 10.2 eV.

Step 1: Identify Energy Formula

Recognize that the energy levels of an electron in the hydrogen atom are given by the formula: \(E_n = \frac{-13.6}{n^2} \text{ eV}\), where \(n\) is the principal quantum number. We need to calculate the energy for the first orbit (\(n=1\)) and the second orbit (\(n=2\)).

Step 2: Calculate Energy for First Orbit (n=1)

Substitute \(n=1\) into the energy formula to find the energy of the electron in the first orbit: \(E_1 = \frac{-13.6}{1^2} \text{ eV} = -13.6 \text{ eV}\).

Step 3: Calculate Energy for Second Orbit (n=2)

To find the energy of the electron in the second orbit, substitute \(n=2\) into the energy formula: \(E_2 = \frac{-13.6}{2^2} \text{ eV} = \frac{-13.6}{4} \text{ eV} = -3.4 \text{ eV}\).

Step 4: Determine Energy Difference

Calculate the energy needed to move the electron from the first orbit to the second orbit. This is the energy difference: \( ext{Energy required} = E_2 - E_1 = (-3.4) - (-13.6) \text{ eV} = 10.2 \text{ eV}\).

Step 5: Final Answer

The energy required to raise an electron from the first orbit to the second orbit is \(10.2\text{ eV}\).

Key Concepts

Principal Quantum Number

The principal quantum number is a fundamental concept when studying the structure of atoms, especially the hydrogen atom. It is denoted by the letter \( n \) and indicates the primary energy levels an electron can occupy. These are sometimes referred to as "shells." In the Bohr model of the hydrogen atom, \( n \) can be any positive integer (\( n = 1, 2, 3, \ldots \)). This number helps us understand the electron's energy and its distance from the nucleus.

For the hydrogen atom, the energy of an electron in a particular energy level or orbit is determined primarily by this number \( n \). As \( n \) increases, the electron is found further from the nucleus and possesses higher energy. The principal quantum number is crucial as it also determines other quantum numbers associated with an electron, giving a comprehensive understanding of its position and energy state in an atom.

Electron Transition

Electron transition is a concept that helps explain how electrons move between different energy levels or shells within an atom. When an electron absorbs or emits energy, it transitions from one level to another, specifically between shells denoted by different principal quantum numbers.

- **Upward Transition**: If the electron gains energy, it moves from a lower energy level to a higher one, e.g., moving from \( n=1 \) to \( n=2 \).
- **Downward Transition**: Conversely, if the electron loses energy, it falls back to a lower energy level.

These transitions play a critical role in phenomena such as the emission spectrum of hydrogen. Each possible electron transition in hydrogen corresponds to a specific change in energy, which then emits or absorbs light at specific wavelengths. In our exercise, raising an electron from the first to the second orbit involves a transition characterized by energy absorption.

Energy Calculation

Energy calculation in the context of an electron in a hydrogen atom involves determining the difference in energy between two principal quantum levels. The formula used is \( E_n = \frac{-13.6}{n^2} \) eV, where \( E_n \) describes the energy at a specific energy level \( n \).

To calculate the energy required for an electron transition from the first orbit (\( n=1 \)) to the second orbit (\( n=2 \)), follow these steps:

• First, determine the energy at \( n=1 \): \( E_1 = -13.6 \, \text{eV} \).
• Second, calculate the energy at \( n=2 \): \( E_2 = \frac{-13.6}{4} \, \text{eV} = -3.4 \, \text{eV} \).
• Finally, find the energy difference to understand the energy required for the transition: \( E_2 - E_1 = (-3.4) - (-13.6) = 10.2 \, \text{eV} \).

The positive value of 10.2 eV indicates the amount of energy needed to move the electron from its ground state to an excited state, specifically from the first to the second orbit.