Problem 86 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The potential energy of electron revolving in the ground state of \(\mathrm{H}\) atom is (a) \(-13.6 \mathrm{eV}\) (b) \(-6.8 \mathrm{eV}\) (c) \(-27.2 \mathrm{eV}\) (d) Zero

Step-by-Step Solution

Answer

-13.6 eV

Step 1: Understanding the concept of potential energy in the hydrogen atom

The potential energy of an electron in a hydrogen atom can be calculated using the formula for the energy of an electron in a particular orbit of a hydrogen atom, which is given by the Bohr model. The energy of an electron in the nth orbit is given by the formula: \(E_n = -\frac{13.6 eV}{n^2}\), where \(n\) is the principal quantum number of the orbit.

Step 2: Determining the ground state energy

For the ground state of the hydrogen atom, the electron is in the first orbit, so the principal quantum number \(n\) is 1. Plug in \(n = 1\) into the energy formula \(E_n = -\frac{13.6 eV}{n^2}\) to find the potential energy.

Step 3: Calculating the potential energy of the electron in the ground state

Now, substitute \(n = 1\) into the energy formula to get \(E_1 = -\frac{13.6 eV}{(1)^2} = -13.6 eV\). So, the potential energy of an electron in the ground state of a hydrogen atom is \(-13.6 eV\).

Key Concepts

Bohr Model

In the early 20th century, Niels Bohr introduced a revolutionary model of the atom that explained how electrons orbit the nucleus and why atoms emit and absorb light at specific wavelengths. This was dubbed the Bohr model. It posits that electrons revolve around the atomic nucleus in fixed orbits, or shells, with each orbit corresponding to a specific energy level.

According to the Bohr model, each electron orbit has a quantized energy state, meaning the electron can only exist in certain discrete energy levels and not in between. This contrasts with classical physics, which suggested that an electron could hold any value of energy. The model also proposed that when an electron jumps from one orbit to a lower energy orbit, it emits energy in the form of a photon with a frequency directly related to the energy difference between the two orbits.

Importance of Quantized Orbits

The concept of quantized orbits was groundbreaking because it provided the foundation for understanding atomic spectra. Each element has its unique signature in the spectrum due to the specific energy levels of its electrons. The Bohr model successfully explained the Rydberg formula for the spectral emission lines of the hydrogen atom.

Principal Quantum Number

The principal quantum number, represented by the symbol 'n', is an integer that determines the energy level of an electron within an atom. It is one of four quantum numbers used in quantum mechanics to describe the unique quantum state of an electron.

According to the Bohr model, the principal quantum number also defines the size of an electron's orbit, with higher values of 'n' corresponding to orbits further away from the nucleus. As 'n' increases, so does the energy of the electron: it becomes less bound to the nucleus and easier to remove from the atom.

Role in Energy Calculation

In calculating the energy of an electron within an atom, the principal quantum number is used within the equation for the energy level of a hydrogen atom. The energy associated with a particular orbit is negative, indicating that the electron is in a bound state with respect to the protons in the nucleus, and it increases (becomes less negative) as 'n' increases.

Ground State Energy

The ground state energy refers to the lowest possible energy level of an electron in an atom when it is at rest. In this state, the electron occupies the closest orbit to the nucleus, which corresponds to the principal quantum number 'n' equal to 1.

The ground state is significant because it represents the most stable state of an electron, where it has the minimum energy. Additionally, no energy can be extracted from an atom in its ground state without adding energy to it (e.g., through absorption of a photon).

Hydrogen's Ground State Energy

For a hydrogen atom, which is the simplest atom with only one electron and one proton, the ground state energy is often given as the reference point for the energy levels of electrons in more complex atoms. In the case of hydrogen, this energy is quantified as -13.6 electronvolts (eV), showing that the electron is in a state of potential energy that would require 13.6 eV of work to free it from the attractive influence of the proton.