The wavelength of the `K_a` line for the uranium is `(Z = 92) (R = 1.0973xx10^7m^(-1)`

Options

A. `1.5 Å`
B. `0.5 Å`
C. `0.15 Å`
D. `2.0 Å`

Explanation

To solve the problem of finding the wavelength of the K_alpha line for uranium (Z = 92), we will use the formula for the wavelength in terms of the Rydberg constant and the principal quantum numbers involved in the transition.

### Step-by-Step Solution:

1. **Identify the Formula**:
The formula for the wavelength of the K_alpha line is given by:
\[ \frac{1}{\lambda} = R \cdot (Z - 1)^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where:
- \( R = 1.0973 \times 10^7 \, \text{m}^{-1} \)
- \( Z = 92 \) (atomic number of uranium)
- For K_alpha line, \( n_1 = 1 \) and \( n_2 = 2 \).

2. **Substitute the Values**:
Substitute \( n_1 \) and \( n_2 \) into the equation:
\[ \frac{1}{\lambda} = R \cdot (Z - 1)^2 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) \]

3. **Calculate the Difference in Terms**:
Calculate \( \frac{1}{1^2} - \frac{1}{2^2} \):
\[ \frac{1}{1} - \frac{1}{4} = 1 - 0.25 = 0.75 \]

4. **Substitute Back into the Equation**:
Now substitute this back into the equation:
\[ \frac{1}{\lambda} = R \cdot (Z - 1)^2 \cdot 0.75 \]
\[ \frac{1}{\lambda} = 1.0973 \times 10^7 \cdot (92 - 1)^2 \cdot 0.75 \]
\[ \frac{1}{\lambda} = 1.0973 \times 10^7 \cdot 91^2 \cdot 0.75 \]

5. **Calculate \( (Z - 1)^2 \)**:
Calculate \( 91^2 \):
\[ 91^2 = 8281 \]

6. **Complete the Calculation**:
Now substitute \( 8281 \) back into the equation:
\[ \frac{1}{\lambda} = 1.0973 \times 10^7 \cdot 8281 \cdot 0.75 \]
\[ \frac{1}{\lambda} = 1.0973 \times 10^7 \cdot 6210.75 \]
\[ \frac{1}{\lambda} \approx 6.826 \times 10^{10} \, \text{m}^{-1} \]

7. **Find the Wavelength**:
Now take the reciprocal to find \( \lambda \):
\[ \lambda = \frac{1}{6.826 \times 10^{10}} \approx 0.146 \times 10^{-10} \, \text{m} \]
\[ \lambda \approx 0.146 \, \text{Å} \]

### Final Answer:
The wavelength of the K_alpha line for uranium is approximately \( 0.146 \, \text{Å} \).