If a proton had a radius R and the charge was uniformly distributed, the ground state energy (in eV) of a H -atom for `R=0.1Å` is

Options

A. `-13.6`
B. `-27.2`
C. `-3.4`
D. `-30.8`

Explanation

To find the ground state energy of a hydrogen atom with a uniformly distributed charge over a proton of radius \( R = 0.1 \, \text{Å} \), we can use the formula derived from Bohr's model of the atom. Here’s a step-by-step solution:

### Step 1: Identify the parameters
- For a hydrogen atom, the atomic number \( Z = 1 \).
- The charge of the proton (which is equal to the charge of the electron) is \( e = 1.6 \times 10^{-19} \, \text{C} \).
- The permittivity of free space \( \epsilon_0 = 8.854 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \).
- The radius \( R = 0.1 \, \text{Å} = 0.1 \times 10^{-10} \, \text{m} = 1 \times 10^{-11} \, \text{m} \).

### Step 2: Use the formula for total energy
The total energy \( E \) of the hydrogen atom in terms of the radius \( R \) is given by:
\[ E = -\frac{Z e^2}{8 \pi \epsilon_0 R} \]

### Step 3: Substitute the values into the formula
Substituting the known values into the equation:
\[ E = -\frac{1 \times (1.6 \times 10^{-19})^2}{8 \pi (8.854 \times 10^{-12}) (1 \times 10^{-11})} \]

### Step 4: Calculate the numerator
Calculating the numerator:
\[ (1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \, \text{C}^2 \]

### Step 5: Calculate the denominator
Calculating the denominator:
\[ 8 \pi (8.854 \times 10^{-12}) (1 \times 10^{-11}) \approx 8 \times 3.14 \times 8.854 \times 10^{-23} \approx 2.48 \times 10^{-21} \]

### Step 6: Calculate the energy
Now substituting back into the equation:
\[ E = -\frac{2.56 \times 10^{-38}}{2.48 \times 10^{-21}} \approx -1.031 \times 10^{-17} \, \text{J} \]

### Step 7: Convert the energy to electron volts
To convert joules to electron volts, we use the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):
\[ E \text{ (in eV)} = \frac{-1.031 \times 10^{-17}}{1.6 \times 10^{-19}} \approx -64.4 \, \text{eV} \]

### Final Answer
The ground state energy of a hydrogen atom for \( R = 0.1 \, \text{Å} \) is approximately:
\[ E \approx -64.4 \, \text{eV} \]

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