If an electron in a hydrogen atom is moving with a kinetic energy of `5.45 xx10^(-19)` j then what will be the energy level for this electron ?

Options

A. 1
B. 2
C. 3
D. 6

Answer

To find the energy level of an electron in a hydrogen atom given its kinetic energy, we can follow these steps:

### Step 1: Understand the relationship between kinetic energy and total energy.
The total energy (E) of an electron in a hydrogen atom is related to its kinetic energy (KE) by the equation:
\[ E = -KE \]

### Step 2: Calculate the total energy.
Given that the kinetic energy of the electron is \( KE = 5.45 \times 10^{-19} \) J, we can find the total energy:
\[ E = -5.45 \times 10^{-19} \, \text{J} \]

### Step 3: Convert total energy from joules to electron volts.
We know that \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \). To convert the total energy into electron volts (eV):
\[ E = \frac{-5.45 \times 10^{-19} \, \text{J}}{1.6 \times 10^{-19} \, \text{J/eV}} \]
\[ E = -\frac{5.45}{1.6} \, \text{eV} \]
\[ E \approx -3.41 \, \text{eV} \]

### Step 4: Use the formula for total energy in a hydrogen atom.
The total energy of an electron in a hydrogen atom is given by the formula:
\[ E = -\frac{13.6 \, Z^2}{n^2} \]
For hydrogen, \( Z = 1 \):
\[ E = -\frac{13.6}{n^2} \]

### Step 5: Set the total energy equation equal to the calculated total energy.
Now, we can set the two expressions for total energy equal to each other:
\[ -\frac{13.6}{n^2} = -3.41 \]

### Step 6: Solve for \( n^2 \).
Removing the negative signs and rearranging gives:
\[ \frac{13.6}{n^2} = 3.41 \]
\[ n^2 = \frac{13.6}{3.41} \]
\[ n^2 \approx 4 \]

### Step 7: Find \( n \).
Taking the square root of both sides:
\[ n \approx 2 \]

### Conclusion:
The energy level for the electron in the hydrogen atom is \( n = 2 \).

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