Problem 72 - A Complete Resource Book in Chemistry for JEE Main

Question

In hydrogen atom, energy of first excited state is \(-3.4\) \(\mathrm{eV}\). The kinetic energy of the same orbit of hydrogen atom would be (a) \(+3.4 \mathrm{eV}\) (b) \(+6.8 \mathrm{eV}\) (c) \(-13.6 \mathrm{eV}\) (d) \(+13.6 \mathrm{eV}\)

Step-by-Step Solution

Answer

The kinetic energy is \(+3.4 \mathrm{eV}\).

Step 1: Understanding the Problem

We are given the energy of the first excited state of a hydrogen atom, which is \(-3.4\, \mathrm{eV}\). We need to determine the kinetic energy (KE) of the electron in the same orbit.

Step 2: Relating Energy States

In a hydrogen atom, the total energy (E) of an electron in an orbit is related to its potential energy (PE) and kinetic energy (KE) by the formula: \(E = KE + PE\). For the electron in a hydrogen atom, potential energy is twice the total energy but negative, i.e., \(PE = 2E\).

Step 3: Determining Kinetic Energy

Since total energy is given as \(-3.4\, \mathrm{eV}\), we use \(KE = -E\) because double the total energy accounts for potential energy and thus leads to \(KE = -(-3.4 \mathrm{eV}) = 3.4 \mathrm{eV}\). Alternatively, since \(KE = -\frac{E}{2}\), substituting in gives \(KE = 3.4 \mathrm{eV}\).

Key Concepts

Energy Levels

In atoms, energy levels are the fixed energies that electrons can have. These levels occur because an electron in an atom is bound by electrostatic forces of attraction to the nucleus.

• The energy levels are quantized, meaning electrons can only exist at certain discrete energy values.
• Each energy level corresponds to an electron's orbit or state around the nucleus.
• These levels are specific to each type of atom.

For a hydrogen atom, which is the simplest atom consisting of one proton and one electron, these energy levels are determined by the formula: \[ E_n = -\frac{13.6}{n^2} \text{ eV} \]Where:- \(E_n\) is the energy of the level \(n\).- \(n\) is the principal quantum number, which can be 1, 2, 3, etc.The principal quantum number \(n = 1\) represents the ground state, and for \(n = 2\) we refer to the first excited state. The value of 13.6 eV is specific to hydrogen due to its unique properties.

Kinetic Energy

The kinetic energy (KE) of an electron in an atom is the energy due to its motion while orbiting the nucleus.

• Kinetic energy represents the motion of an electron.
• In the case of a hydrogen atom, it is influenced by the negative total energy.
• For electrons, being negatively charged, the kinetic energy is a positive value reflecting active motion.

For a given energy level in a hydrogen atom, the relationship between kinetic energy and total energy is given by:\[ KE = -E \]Thus, if the total energy at the first excited state is \(-3.4\, \text{eV}\), the kinetic energy will be positive and equal to \(3.4\, \text{eV}\). This relationship can be analyzed as:\[ KE = -(-3.4\, \text{eV}) = 3.4\, \text{eV} \]

Potential Energy

In physics, potential energy (PE) in a hydrogen atom is primarily due to the electrostatic interactions between the negatively charged electron and positively charged nucleus.

• Potential energy describes the energy stored due to these interactions.
• It's inherently negative, depicting the binding of the electron to the nucleus.
• Reflects the potential to do work as the electron moves.

The potential energy of an electron in a hydrogen atom is related to the total energy by:\[ PE = 2E \]Where \(E\) is the total energy. Given the total energy of the first excited state \(-3.4\, \text{eV}\), the potential energy would be:\[ PE = -6.8\, \text{eV} \]This negative potential energy further indicates the electron's strong attraction to the nucleus, a defining characteristic of electron behavior in atoms.

First Excited State

The first excited state of an atom refers to the first energy level above the ground state. In the context of a hydrogen atom:

• The ground state is when the electron is at its lowest energy level, \(n = 1\).
• The first excited state is reached when the electron jumps to the next energy level, \(n = 2\).

For a hydrogen atom, transitioning from the ground state (\(-13.6\, \text{eV}\)) to the first excited state (\(-3.4\, \text{eV}\)) requires energy absorption. This change is essential in understanding atomic transitions:- When an electron moves to a higher energy level, it absorbs energy (endothermic process).- Conversely, moving to a lower energy level releases energy (exothermic process).The first excited state, being at \(-3.4\, \text{eV}\), indicates that the electron has absorbed a certain amount of energy and now possesses a higher potential to return to the ground state, emitting energy in the process.