Question

If the binding energy of electron in a hydrogen atom is \(13.6 \mathrm{eV}\), the energy required to remove the electron form the first state of \(\mathrm{Li}^{2+}\) is. (A) \(13.6 \mathrm{eV}\) (B) \(30.6 \mathrm{eV}\) (C) \(122.4 \mathrm{eV}\) (D) \(3.4 \mathrm{eV}\)

Short answer

The energy required to remove the electron from the first state of $\mathrm{Li}^{2+}$ can be found using the Rydberg formula: $E = -\frac{13.6 Z^2}{n^2}$. For $\mathrm{Li}^{2+}$, we have $Z = 3$ and $n = 1$. Substituting these values, we get $E_{\mathrm{Li}^{2+}} = -\frac{13.6 \cdot 9}{1} = -122.4\,\mathrm{eV}$. To remove the electron, we need the positive value of this energy, which is $E_{\mathrm{removal}} = 122.4\,\mathrm{eV}$. Therefore, the correct answer is (C) $122.4\,\mathrm{eV}$.

Step-by-step solution

The Rydberg Formula

The binding energy (in eV) of an electron in a hydrogen-like atom can be given by the Rydberg formula: \[E = -\frac{13.6 Z^2}{n^2}\] Where, E is the binding energy, Z is the atomic number, and n is the principal quantum number.

Substitute the values for Li²⁺

For Li²⁺, the atomic number (Z) is 3 and the electron is in the first state i.e., n=1. Now using the Rydberg formula, we can find the binding energy for Li²⁺: \[E_{Li^{2+}} = -\frac{13.6\cdot 3^2}{1^2}\]

Calculate the binding energy

Now, calculate the binding energy: \[E_{Li^{2+}} = -\frac{13.6 \cdot 9}{1}\] \[E_{Li^{2+}} = -122.4 \mathrm{eV}\]

Energy required to remove the electron

To remove the electron, we need to provide the exact amount of energy that is required to balance the binding energy but in the opposite direction (positive), that is: \[E_{removal} = 122.4 \mathrm{eV}\] So, the correct answer is (C) 122.4 eV.

Key concepts

Binding Energy

Binding energy is the energy needed to remove an electron from an atom or ion. It reflects the strength of attraction between the electron and the nucleus. In the context of a hydrogen atom, the binding energy of an electron in its ground state is known to be 13.6 eV.
This value indicates how much energy is required to free the electron from the influence of the nucleus.

For hydrogen-like atoms, which are atoms with only one electron, the binding energy can be calculated using the Rydberg formula. The negative sign in the equation indicates that energy must be added to overcome the force holding the electron bound to the nucleus. Here is an important takeaway about binding energy:

• Higher binding energy suggests a stronger attraction between the electron and nucleus.
• It is inversely related to the principal quantum number; the higher the principal quantum number, the lower the binding energy.

Hydrogen-like Atom

A hydrogen-like atom is a simple model of an atom that consists of a single electron orbiting a nucleus, just like in a hydrogen atom. These atoms are useful for understanding more complex interactions in quantum physics.

Hydrogen-like atoms are characterized by having only one electron, but they may have a different number of protons in the nucleus compared to hydrogen, resulting in different atomic numbers.
Because of their single electron, the behavior of hydrogen-like atoms can be studied using equations derived for the hydrogen atom, especially using the Rydberg formula for energy levels.

Here are some features to remember about hydrogen-like atoms:

• They include ions such as He⁺ or ext{Li}^{2+}, which have lost all but one electron.
• They are ideal systems to apply quantum mechanics principles.

Principal Quantum Number

The principal quantum number, denoted by \(n\), is a critical quantum number that specifies the energy level of an electron in an atom. It determines the electron's distance from the nucleus and is mostly associated with the size of the electron orbit.

The principal quantum number \(n\) can take positive integer values (1, 2, 3,...), with lower values indicating electrons closer to the nucleus and typically higher energies associated with higher \(n\) values.
This number plays a crucial role in defining the binding energy in a hydrogen-like atom using the Rydberg formula.

Key points about the principal quantum number include:

• It determines the energy level or shell of an electron.
• Higher \(n\) values imply electrons are in higher energy states and orbit further from the nucleus.

Atomic Number

The atomic number, symbolized by \(Z\), is the number of protons found in the nucleus of an atom. This number not only identifies the element but also determines the positive charge of the nucleus.

In the case of hydrogen-like atoms, the atomic number plays a crucial role in determining various properties, including the binding energy of the electron according to the Rydberg formula.
If the atomic number is higher, the positively charged nucleus exerts a stronger force on the single electron, increasing the binding energy.

A few important aspects of atomic number include:

• It determines the identity of the element.
• For hydrogen-like atoms, it affects the magnitude of the binding energy and influences atomic behavior.