The ratio of the kinetic energy and the potential energy of electron in the hydrogen atom will be

Options

A. `1:2`
B. `-1:2`
C. `2:1`
D. `-2:1`

Explanation

To find the ratio of the kinetic energy (KE) and potential energy (PE) of an electron in a hydrogen atom, we can follow these steps:

### Step 1: Understand the Total Energy of the Hydrogen Atom
The total energy (E) of an electron in the hydrogen atom at the nth energy level is given by the formula:
\[ E = -\frac{13.6 \, Z^2}{n^2} \]
For hydrogen, the atomic number \( Z = 1 \), so the formula simplifies to:
\[ E = -\frac{13.6}{n^2} \]

### Step 2: Determine the Kinetic Energy
The kinetic energy of the electron in the hydrogen atom is related to the total energy. It can be expressed as:
\[ KE = -\frac{1}{2} E \]
Substituting the expression for total energy:
\[ KE = -\frac{1}{2} \left(-\frac{13.6}{n^2}\right) = \frac{13.6}{2n^2} = \frac{6.8}{n^2} \]

### Step 3: Determine the Potential Energy
The potential energy (PE) of the electron in the hydrogen atom is given by:
\[ PE = 2 \times KE \]
Substituting the expression for kinetic energy:
\[ PE = 2 \times \frac{6.8}{n^2} = \frac{13.6}{n^2} \]
However, since potential energy is negative in this context, we have:
\[ PE = -\frac{13.6}{n^2} \]

### Step 4: Calculate the Ratio of Kinetic Energy to Potential Energy
Now, we can find the ratio of kinetic energy to potential energy:
\[ \text{Ratio} = \frac{KE}{PE} \]
Substituting the expressions for KE and PE:
\[ \text{Ratio} = \frac{\frac{6.8}{n^2}}{-\frac{13.6}{n^2}} = \frac{6.8}{-13.6} = -\frac{1}{2} \]

### Conclusion
Thus, the ratio of the kinetic energy to the potential energy of an electron in a hydrogen atom is:
\[ \text{Ratio} = -\frac{1}{2} \]