Question

The total energy of the electron in the first excited state of hydrogen is \(-3.4 \mathrm{eV}\). what is the kinetic energy of the electron in this state? (A) \(6.8 \mathrm{eV}\) (B) \(3.4 \mathrm{eV}\) (C) \(-3.4 \mathrm{eV}\) \((\mathrm{D})-6.8 \mathrm{eV}\)

Short answer

The kinetic energy of the electron in the first excited state of hydrogen is \(3.4 \mathrm{eV}\). Therefore, the correct answer is (B) \(3.4 \mathrm{eV}\).

Step-by-step solution

Understand the relationship between kinetic and potential energy in a hydrogen atom

In a hydrogen atom, the potential energy (PE) between the electron and proton is twice the negative of the kinetic energy (KE) of the electron. This can be mathematically expressed as: \[PE = -2 \times KE\]

Find the potential energy using total energy

The total energy of the electron (TE) is given as -3.4 eV. The total energy is the sum of potential energy and kinetic energy. So, \[TE = PE + KE\] Replacing the value of the potential energy from step 1, we get: \[-3.4\ eV = (-2 \times KE) + KE\]

Calculate the kinetic energy

To find the value of KE, first simplify the equation from step 2, \[-3.4\ eV = -KE\] Now, multiply both sides of the equation by -1, \[KE = 3.4\ eV\] The kinetic energy of the electron in the first excited state of hydrogen is 3.4 eV. Therefore, the correct answer is (B) \(3.4 \mathrm{eV}\).

Key concepts

Hydrogen Atom

The hydrogen atom is the simplest and most fundamental atom. It consists of only one proton in its nucleus and one electron orbiting around it. This simplicity makes it an ideal subject for studying basic quantum mechanics and atomic structure.
In this atom, the proton exerts a force of attraction on the electron, keeping it in orbit. This creates potential energy due to the electrostatic attraction between positive and negative charges. Simultaneously, the electron moves, giving it kinetic energy.

• There is only one electron and one proton.
• Studying it helps understand bigger atomic structures.
• Potential and kinetic energies play vital roles.

The hydrogen atom serves as a crucial model for understanding energy levels in all atoms.

Total Energy

Total energy in the context of the hydrogen atom refers to the sum of its potential energy (PE) and kinetic energy (KE). This energy is quantized, meaning it can only take certain discrete values. In the simple hydrogen atom, each orbit of the electron corresponds to a specific energy level with a quantized total energy.
In quantum mechanics, the total energy of an electron in an atom is negative. This indicates a bound state, meaning the electron is held by the attractive forces of the nucleus and isn't free to drift away. The formula connecting these energies is:
\[\text{Total Energy} = \text{PE} + \text{KE}\]

• Sum of kinetic and potential energies.
• Negative due to binding of the electron.
• Determines the stability of the atom.

Understanding total energy levels is essential for grasping atomic stability.

Potential Energy

Potential energy in a hydrogen atom is related to the position and the interaction of the electron with the proton. It is crucial to note that this energy is negative, reflecting the attractive nature of the force between the electron and the proton.
The potential energy of the electron becomes more negative as the electron gets closer to the proton. This is governed by the equation:
\[\text{PE} = -2 \times \text{KE}\]This equation highlights that the forces attracting the electron are twice as strong as the electron's kinetic energy. This concept ensures that the electron remains in orbit around the nucleus.

• Negative due to electron-proton attraction.
• Related to electron's position.
• Keeps electron bound to nucleus.

Understanding potential energy helps explain why electrons remain bound in atoms.

First Excited State

In a hydrogen atom, the first excited state refers to the energy level above the ground state where the electron can reside. In terms of total energy, this is a less negative value compared to the ground state.
When an electron gains energy, it moves to this higher energy level, which changes its potential and kinetic energies. For the hydrogen atom, the first excited state energy level is given as \(-3.4 \, \text{eV}\). This value demonstrates that while the electron has more energy, it is still bound to the atom.

• Higher than the ground state.
• Energy gains lead to this level.
• Indicates electron excitement.

Understanding the first excited state is key to grasping atomic reactions and energy transitions.