An electron in Hydrogen atom ( ground state) jumps to higher energy level x, such that the potential energy of electron becomes half of itstotal energy at ground state. What is the value of x ?

Answer

To solve the problem, we need to determine the value of the energy level \( x \) to which an electron in a hydrogen atom jumps such that the potential energy becomes half of its total energy at the ground state.

### Step-by-Step Solution:

1. **Understand the Energy Levels in Hydrogen Atom:**
The total energy \( E_n \) of an electron in a hydrogen atom at a principal quantum number \( n \) is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number.

2. **Calculate Total Energy at Ground State:**
For the ground state (\( n = 1 \)):
\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]

3. **Determine the Condition Given in the Problem:**
The problem states that the potential energy \( U \) in the excited state becomes half of the total energy in the ground state. The total energy in the ground state is \( -13.6 \, \text{eV} \), so:
\[ U = \frac{1}{2} E_1 = \frac{1}{2} (-13.6 \, \text{eV}) = -6.8 \, \text{eV} \]

4. **Relate Potential Energy and Total Energy in Excited State:**
In a stationary state, the potential energy \( U \) is related to the total energy \( E \) by:
\[ U = 2E \]
Therefore, if \( E_x \) is the total energy at the excited state \( n = x \):
\[ U_x = -\frac{13.6 \, \text{eV}}{x^2} \quad \text{and} \quad E_x = -\frac{13.6 \, \text{eV}}{x^2} \]
Thus, we have:
\[ U_x = 2E_x \]

5. **Set Up the Equation:**
From the relationship:
\[ -\frac{13.6 \, \text{eV}}{x^2} = 2 \left(-\frac{13.6 \, \text{eV}}{x^2}\right) \]
This simplifies to:
\[ -\frac{13.6 \, \text{eV}}{x^2} = -\frac{27.2 \, \text{eV}}{x^2} \]

6. **Solve for \( x \):**
Rearranging gives:
\[ -13.6 = -\frac{27.2}{x^2} \]
Cross-multiplying leads to:
\[ 13.6x^2 = 27.2 \]
Dividing both sides by 13.6:
\[ x^2 = 2 \]
Taking the square root:
\[ x = \sqrt{2} \approx 1.414 \]

7. **Conclusion:**
The value of \( x \) is approximately \( 1.414 \).