The potential energy of the orbital electron in the ground state of hydrogen atoms is -E, what is the kinetic energy?

Options

A. `4E`
B. `2E`
C. `E/2`
D. `E/4`

Explanation

To find the kinetic energy of the orbital electron in the ground state of a hydrogen atom given that the potential energy is -E, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Potential and Kinetic Energy**:
In a hydrogen atom, the total mechanical energy (E_total) is the sum of the kinetic energy (K.E) and potential energy (P.E). The relationship can be expressed as:
\[ E_{\text{total}} = K.E + P.E \]

2. **Use the Given Potential Energy**:
We know from the problem that the potential energy (P.E) of the electron in the ground state is given as:
\[ P.E = -E \]

3. **Express Total Energy in Terms of Kinetic Energy**:
For a hydrogen atom in the ground state, it is a known result that the total energy is half of the potential energy (but negative). Therefore, we can express the total energy as:
\[ E_{\text{total}} = \frac{K.E}{2} \]

4. **Substituting for Total Energy**:
Since we know that the total energy is also equal to the potential energy plus kinetic energy, we can write:
\[ E_{\text{total}} = K.E - E \]
Setting the two expressions for total energy equal gives:
\[ K.E - E = \frac{K.E}{2} \]

5. **Solving for Kinetic Energy**:
Rearranging the equation:
\[ K.E - \frac{K.E}{2} = E \]
This simplifies to:
\[ \frac{K.E}{2} = E \]
Therefore, multiplying both sides by 2 gives:
\[ K.E = 2E \]

6. **Final Answer**:
Thus, the kinetic energy of the electron in the ground state of the hydrogen atom is:
\[ K.E = \frac{E}{2} \]