Problem 117 - A Complete Resource Book in Chemistry for JEE Main

Question

If ionization potential for hydrogen atom is \(13.6 \mathrm{eV}\), then ionization potential for He \(^{+}\)will be (a) \(6.5 \mathrm{eV}\) (b) \(54.4 \mathrm{eV}\) (c) \(13.6 \mathrm{eV}\) (d) \(24.5 \mathrm{eV}\)

Step-by-Step Solution

Answer

The ionization potential for \(\text{He}^+\) is \(54.4 \text{ eV}\), matching option (b).

Step 1: Understanding Ionization Potential

Ionization potential is the energy required to remove an electron from an atom or ion. For hydrogen, this is given as \(13.6 \text{ eV}\). We need to calculate it for \(\text{He}^+\) (Helium ion with a single electron).

Step 2: Using the Formula for Ionization Potential

The ionization potential can be calculated using the formula: \(E_n = -13.6 \cdot \frac{Z^2}{n^2} \text{ eV}\), where \(Z\) is the atomic number, and \(n\) is the principal quantum number. For ground state \(n=1\).

Step 3: Substituting Values for Helium Ion

For \(\text{He}^+\), \(Z = 2\). Substitute into the formula: \[E_1 = -13.6 \cdot \frac{(2)^2}{(1)^2} = -13.6 \cdot 4 \text{ eV}\].

Step 4: Calculating Ionization Potential

Calculate the energy: \[E_1 = -13.6 \cdot 4 = -54.4 \text{ eV}\]. The negative sign indicates energy is required to remove the electron.

Step 5: Comparing with Options

Given options are: (a) \(6.5 \text{ eV}\), (b) \(54.4 \text{ eV}\), (c) \(13.6 \text{ eV}\), (d) \(24.5 \text{ eV}\). The calculated ionization potential is \(54.4 \text{ eV}\), which matches option b.

Key Concepts

Hydrogen Atom

The hydrogen atom is the simplest atom and serves as a fundamental building block for understanding atomic structure. It consists of one proton and one electron. This simplicity makes it an ideal model for studying quantum mechanics and atomic physics.
In a hydrogen atom, the electron orbits around the proton. This is primarily due to the electrostatic force of attraction between the positively charged proton and the negatively charged electron. The energy levels of the electron are quantized, meaning the electron can only exist in specific energy states.

• The ground state, or the lowest energy level, is the most stable state for the electron.
• When energy is added, the electron can move to a higher energy level or become ionized, which means it is no longer bound to the atom.

Understanding the hydrogen atom is vital as it helps in calculating ionization potentials and explaining the behavior of more complex atoms.

Helium Ion

A helium ion, represented as He\(^+\), is formed when one electron is removed from a helium atom, leaving behind a singly charged ion with one electron around the nucleus. Unlike neutral helium, which has two electrons, He\(^+\) behaves similarly to a hydrogen atom because it has only one electron left.
Helium ions are important in physics and chemistry because they help us understand ionization energy and atomic structure more deeply. Unlike hydrogen, the helium ion has a higher atomic number which affects its ionization potential.

• The presence of two protons increases the electrostatic force pulling the remaining electron towards the nucleus.
• This creates a stronger attraction compared to the single proton found in hydrogen.

As a result, it takes more energy to remove the electron from a helium ion than from a hydrogen atom.

Atomic Number

The atomic number, often represented by the symbol \(Z\), is a fundamental property of an element. It denotes the number of protons found in the nucleus of an atom. This number is crucial because it determines the identity and properties of the element.
In the context of ionization potential, the atomic number plays a significant role since it influences the strength of the electrostatic force between electrons and the nucleus.

• For hydrogen, the atomic number is 1, meaning there is one proton in its nucleus.
• For helium, the atomic number is 2, indicating two protons.

This difference in the atomic number affects the calculation of ionization energy. As the atomic number increases, the nucleus exerts a stronger pull on the electrons, making them harder to remove.

Principal Quantum Number

The principal quantum number, denoted by \(n\), is one of the quantum numbers used to describe the properties of electrons in atoms. It indicates the main energy level occupied by an electron.
Larger values of \(n\) correspond to higher energy levels, meaning the electron is further from the nucleus.

• For hydrogen and helium ions, when calculating ionization potential, the electron is usually considered in the ground state, where \(n = 1\).
• The principal quantum number affects the size of the electron cloud and the energy required to remove the electron.

Understanding the principal quantum number is essential for solving ionization potential problems, as it directly influences the energy calculations involved in removing an electron from an atom or ion.