The kinectic energy of the electron present in the ground state of Li2+ ion is represented by

1) (3e^2)÷(8piε0r)

2) (3e^2)÷(8piε0r)

Could you please explain how you found it?

Thanks

To determine the kinetic energy of the electron in the ground state of the Li²⁺ ion, we can use principles from quantum mechanics and electrostatics. The Li²⁺ ion is a hydrogen-like atom, meaning it has only one electron and a nucleus with a charge of +3e (since lithium has three protons). This allows us to apply similar formulas that we use for hydrogen, but with some adjustments for the increased nuclear charge.

Understanding the System

In a hydrogen-like atom, the kinetic energy (T) of the electron can be derived from its potential energy (U) due to the electrostatic attraction between the positively charged nucleus and the negatively charged electron. The potential energy for a two-charge system is given by:

• U = - (k * Q * q) / r

Here, k is Coulomb's constant, Q is the charge of the nucleus, q is the charge of the electron, and r is the distance between them. For Li²⁺, we have:

• Q = +3e
• q = -e

Calculating Potential Energy

Substituting these values into the potential energy formula gives us:

• U = - (k * (3e) * (-e)) / r = (3ke^2) / r

Now, we know from quantum mechanics that for a hydrogen-like atom, the kinetic energy is related to the potential energy. Specifically, the kinetic energy is half the magnitude of the potential energy:

• T = - (1/2) U

Finding Kinetic Energy

Substituting our expression for U into this equation, we get:

• T = - (1/2) * (3ke^2) / r

Since k can be expressed as:

• k = 1 / (4πε₀)

We can rewrite the kinetic energy as:

• T = - (3e^2) / (8πε₀r)

Final Expression

Thus, the kinetic energy of the electron in the ground state of the Li²⁺ ion is:

• T = (3e^2) / (8πε₀r)

This matches the expression you provided. The key takeaway is that the kinetic energy is derived from the potential energy relationship in a hydrogen-like atom, adjusted for the charge of the nucleus. This method can be applied to any hydrogen-like ion by substituting the appropriate nuclear charge.