Problem 47 - Master Resource Book in JEE Main Physics

Question

The binding energy of an electron in the ground state of He is equal to \(24.6 \mathrm{eV}\). The energy required to remove both the electrons is (a) \(49.2 \mathrm{eV}\) (b) \(24.6 \mathrm{eV}\) (c) \(38.2 \mathrm{eV}\) (d) \(79.0 \mathrm{eV}\)

Step-by-Step Solution

Answer

The energy required to remove both electrons is 49.2 eV (option a).

Step 1: Understanding the given data

The problem states that the binding energy of an electron in the ground state of the Helium atom (He) is 24.6 eV. This value represents the energy required to remove one electron from the Helium atom.

Step 2: Exploring Helium's electron setup

A Helium atom in its neutral state has two electrons. Removing one electron requires an energy expenditure that matches its binding energy. Since we have the energy for one electron, we need to think about what it takes to remove both electrons.

Step 3: Calculating the total energy required

Since the atom has two electrons, the total energy required to remove both electrons should be twice the binding energy of one electron. Thus, calculate the total as follows:\[ \text{Total energy} = 2 \times 24.6 \text{ eV} = 49.2 \text{ eV} \]

Step 4: Final verification

Compare the calculated energy with the given choices. Check that the energy calculated, 49.2 eV, matches option (a) of the given options. Thus, option (a) 49.2 eV is correct.

Key Concepts

Helium Atom

The Helium atom is one of the simplest and most abundant elements in the universe. It is the second lightest element and is represented by the symbol \(^4\text{He}\) in the periodic table. Helium has two protons in its nucleus, which makes it electrically neutral, as its two electrons balance the positive charge. These electrons orbit around the nucleus in the atom's outer shell.
Helium is a noble gas, which means it's extremely stable and unreactive under normal conditions. This is due to its full outer electron shell, giving it no tendency to gain or lose electrons. Helium is colorless, odorless, and tasteless, making it relatively inert. Because of its low atomic mass, it is lighter than air, which is why helium is used in balloons to make them float.
In quantum physics, the study of Helium is crucial due to its simplicity, and it often provides fundamental insights into understanding more complex atoms. Each electron in a Helium atom influences how the atom behaves, particularly in its ground state and during electron removal processes.

Electron Removal

Electron removal, often called ionization, occurs when an electron is provided enough energy to overcome its binding energy and escape from the atom. Binding energy is the measure of how strongly an electron is held within an atom; thus, higher values mean the electron is held more tightly.
In the case of Helium, we are dealing with two electrons. Each of these electrons has a binding energy of 24.6 eV in its ground state. When attempting to remove one electron, energy equivalent to its binding energy must be supplied. For Helium, that's 24.6 eV for one electron.
To remove both electrons from a Helium atom, you'd have to provide twice the energy, assuming the second electron requires the same amount of energy as the first. This linear relationship helps in calculating the total energy needed. The concept plays an essential role in understanding the behavior of electrons in atoms, especially in atomic and nuclear physics.

Ground State

The ground state of an atom is its lowest energy state, meaning electrons are as close to the nucleus as possible. For a Helium atom, which has two electrons, the ground state is when both electrons are located in the closest orbital to the nucleus, the 1s orbital.
In this state, the Helium atom is in its most stable form. This is because the electrons are at their lowest potential energy configuration. This stability is what makes the ground state significant in various scientific fields.
When electrons are in the ground state, they have the minimum possible energy required by their binding force in the atom. That is why, in Helium, the removal of a ground state electron requires exactly its binding energy, 24.6 eV. The concept of ground state is fundamental in quantum mechanics and is critical in explaining the behavior of not just Helium, but other elements as well.