In a hypotherical Bohr hydrogen, the mass of the electron is doubled. The energy `E_(0)` and the radius `r_(0)` of the first orbit will be (`a_(0)` is the Bohr radius)

Options

A. `E_(0)=-27.eV, r_(0)=a_(0)//2`
B. `E_(0)=-27.2 eV, r_(0)=a_(0)=a_(0)`
C. `E_(0)=-13.6eV, r_(0)=a_(0)//2`
D. `E_(0)=-13.6 eV r_(0)=a_(0)`

Explanation

To solve the problem of how the energy \( E_0 \) and the radius \( r_0 \) of the first orbit change when the mass of the electron is doubled in a hypothetical Bohr hydrogen atom, we will follow these steps:

### Step 1: Understand the Original Bohr Model
In the original Bohr model for hydrogen, the energy of the first orbit is given by:
\[ E_0 = -\frac{Z^2 e^4 m}{2 \hbar^2} = -\frac{Z^2 e^4}{8 \epsilon_0 a_0} \]
where \( Z \) is the atomic number (1 for hydrogen), \( e \) is the charge of the electron, \( m \) is the mass of the electron, \( \hbar \) is the reduced Planck's constant, \( \epsilon_0 \) is the permittivity of free space, and \( a_0 \) is the Bohr radius.

The radius of the first orbit is given by:
\[ r_0 = \frac{n^2 \hbar^2}{Z e^2 m} \]
For the first orbit (\( n = 1 \)), this simplifies to:
\[ r_0 = \frac{\hbar^2}{Z e^2 m} \]

### Step 2: Modify the Mass of the Electron
Now, if the mass of the electron is doubled, we denote the new mass as \( m' = 2m \).

### Step 3: Calculate the New Energy \( E_0' \)
Substituting \( m' \) into the energy formula:
\[ E_0' = -\frac{Z^2 e^4 m'}{2 \hbar^2} = -\frac{Z^2 e^4 (2m)}{2 \hbar^2} = -\frac{Z^2 e^4 m}{\hbar^2} \]
This shows that the new energy \( E_0' \) is twice the original energy:
\[ E_0' = 2E_0 \]

### Step 4: Calculate the New Radius \( r_0' \)
Substituting \( m' \) into the radius formula:
\[ r_0' = \frac{\hbar^2}{Z e^2 m'} = \frac{\hbar^2}{Z e^2 (2m)} = \frac{1}{2} \cdot \frac{\hbar^2}{Z e^2 m} = \frac{1}{2} r_0 \]

### Conclusion
Thus, when the mass of the electron is doubled:
- The energy of the first orbit \( E_0' \) becomes \( 2E_0 \).
- The radius of the first orbit \( r_0' \) becomes \( \frac{1}{2} r_0 \).

### Final Answer
- The energy \( E_0' = 2E_0 \)
- The radius \( r_0' = \frac{1}{2} r_0 \)