Question

The binding energy of the electron in the lowest orbit of the hydrogen atom is \(13.6 \mathrm{cV}\). The energies required in \(\mathrm{cV}\) to remove an electron from three lowest orbits of the hydrogen atom arc (1) \(13.6,6.8,8.4 \mathrm{eV}\) (2) \(13.6,10.2,3.4 \mathrm{eV}\) (3) \(13.6,27.2,40.8 \mathrm{eV}\) (4) \(13.6,3.4,1.5 \mathrm{eV}\)

Short answer

Option (4) is correct.

Step-by-step solution

Understand the problem

The task is to find the binding energy required to remove an electron from the three lowest orbits (n=1, n=2, and n=3) of the hydrogen atom. The given binding energy for the lowest orbit (n=1) is 13.6 eV.

Use the formula for binding energy

The binding energy of the electron in the nth orbit of hydrogen can be calculated using the formula \(E_n = \frac{13.6 \text{ eV}}{n^2}\).

Calculate binding energies for n=1, n=2, and n=3

Substitute n=1, n=2, and n=3 into the formula: \(E_1 = \frac{13.6 \text{ eV}}{1^2} = 13.6 \text{ eV}\) \(E_2 = \frac{13.6 \text{ eV}}{2^2} = \frac{13.6 \text{ eV}}{4} = 3.4 \text{ eV}\) \(E_3 = \frac{13.6 \text{ eV}}{3^2} = \frac{13.6 \text{ eV}}{9} \text{ eV} ≈ 1.51 \text{ eV}\)

Find the corresponding answer from the given choices

The calculated binding energies are: 13.6 eV for n=1, 3.4 eV for n=2, and 1.5 eV for n=3. Compare these with the given options.

Match with the correct option

Option (4) matches the calculated binding energies of 13.6 eV, 3.4 eV, and 1.5 eV.

Key concepts

electron orbitals

In an atom, electrons reside in specific regions called orbitals. These orbitals are areas around the nucleus where the probability of finding an electron is highest. Each orbital can hold a certain number of electrons and is defined by its energy level and shape as described by quantum numbers.
Electrons in the lowest energy levels are found closer to the nucleus, while those in higher energy levels are farther away. Orbitals are shaped differently, such as spherical (s-orbitals), dumbbell-shaped (p-orbitals), and more complex shapes (d- and f-orbitals). In the case of the hydrogen atom, the simplest atom, the electron in the lowest energy level or ground state occupies the 1s orbital.
Understanding electron orbitals is crucial as they help determine the chemical properties of elements, including how they bond with other atoms.

energy levels

Energy levels (or electron shells) are the fixed distances from the nucleus of an atom where electrons can reside. Each energy level corresponds to a specific amount of energy that an electron must have to be in that level.
In the context of the hydrogen atom, when we talk about n=1, n=2, n=3, etc., we refer to the principal quantum number that denotes these energy levels. The lowest energy level, where n=1, is closest to the nucleus and has the lowest energy. Higher energy levels (n=2, n=3, and so on) are further from the nucleus and correspondingly have higher energy.
Transitions between these energy levels involve the absorption or emission of energy. For example, moving an electron from a lower energy level (n=1) to a higher one (n=2) requires energy absorption, while the opposite transition releases energy.

quantum mechanics

Quantum mechanics is the branch of physics that explains the behavior of particles at the atomic and subatomic levels. It is essential for describing how electrons move and interact within an atom.
One of the fundamental principles of quantum mechanics is that energy is quantized. This means that electrons can only occupy specific energy levels and cannot exist in between these levels. The energies associated with different orbitals are thus discrete, not continuous.
Quantum mechanics also introduces the concept of wave-particle duality, where particles such as electrons exhibit both wave-like and particle-like properties. Additionally, it employs mathematical functions called wave functions to describe the probability of finding an electron in a particular region around the nucleus.

binding energy calculation

Binding energy is the energy required to remove an electron from an atom. For the hydrogen atom, this is calculated using the formula: \(E_n = \frac{13.6 \text{ eV}}{n^2}\).
The constant 13.6 eV represents the binding energy of the electron in the lowest orbital (n=1). By plugging in values for n, you can determine the binding energies for higher energy levels. For example:
\[ E_1 = \frac{13.6 \text{ eV}}{1^2} = 13.6 \text{ eV}\]
\[ E_2 = \frac{13.6 \text{ eV}}{2^2} = \frac{13.6 \text{ eV}}{4} = 3.4 \text{ eV}\]
\[ E_3 = \frac{13.6 \text{ eV}}{3^2} = \frac{13.6 \text{ eV}}{9} \text{ eV} ≈ 1.51 \text{ eV}\]
These calculations show the energy needed to remove an electron from the hydrogen atom's first, second, and third energy levels, helping you understand why Option (4) is the correct answer in the exercise.