In hydrogen atom, the total energy of an electron in a given orbit is -1.5eV. The potential energy in the same orbit is

Options

A. 1.5eV
B. 3.0eV
C. `-3.0eV`
D. `-1.5eV`

Explanation

To find the potential energy of an electron in a hydrogen atom given that the total energy is -1.5 eV, we can follow these steps:

### Step 1: Understand the relationship between total energy and potential energy
In a hydrogen atom, the total energy (E) of an electron in a given orbit can be expressed as:
\[ E = K + U \]
where \( K \) is the kinetic energy and \( U \) is the potential energy.

### Step 2: Use the formula for total energy
The total energy of an electron in a hydrogen atom is given by the formula:
\[ E_n = -\frac{13.6 \, Z^2}{n^2} \]
For hydrogen, \( Z = 1 \), so:
\[ E_n = -\frac{13.6}{n^2} \]

### Step 3: Calculate the value of n
Given that the total energy \( E_n = -1.5 \, eV \), we can set up the equation:
\[ -\frac{13.6}{n^2} = -1.5 \]
This simplifies to:
\[ \frac{13.6}{n^2} = 1.5 \]

### Step 4: Solve for n^2
Rearranging gives:
\[ n^2 = \frac{13.6}{1.5} \]
Calculating this gives:
\[ n^2 \approx 9.0667 \]
Taking the square root:
\[ n \approx 3 \]

### Step 5: Use the potential energy formula
The potential energy \( U \) in the same orbit is given by:
\[ U = -\frac{27.2 \, Z^2}{n^2} \]
Substituting \( Z = 1 \) and \( n = 3 \):
\[ U = -\frac{27.2}{3^2} = -\frac{27.2}{9} \]
Calculating this gives:
\[ U = -3.0222 \, eV \]
Rounding this to two decimal places, we find:
\[ U \approx -3 \, eV \]

### Final Answer
The potential energy in the same orbit is approximately:
\[ U = -3 \, eV \]

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