Problem 156 - A Complete Resource Book in Chemistry for JEE Main

Question

The ionization enthalpy of hydrogen atom is \(1.312 \times\) \(10^{6} \mathrm{~J} \mathrm{~mol}^{-1}\). The energy required to excite the electron in the atom from \(n=1\) to \(n=2\) is (a) \(8.51 \times 10^{5} \mathrm{~J} \mathrm{~mol}^{-1}\) (b) \(6.56 \times 10^{5} \mathrm{~J} \mathrm{~mol}^{-1}\) (c) \(7.56 \times 10^{5} \mathrm{~J} \mathrm{~mol}^{-1}\) (d) \(9.84 \times 10^{5} \mathrm{~J} \mathrm{~mol}^{-1}\)

Step-by-Step Solution

Answer

The energy required is \(9.84 \times 10^5 \mathrm{~J/mol}\), option (d).

Step 1: Understand the Ionization Enthalpy

The ionization enthalpy for hydrogen is given as \(1.312 \times 10^6\, \text{J/mol}\). This represents the energy required to remove an electron from the hydrogen atom when it is in its ground state (\(n=1\)) to infinity (completely remove it from the atom).

Step 2: Energy Level Formula for Hydrogen Atom

Hydrogen atom energy levels can be calculated using the formula: \( E_n = - \frac{R_H}{n^2} \), where \(R_H\) is the Rydberg constant \(2.18 \times 10^{-18}\, \text{J}\), and \(n\) is the principal quantum number. For a mole, multiply by Avogadro's number.

Step 3: Calculate Energy for n=1 State

Substitute \(n=1\) into the formula:\[ E_1 = - \frac{R_H}{1^2} = -2.18 \times 10^{-18}\, \text{J} \] For a mole, it becomes:\[ E_{1,\text{mol}} = -2.18 \times 10^{-18} \times 6.022 \times 10^{23} = -1.312 \times 10^6\, \text{J/mol} \]

Step 4: Calculate Energy for n=2 State

Substitute \(n=2\) into the formula:\[ E_2 = - \frac{R_H}{2^2} = - \frac{2.18 \times 10^{-18}}{4} \]\[ E_2 = -0.545 \times 10^{-18} \text{J} \] For a mole, it becomes:\[ E_{2,\text{mol}} = -0.545 \times 10^{-18} \times 6.022 \times 10^{23} = -3.28 \times 10^5\, \text{J/mol} \]

Step 5: Calculate the Energy Difference

The energy required to excite an electron from \(n=1\) to \(n=2\) is the difference between \(E_2\) and \(E_1\):\[ \Delta E = E_2 - E_1 \] \[ \Delta E = (-3.28 \times 10^5) - (-1.312 \times 10^6) = 9.84 \times 10^5\, \text{J/mol} \]

Step 6: Choose the Correct Option

Compare the calculated energy difference \(9.84 \times 10^5\, \text{J/mol}\) with the given options. The correct option is (d) \(9.84 \times 10^5\, \text{J/mol}\).

Key Concepts

Ionization Enthalpy

Ionization enthalpy is a crucial concept when discussing the ionization of a hydrogen atom. It refers to the amount of energy needed to completely remove the electron from an atom's ground state, bringing it to a point infinitely far away from the nucleus. For hydrogen atoms, this energy is specifically calculated as \(1.312 \times 10^6 \text{ J/mol}\). This large amount of energy indicates how tightly the electron is bound to the nucleus in its ground state.

• Ground State: The lowest energy state of an electron in an atom.
• Infinite Distance: Reaching a point where the electron is no longer influenced by the nucleus's force.

Understanding ionization enthalpy helps in analyzing energy transitions and is directly tied to the concepts of atomic orbital energies and levels.

Rydberg Constant

The Rydberg constant is fundamental to the analysis of atomic spectra for hydrogen-like atoms. It represents the ultimate limit of the series for atomic hydrogen in spectral lines. With a value of \(2.18 \times 10^{-18} \text{ J}\), it captures the energy difference between transitions of an electron from one energy level to another.

• Significance: Central to calculating bonding energies between electrons and nuclei.
• Usability: Applied in several formulas describing the energy levels of one-electron atoms or ions.

Each time an electron transitions between layers (energy levels), the Rydberg constant helps find the precise energy variation, which is measured as light in spectroscopic experiments.

Energy Levels

Energy levels in an atom represent the fixed energies that electrons can have. For the hydrogen atom, these can be calculated using the formula: \( E_n = - \frac{R_H}{n^2} \). Here, \(R_H\) is the Rydberg constant and \(n\) is the principal quantum number.

• Calculation: Dependent on the principal quantum number \(n\).
• States: Electrons move to higher energy levels by absorbing energy and vice versa.

Electrons can "jump" from one energy level to another, absorbing or releasing energy in the process. This is akin to climbing and descending a staircase, where each step represents a specific energy level or state. The energy needed to move between different states provides insightful information into the atom's structure and behavior.

Principal Quantum Number

The principal quantum number \(n\), is an essential quantum mechanical concept that indicates the main energy level occupied by an electron. Starting with \(n = 1\) for an electron closest to the nucleus, each increase in \(n\) signifies a higher energy level further out.

• Level Indication: The primary number showing which main energy shell is occupied.
• Energy Relation: Directly influences the size and energy of the electron's orbital.

For example, when an electron in a hydrogen atom moves from \(n=1\) to \(n=2\), it requires energy absorption. This relationship is used to determine how much energy is needed for electronic transitions, making the principal quantum number a critical part of atomic theory and quantum physics. Understanding this helps comprehend atomic spectra and the physical processes of electron excitation.