The element which has a `K_(alpha)` X-rays line of wavelength `1.8 Å` is
`(R = 1.1 xx 10^(7) m^(-1), b = 1 and sqrt(5//33) = 0.39)`

Options

A. `C o, Z = 27`
B. `Iron, Z = 26`
C. `M n, Z = 25`
D. `N i, Z = 28`

Explanation

To solve the problem of identifying the element that has a K-alpha X-ray line of wavelength 1.8 Å, we will use the formula derived from Moseley's law. Let's go through the solution step by step.

### Step 1: Understanding the K-alpha X-ray Transition
The K-alpha X-ray transition occurs when an electron transitions from the n=2 energy level to the n=1 energy level. In Moseley’s law, the wavelength (λ) of the emitted X-ray can be related to the atomic number (Z) of the element.

### Step 2: Use the Formula for K-alpha X-rays
The formula for the wavelength of K-alpha X-rays is given by:
\[ \frac{1}{\lambda} = R \cdot (Z - b)^2 \cdot \left( \frac{3}{4} \right) \]
where:
- \( R \) is the Rydberg constant (given as \( 1.1 \times 10^7 \, m^{-1} \)),
- \( b \) is a constant (given as \( 1 \)),
- \( \lambda \) is the wavelength (given as \( 1.8 \, Å = 1.8 \times 10^{-10} \, m \)).

### Step 3: Rearranging the Formula
Rearranging the formula to solve for \( Z \):
\[ Z - b = \sqrt{\frac{4}{3} R \cdot \lambda} \]
Substituting \( b = 1 \):
\[ Z - 1 = \sqrt{\frac{4}{3} R \cdot \lambda} \]

### Step 4: Substitute the Values
Now, substituting the values of \( R \) and \( \lambda \):
\[ Z - 1 = \sqrt{\frac{4}{3} \cdot (1.1 \times 10^7) \cdot (1.8 \times 10^{-10})} \]

### Step 5: Calculate the Right Side
Calculating the right side:
1. Calculate \( R \cdot \lambda \):
\[ R \cdot \lambda = 1.1 \times 10^7 \cdot 1.8 \times 10^{-10} = 1.98 \times 10^{-3} \]
2. Then calculate \( \frac{4}{3} R \cdot \lambda \):
\[ \frac{4}{3} \cdot 1.98 \times 10^{-3} = 2.64 \times 10^{-3} \]
3. Now take the square root:
\[ Z - 1 = \sqrt{2.64 \times 10^{-3}} \approx 0.0514 \]

### Step 6: Solve for Z
Now, adding 1 to both sides:
\[ Z \approx 1 + 0.0514 \approx 27 \]

### Step 7: Identify the Element
The atomic number \( Z = 27 \) corresponds to the element Cobalt (Co).

### Final Answer
The element which has a K-alpha X-ray line of wavelength 1.8 Å is **Cobalt (Co)**.

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