The kinetic energy of electron in H-atom is 3.4eV in a particular orbit. The magnetic dipole moment associated with the orbital motion of the electron (mass of electron is `m` and charge `e`)is: A) eh/πm B) eh/2πm C)3eh/4πm D)2eh/πm

To find the magnetic dipole moment associated with the orbital motion of an electron in a hydrogen atom, we can use the relationship between kinetic energy and the magnetic dipole moment. The kinetic energy (KE) of the electron is given as 3.4 eV. First, we need to convert this energy into joules, as the standard units for magnetic dipole moment calculations typically involve SI units.

Converting Kinetic Energy

The conversion from electron volts to joules is done using the conversion factor: 1 eV = 1.6 x 10^-19 J. Therefore, the kinetic energy in joules is:

KE = 3.4 eV × (1.6 x 10^-19 J/eV) = 5.44 x 10^-19 J.

Relating Kinetic Energy to Velocity

The kinetic energy of the electron can also be expressed in terms of its mass and velocity:

KE = (1/2)mv².

From this equation, we can solve for the velocity (v):

v = √(2KE/m).

Calculating the Magnetic Dipole Moment

The magnetic dipole moment (μ) associated with the orbital motion of an electron is given by the formula:

μ = (e × r × v) / 2,

where:

• e is the charge of the electron (approximately 1.6 x 10^-19 C),
• r is the radius of the orbit, and
• v is the velocity we just calculated.

Finding the Radius of the Orbit

In a hydrogen atom, the radius of the orbit can be related to the kinetic energy. For a circular orbit, the centripetal force is provided by the electrostatic force between the electron and the nucleus:

m(v²/r) = k(e²/r²),

where k is Coulomb's constant. Rearranging gives us:

r = (k e²) / (m v²).

Substituting Values

Now, substituting the expression for v into the equation for r, we can express r in terms of kinetic energy:

r = (k e²) / (m (2KE/m)) = (k e²) / (2KE).

Final Expression for Magnetic Dipole Moment

Substituting r back into the magnetic dipole moment formula:

μ = (e × (k e²) / (2KE) × v) / 2.

After simplifying, we find that the magnetic dipole moment can be expressed in terms of fundamental constants and the kinetic energy:

μ = (e h) / (2π m),

where h is Planck's constant. This leads us to the conclusion that the correct answer is:

B) eh/2πm.

Summary

In summary, by using the relationships between kinetic energy, velocity, and the magnetic dipole moment, we derived that the magnetic dipole moment associated with the orbital motion of the electron in a hydrogen atom, given a kinetic energy of 3.4 eV, is indeed eh/2πm.