The ratio of the speed of the electrons in the ground state of hydrogen to the speed of light in vacuum is

Options

A. `(1)/(2)`
B. `(2)/(237)`
C. `(1)/(137)`
D. `(1)/(237)`

Explanation

To find the ratio of the speed of the electrons in the ground state of hydrogen to the speed of light in vacuum, we can follow these steps:

### Step 1: Use the formula for the speed of electrons in the n-th state of hydrogen.
The speed \( V \) of an electron in the n-th state of a hydrogen atom is given by the formula:
\[ V = \frac{e^2}{2nH\epsilon_0} \]
where:
- \( e \) is the charge of the electron,
- \( n \) is the principal quantum number,
- \( H \) is Planck's constant,
- \( \epsilon_0 \) is the permittivity of free space.

### Step 2: Substitute values for the ground state (n=1).
For the ground state, \( n = 1 \). Thus, the formula simplifies to:
\[ V = \frac{e^2}{2H\epsilon_0} \]

### Step 3: Insert the known values.
We know:
- \( e = 1.6 \times 10^{-19} \, \text{C} \) (charge of the electron),
- \( H = 6.63 \times 10^{-34} \, \text{Js} \) (Planck's constant),
- \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \) (permittivity of free space).

Now substituting these values into the equation:
\[ V = \frac{(1.6 \times 10^{-19})^2}{2 \times (6.63 \times 10^{-34}) \times (8.85 \times 10^{-12})} \]

### Step 4: Calculate the speed \( V \).
Calculating the numerator:
\[ (1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \]
Calculating the denominator:
\[ 2 \times (6.63 \times 10^{-34}) \times (8.85 \times 10^{-12}) = 1.175 \times 10^{-45} \]
Now, substituting these values:
\[ V = \frac{2.56 \times 10^{-38}}{1.175 \times 10^{-45}} \approx 0.0219 \times 10^{8} \, \text{m/s} \]

### Step 5: Find the speed of light \( C \).
The speed of light in vacuum is given by:
\[ C = 3 \times 10^{8} \, \text{m/s} \]

### Step 6: Calculate the ratio of the speed of the electron to the speed of light.
Now, we calculate the ratio:
\[ \text{Ratio} = \frac{V}{C} = \frac{0.0219 \times 10^{8}}{3 \times 10^{8}} = \frac{0.0219}{3} \approx \frac{1}{137} \]

### Final Answer:
The ratio of the speed of the electrons in the ground state of hydrogen to the speed of light in vacuum is approximately \( \frac{1}{137} \).

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