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? [2015] (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 an excited state of hydrogen is \(-3.4 \) eV.

Step 1: Understand the Hydrogen Atom Energy Levels

The energy levels of the hydrogen atom are given by the formula \( E_n = -\frac{13.6}{n^2} \) eV, where \( n \) is the principal quantum number, which can be any positive integer (1, 2, 3,...). An excited state is any state with \( n > 1 \).

Step 2: Calculate Possible Energy Levels

First, calculate the energy for \( n = 2 \). Using the formula \( E_2 = -\frac{13.6}{2^2} = -3.4 \) eV. Similarly, for \( n = 3 \), \( E_3 = -\frac{13.6}{3^2} = -1.51 \) eV. Note that these energies must be negative for bound states.

Step 3: Identify the Excited State Energy from Choices

Out of the choices given: (a) \(+13.6 \) eV, (b) \(-6.8 \) eV, (c) \(-3.4 \) eV, and (d) \(+6.8 \) eV, the only energy that matches an excited state of hydrogen (from \( n = 2 \) calculation) is \(-3.4 \) eV, as derived from previous calculations.

Key Concepts

Excited states

In the study of hydrogen atoms, understanding energy levels is key. An excited state refers to a condition where an electron occupies a higher energy level than its ground state. The ground state of a hydrogen atom is when the electron is in the lowest energy level, characterized by the principal quantum number, or \( n = 1 \). Excited states occur when \( n \) is greater than 1, such as \( n = 2, 3, 4, \) and so on.

These states are crucial in phenomena like light emission, as an electron dropping from an excited state to a lower energy state releases energy in the form of light. This is why the excited states are often associated with the unique spectral lines of hydrogen. The transitions between different levels offer insights into the electronic structure and help us understand the atom's behavior under various conditions.

Principal quantum number

The principal quantum number, denoted as \( n \), is one of the key numbers used to describe the quantum state of an electron in an atom. It indicates the relative size and energy of atomic orbitals. It can take any positive integer value, such as 1, 2, 3, etc.

• When \( n = 1 \), the electron is in the ground state, the most stable and least energetic state of the atom.
• Values of \( n \) greater than 1 indicate excited states, where electrons have more energy.

The larger the value of \( n \), the further the electron is from the nucleus on average, and the more energy it possesses. This principle is foundational in quantum mechanics, explaining not just hydrogen atom behavior, but other elements' electronic configurations as well.

Bound state energies

Bound state energies are the energies of electrons that are held within an atom, as opposed to being free or knocked out of the atom. In the hydrogen atom, these energies are quantized according to the formula:
\[E_n = -\frac{13.6}{n^2} \text{ eV}\]Here, \( E_n \) is the energy at a given principal quantum number \( n \).

• For \( n = 1 \), the energy level is \( -13.6 \) eV, representing the ground state.
• For any \( n > 1 \), the states are excited states with less negative energy values like \( -3.4 \) eV for \( n = 2 \), or \( -1.51 \) eV for \( n = 3 \).

Note that these energies are negative, signifying that the electrons are in bound states, as opposed to being unbound or free. Negative energy indicates that work must be done to free the electron from the atom, important for understanding reactions and photon interaction with atoms.