Question

In a certain Bohr orbit the total energy is \(-4.9 \mathrm{eV}\), for this orbit, the kinetic energy and potential energy are respectively (1) \(9.8 \mathrm{eV},-4.9 \mathrm{eV}\) (2) \(4.9 \mathrm{eV},-9.8 \mathrm{eV}\) (3) \(4.9 \mathrm{eV}_{3}-4.9 \mathrm{eV}\) (4) \(9.8 \mathrm{eV},-9.8 \mathrm{eV}\)

Short answer

Option (2): 4.9 eV, -9.8 eV

Step-by-step solution

Recall the Relationship of Total Energy

In a Bohr orbit, the total energy (E) is related to the kinetic energy (K) and potential energy (U) by the equation: \[ E = K + U \]

Property of Kinetic Energy in Bohr Orbit

The kinetic energy (K) in a Bohr orbit is equal to the negative of the total energy: \[ K = -E \]Given that the total energy (E) is -4.9 eV, we have \[ K = -(-4.9) = 4.9 \text{ eV} \]

Property of Potential Energy in Bohr Orbit

The potential energy (U) in a Bohr orbit is twice the total energy: \[ U = 2E \]Given that the total energy (E) is -4.9 eV, we have \[ U = 2(-4.9) = -9.8 \text{ eV} \]

State the Results for Kinetic and Potential Energy

So the kinetic energy is 4.9 eV, and the potential energy is -9.8 eV. Option (2) correctly matches these values.

Key concepts

Total Energy in Bohr Orbit

In atomic physics, the Bohr model plays a crucial role in understanding the energy levels of electrons in an atom. The total energy (E) of an electron in a Bohr orbit is the sum of its kinetic energy (K) and potential energy (U). This relationship is given by the equation:











E = K + U

For example, if the total energy is given to be -4.9 eV, this means that the energy level of the electron in that particular Bohr orbit is -4.9 eV. Total energy combines both the kinetic energy and potential energy of the electron. An important aspect to note is that the total energy in a Bohr orbit is always negative, indicating that the electron is bound to the nucleus.

Kinetic Energy in Bohr Orbit

The kinetic energy (K) of an electron in a Bohr orbit comes from its motion around the nucleus. A unique property of the Bohr model is that the kinetic energy is equal to the negative of the total energy. This can be expressed with the formula:





E

K = -E

This means if the total energy (E) is -4.9 eV, the kinetic energy will be:




K =-(-4.9) = 4.9 eV

In this case, the electron has a kinetic energy of 4.9 eV, indicating how much energy it has due to its motion around the nucleus. This relationship underscores how tightly bound the electron is within the atom.

Potential Energy in Bohr Orbit

The potential energy (U) of an electron in a Bohr orbit is related to the electrostatic force between the negatively charged electron and the positively charged nucleus. The Bohr model reveals that the potential energy is twice the total energy:

















U = 2E

When the total energy (E) is -4.9 eV, the potential energy (U) can be calculated as follows:


U = 2(-4.9) = -9.8 eV

Thus, the electron in this Bohr orbit has a potential energy of -9.8 eV. The negative sign signifies that this energy is needed to overcome the electrostatic attraction and remove the electron from the atom.