Problem 1 - Chemistry in 30 days

Question

The energy of hydrogen atom in its ground state is \(-13.6 \mathrm{eV}\). The energy of the level corresponding to \(n=5\) is: (a) \(-0.54 \mathrm{eV}\) (b) \(-5.40 \mathrm{cV}\) (c) \(-0.85 \mathrm{eV}\) (d) \(-2.71 \mathrm{eV}\)

Step-by-Step Solution

Answer

The energy of the level for \(n=5\) is \(-0.54 \mathrm{eV}\) (option a).

Step 1: Understanding the Energy Levels Formula

The energy of an electron in a hydrogen atom at a principal quantum number \( n \) is given by the formula: \( E_n = \frac{-13.6}{n^2} \mathrm{eV} \). This formula allows us to determine the energy level for any state \( n \).

Step 2: Apply the Formula to Find \( n=5 \) Energy

Substitute \( n = 5 \) into the formula: \( E_5 = \frac{-13.6}{5^2} \mathrm{eV} \). Simplify the expression \( E_5 = \frac{-13.6}{25} \mathrm{eV} \).

Step 3: Calculate the Energy Value

Perform the division: \( E_5 = \frac{-13.6}{25} \mathrm{eV} = -0.544 \mathrm{eV} \). For rounding purposes, this is approximately \(-0.54 \mathrm{eV}\).

Step 4: Match the Answer with Options

Compare \(-0.54 \mathrm{eV}\) with the given options. The correct match is option (a) \(-0.54 \mathrm{eV}\).

Key Concepts

Principal Quantum Number

The principal quantum number, symbolized as \( n \), is a crucial part of quantum mechanics used to describe the energy levels of electrons in an atom. It primarily indicates the main energy level occupied by an electron. For hydrogen, \( n \) can be any positive integer: 1, 2, 3, and onwards. The value of \( n \) reflects:

• The electron's energy level or shell.
• The distance of the electron from the nucleus. A larger \( n \) means the electron is further away.

In the specific problem from the exercise, \( n = 5 \) signifies that the electron is in the fifth energy level. This concept is critical in determining the potential energy of the electron within the atom.

Energy of Electron

The energy of an electron in a hydrogen atom can be calculated using the formula \( E_n = \frac{-13.6}{n^2} \mathrm{eV} \). This formula derives from the Bohr model of the hydrogen atom, which considers the quantized nature of electron orbits. Here, the constant \(-13.6 \mathrm{eV}\) is the energy in the ground state (\( n = 1 \)).
As \( n \) increases, the energy becomes less negative, indicating:

• The electron is less tightly bound to the nucleus.
• The electron is at a higher energy level.

For the given exercise, substituting \( n = 5 \) into the formula results in an energy value of approximately \(-0.54 \mathrm{eV}\), demonstrating how such calculations predict the specific energy levels of electrons.

Quantum Mechanics

Quantum mechanics is the fundamental theory in physics that describes nature at the smallest scales, such as atoms and subatomic particles. It is the backbone for explaining the behavior of electrons in atoms, including their energy levels and movement. This theory revolutionized our understanding by introducing the concept that energy is quantized, meaning electrons can only exist in specific energy levels.

• Absorption of energy can cause electrons to jump to a higher level.
• Releasing energy can bring them to a lower energy state.

By applying the principles of quantum mechanics, we gain insight into how an electron in a hydrogen atom behaves when \( n \) shifts from one energy level to another, showcasing the quantized nature of electron positions and energies.

Electron Configuration

Electron configuration is the arrangement of electrons in the orbitals of an atom. For hydrogen, the simplest atom with one electron, its configuration heavily depends on the principal quantum number \( n \). This configuration impacts:

• The chemical properties of an element.
• The way atoms interact in chemical reactions.

For higher elements, electron configuration becomes more complex with additional principles like the Pauli exclusion principle and Hund’s rule. However, in hydrogen, as \( n \) increases, it leads us to a straightforward progression of an electron to higher energy levels, as represented by \( n = 5 \) in the exercise. Understanding these configurations explains the stability and reactivity of elements.