Problem 180 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following is the energy of a possible excited state of hydrogen? (a) \(+13.6 \mathrm{eV}\) (b) \(-6.8 \mathrm{eV}\) (c) \(-3.4 \mathrm{eV}\) (d) \(+6.8 \mathrm{eV}\)

Step-by-Step Solution

Answer

The energy of a possible excited state of hydrogen is \(-3.4 \, \text{eV}\) (option c).

Step 1: Understanding Hydrogen Atom Energy Levels

The energy levels of a hydrogen atom are given by the formula: \[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]where \( n \) is the principal quantum number (\( n = 1, 2, 3, \ldots\)). This formula tells us that the energy levels are negative and decrease in magnitude as \( n \) increases. Only negative energy values correspond to bound states, while positive values imply the electron is free or outside the atom.

Step 2: Calculate Possible Energy Level Values

Using the formula \( E_n = -\frac{13.6 \, \text{eV}}{n^2} \), let's calculate the energies for the first few levels:- For \( n = 1 \), \( E_1 = -13.6 \, \text{eV} \) - For \( n = 2 \), \( E_2 = -\frac{13.6}{4} = -3.4 \, \text{eV} \) - For \( n = 3 \), \( E_3 = -\frac{13.6}{9} \approx -1.51 \, \text{eV} \)Notice that the negative energies and their magnitudes reduce as \( n \) increases.

Step 3: Identify the Correct Option

From the calculated values, we see that the energy \(-3.4 \, \text{eV}\) corresponds to the energy of the \( n=2 \) excited state. The options only include one matching negative energy: (c) \(-3.4 \, \text{eV}\), which fits with our calculations.

Key Concepts

Principal Quantum Number

In the fascinating world of quantum mechanics, the principal quantum number, denoted as \( n \), plays a crucial role. It is a positive integer representing the energy level of an electron within an atom. You can think of it as the shell in which an electron orbits the nucleus in a hydrogen atom. The principal quantum number takes on values of 1, 2, 3, and so forth. These numbers indicate different energy levels or shells:

• \( n = 1 \): This is the closest shell to the nucleus and has the lowest energy.-
• Increasing \( n \): As \( n \) increases to 2, 3, or beyond, the electron resides in higher energy shells away from the nucleus.-

The formula for the energy levels of these shells in hydrogen is \( E_n = -\frac{13.6 \, \text{eV}}{n^2} \). This tells us that lower levels (with smaller \( n \)) are more negative and thus more bound, while higher \( n \) values have less negative energies.

Excited States

An excited state is when an electron occupies an energy level higher than the lowest possible level, i.e., it's not in its ground state. For hydrogen atoms, the ground state corresponds to \( n = 1 \), where the electron is at its most stable configuration close to the nucleus. When the electron absorbs energy, it can jump to higher \( n \) levels such as 2, 3, or higher, creating excited states. Key points about excited states include:

• Higher \( n \) values: These states are more precarious as electrons can easily return to lower energy states, releasing energy usually as light.-
• Temporary existence: Electrons do not stay in excited states for long. They tend to fall back to lower levels, a process known as emission or de-excitation.-

Excited states are essential for understanding phenomena such as spectral lines, where specific wavelengths of light are absorbed or emitted by atoms like hydrogen.

Energy Calculations

Calculating energy levels for a hydrogen atom involves a precise formula: \( E_n = -\frac{13.6 \, \text{eV}}{n^2} \). This formula provides an intuitive understanding of the energy associated with different quantum levels:

• Negative Energy Values: These indicate that the electron is in a bound state within the atom.-
• Energy Increases with Decreasing Negative: As the principal quantum number \( n \) rises, the value of energy becomes less negative, indicating less binding.-

To find out the energy of an excited state, you plug the desired \( n \) value into the formula. For instance, \( n = 2 \) gives \( -3.4 \, \text{eV} \), showing that as \( n \) increases, the energies align with predictions from the model.