Problem 90 - A Complete Resource Book in Chemistry for JEE Main

Question

The ionization energy of hydrogen atom is \(13.6 \mathrm{eV}\). What will be the ionization energy of \(\mathrm{He}^{+}\)? (a) \(13.6 \mathrm{eV}\) (b) \(54.4 \mathrm{eV}\) (c) \(122.4 \mathrm{eV}\) (d) zero

Step-by-Step Solution

Answer

The ionization energy of \( \text{He}^+ \) is 54.4 eV.

Step 1: Understanding Ionization Energy

Ionization energy is the amount of energy required to remove an electron from an atom or ion. For hydrogen, this energy is provided as 13.6 eV.

Step 2: Comparing Hydrogen and Helium Ionization

The ionization energy of an atom depends on the charge of the nucleus. Hydrogen has one proton in the nucleus, while \\( ext{He}^+ \) has two protons, making its nucleus twice as positive as hydrogen's. This means the electron in \\( ext{He}^+ \) is more tightly bound.

Step 3: Using Ionization Energy Formula

The formula for ionization energy is \\( E = 13.6 imes Z^2 \) for hydrogen-like species, where \\( Z \) is the atomic number. Thus, for \\( ext{He}^+ \), with \\( Z = 2 \), the energy becomes \\( 13.6 imes 2^2 = 54.4 \ \text{eV} \).

Step 4: Understanding the Calculation

By using the formula, we see that because helium has a double charge compared to hydrogen, its ionization energy is four times greater. This is because \\( 2^2 \) equals 4, hence, \\( 13.6 imes 4 = 54.4 \ \text{eV} \).

Step 5: Finalizing the Solution

Conclude that the ionization energy of \\( ext{He}^+ \) is indeed \\( 54.4 \ \text{eV} \), as the nucleus holds the electron more tightly due to the increased positive charge.

Key Concepts

Hydrogen Atom

The hydrogen atom is the simplest and most abundant atom in the universe. It consists of a single proton in its nucleus and one electron.

• The lone electron orbits the proton, attracted by the electrical charge of the proton.
• Its simplicity makes hydrogen very important in quantum mechanics and atomic theory studies.

Hydrogen's ionization energy is 13.6 eV, which is the energy needed to remove its single electron. This property makes hydrogen an excellent reference point in understanding more complex atoms.

Helium Ion

A helium ion, specifically \( ext{He}^+ \), starts as a neutral helium atom with two protons and two electrons. When one electron is removed, it becomes positively charged.

• This ionization leaves one electron orbiting the nucleus, much like a hydrogen atom but with greater complexity.
• The two protons exert a greater attractive force on the remaining electron.

Because of this increased attraction, the ionization energy needed to remove the second electron from \( ext{He}^+ \) is significantly higher than hydrogen's.

Nuclear Charge

The nuclear charge refers to the total charge of the protons in an atom's nucleus. It is a key factor in determining the ionization energy of an atom.

• This charge is directly proportional to the number of protons in the nucleus.
• For hydrogen, this charge is +1 due to its single proton, whereas for helium, it's +2.

The increased positive charge in helium means its nucleus pulls electrons more tightly, increasing the ionization energy. By understanding nuclear charge, we can predict how strongly an atom holds its electrons.

Electron Removal

Electron removal, or ionization, involves supplying enough energy to overcome the attractive force between an electron and the nucleus it orbits.

• In hydrogen, the energy required is relatively low at 13.6 eV due to its single proton.
• For helium ions, stronger nuclear attraction means more energy is needed - specifically, 54.4 eV.

When removing an electron from an atom, it turns from a neutral state to a positively charged ion, since it loses a negatively charged electron while the positive protons remain.

Atomic Number

An element's atomic number is the count of protons in its nucleus, serving as its unique identifier in the periodic table.

• For hydrogen, the atomic number is 1, indicating a single proton.
• Helium has an atomic number of 2, with two protons.

The atomic number also determines the overall nuclear charge, influencing how tightly an atom binds its electrons. Thus, the difference in atomic numbers between hydrogen and helium explains the variance in their ionization energies, since a higher atomic number means a stronger attraction to electrons.