Question

The Bohr's energy of a stationary state of hydrogen atom is given as \(E_{n}=\frac{-2 \pi^{2} m e^{4}}{n^{2} h^{2}}\). Putting the values of \(m\) and \(e\) for \(n^{\text {th }}\) energy level which is not the correct value? (a) \(E_{n}=\frac{-21.8 \times 10^{-19}}{n^{2}} \mathrm{~J}\) atom \(^{-1}\). (b) \(E_{n}=\frac{-13.6}{n^{2}} \mathrm{eV}\) atom \(^{-1}\) (c) \(E_{n}=\frac{-1312}{n^{2}} \mathrm{~kJ} \mathrm{~mol}^{-1}\) (d) \(E_{n}=\frac{-12.8 \times 10^{-19}}{n^{2}}\) erg \(\mathrm{atom}^{-1}\)

Short answer

(d) \(-12.8 \times 10^{-19}\) erg atom\(^{-1}\) is not the correct value.

Step-by-step solution

- Understanding Bohr's energy formula

The formula given, \(E_{n}=\frac{-2 \pi^{2} me^{4}}{n^{2}h^{2}}\), represents the energy of an electron in the nth energy level of a hydrogen atom in the Bohr model. The formula includes Planck's constant (h), the mass of an electron (m), the charge of an electron (e), and the principle quantum number (n).

- Calculation for the correct value

To find the correct value, substitute the constants into the formula. We need the values of the mass of an electron (\(m\)) which is approximately \(9.11 \times 10^{-31} \) kg, the charge of an electron (\(e\)) \(1.6 \times 10^{-19} \) C, Planck's constant (\(h\)) \(6.626 \times 10^{-34} \) Js, and the value of \(\pi\).

- Conversion of units

To check the validity of the given options, we may need to convert the resulting energy from Joules to electronvolts (eV), kilojoules per mole (kJ/mol), and ergs to match the given units.

- Evaluate the options

Calculate the energy using Bohr's formula and convert the units to compare with the given options. If an option's numerical coefficient doesn't match the calculated value after simplifying and converting units, it is incorrect. The conversion factors are: \(1\) eV \(= 1.602 \times 10^{-19}\) J, \(1\) erg \(= 10^{-7}\) J, and \(1\) kJ/mol is \(\frac{10^{3}}{N_{A}}\) J per atom, where \(N_{A}\), Avogadro's number, is \(6.022 \times 10^{23}\) mol\(^{-1}\).

- Identify the incorrect value

Using the constants provided, the energy for one atom is approximately \(-13.6\) eV when converted from Joules. This matches option (b). When using the conversions, if one of the provided options doesn't align with this value, it will be identified as incorrect.

Key concepts

Hydrogen Atom Energy Levels

Understanding the energy levels of a hydrogen atom is essential for comprehending the behavior of electrons in atoms. The energy levels, or orbits, in a hydrogen atom are quantized, meaning electrons can only exist at certain discrete energy levels.

The simplest atom, hydrogen, has one electron surrounding the nucleus, and according to Niels Bohr's model, this electron can occupy certain permitted orbits. Each orbit corresponds to a specific energy level, denoted by the principal quantum number, 'n'. As 'n' increases, the electron's energy also increases, and it resides further from the nucleus.

Calculating Energy Levels

Using Bohr's formula, En = -13.6 eV/n^2, we can calculate the energy of an electron at any given level 'n'. For example, the first energy level (n=1) has an energy of -13.6 eV, the second (n=2) has an energy of -3.4 eV, indicating that electrons require energy to move to higher levels. When they drop to lower energy levels, they emit energy in the form of electromagnetic radiation.

This quantization explains the distinct lines observed in the atomic emission spectra and is fundamental to understanding atomic and quantum physics.

Planck's Constant

Planck's constant is a significant number in quantum mechanics. Named after physicist Max Planck, it represents the quantization of energy in mathematical form and is denoted by the symbol 'h'.

The Role in Quantum Mechanics

Planck's constant links the energy (E) of a photon and the frequency (f) of its electromagnetic wave through the relation E = hf. This constant sets the scale for the quantum world and is critical in the formula for calculating the energy levels of the hydrogen atom.

The value of Planck's constant is approximately 6.626 x 10^-34 Js (Joule seconds), underscoring the very subtle action scales in quantum phenomena. It is this 'magic number' that made the connection between the macroscopic and quantum worlds and paved the way for the development of Quantum Mechanics.

Electron Charge

The charge of an electron is one of the fundamental properties of particles in the universe. Representing a basic unit of electric charge, the electron charge is a constant and is denoted by 'e'.

Importance of Electron Charge in Formulas

In Bohr's energy formula, the electron charge appears as e^4, highlighting its pivotal role in determining the energy of hydrogen atom orbits. The magnitude of a single electron charge is approximately 1.6 x 10^-19 Coulombs (C), which is negative, indicating that an electron has a negative charge.

This constant is fundamental in many equations and laws across physics, including the classical Coulomb's law, which describes the electrostatic force between two charged particles, and in defining the ampere, the standard unit of current in the International System of Units (SI).

Quantum Numbers

Quantum numbers are sets of numerical values that provide detailed information about the energy and position of an electron within an atom. They are crucial for the understanding of the arrangement of electrons in atoms, known as the electron configuration.

Types of Quantum Numbers

There are four types of quantum numbers:

• The principal quantum number, 'n', describes the energy level of an electron in an atom.
• The azimuthal quantum number, 'l', defines the shape of the electron's orbital.
• The magnetic quantum number, 'm_l', specifies the orientation of the orbital in space.
• The spin quantum number, 'm_s', describes the electron's spin direction in the magnetic field.

In the context of the Bohr model, the principal quantum number 'n' determines the energy of an electron in a specific orbit. As 'n' increases, so does the electron's energy and average distance from the nucleus. Each value of 'n' corresponds to a 'shell' or energy level in the atom where an electron can be found.