The ratio of the radius of the Bohr orbit for the electron orbiting the hydrogen nucleus that of the electron orbiting the deuterium nucleus is approximately

Options

A. `1:1`
B. `1:2`
C. `2:1`
D. `1:4`

Answer

To find the ratio of the radius of the Bohr orbit for the electron orbiting the hydrogen nucleus to that of the electron orbiting the deuterium nucleus, we can follow these steps:

### Step 1: Understand the Formula for Bohr's Radius
The radius of the Bohr orbit can be expressed using the formula:
\[ r = \frac{n^2 h^2}{4 \pi^2 m k z e^2} \]
where:
- \( n \) = principal quantum number
- \( h \) = Planck's constant
- \( m \) = mass of the electron
- \( k \) = Coulomb's constant
- \( z \) = atomic number
- \( e \) = charge of the electron

### Step 2: Simplify for Hydrogen and Deuterium
For hydrogen (H), the atomic number \( z = 1 \), and for deuterium (D), which is an isotope of hydrogen, the atomic number is also \( z = 1 \).

### Step 3: Identify the Principal Quantum Number
In both cases, we will consider the ground state, so the principal quantum number \( n = 1 \).

### Step 4: Write the Expression for Both Atoms
Using the simplified formula for Bohr's radius:
\[ r_H = r_0 \cdot \frac{n^2}{z} \]
\[ r_D = r_0 \cdot \frac{n^2}{z} \]
where \( r_0 \) is the Bohr radius constant (approximately 0.53 Å).

### Step 5: Calculate the Ratio of the Radii
Since both hydrogen and deuterium have the same values for \( n \) and \( z \):
\[ \text{Ratio} = \frac{r_H}{r_D} = \frac{r_0 \cdot \frac{1^2}{1}}{r_0 \cdot \frac{1^2}{1}} = 1 \]

### Conclusion
The ratio of the radius of the Bohr orbit for the electron orbiting the hydrogen nucleus to that of the electron orbiting the deuterium nucleus is approximately:
\[ \text{Ratio} = 1 : 1 \]

### Final Answer
The correct option is 1:1.

---