The characteristic X-rays for the lines of `K_(a)` series in element X and Y are `9.87 Å and 14.6 Å` respectively .If Moseley's equation `sqrt(v) = 4.9 xx 10^(7) (Z - 0.75)` is followed:
The atomic number of X is

Options

A. 8
B. 10
C. 12
D. 16

Explanation

To find the atomic number of element X using the given information, we will follow these steps:

### Step 1: Understand the problem
We are given the characteristic X-ray wavelengths for the Kα series of two elements, X and Y, which are 9.87 Å and 14.6 Å, respectively. We need to find the atomic number (Z) of element X using Moseley's equation.

### Step 2: Convert wavelength to frequency
Moseley's equation relates the frequency (ν) of the X-ray to the atomic number (Z). We first need to convert the wavelength (λ) of element X into frequency using the equation:
\[ \nu = \frac{c}{\lambda} \]
where \( c \) is the speed of light (approximately \( 3 \times 10^8 \) m/s).

For element X:
\[ \lambda = 9.87 \, \text{Å} = 9.87 \times 10^{-10} \, \text{m} \]

Now, substituting the values:
\[ \nu = \frac{3 \times 10^8 \, \text{m/s}}{9.87 \times 10^{-10} \, \text{m}} \]

### Step 3: Calculate the frequency
Calculating the frequency:
\[ \nu = \frac{3 \times 10^8}{9.87 \times 10^{-10}} \approx 3.03 \times 10^{18} \, \text{s}^{-1} \]

### Step 4: Apply Moseley's equation
Moseley's equation is given by:
\[ \sqrt{\nu} = 4.9 \times 10^7 (Z - 0.75) \]

First, we need to calculate \( \sqrt{\nu} \):
\[ \sqrt{\nu} = \sqrt{3.03 \times 10^{18}} \approx 5.50 \times 10^9 \, \text{s}^{-1} \]

### Step 5: Substitute into Moseley's equation
Now we substitute \( \sqrt{\nu} \) into Moseley's equation:
\[ 5.50 \times 10^9 = 4.9 \times 10^7 (Z - 0.75) \]

### Step 6: Solve for Z
To isolate Z, we first divide both sides by \( 4.9 \times 10^7 \):
\[ Z - 0.75 = \frac{5.50 \times 10^9}{4.9 \times 10^7} \]

Calculating the right-hand side:
\[ Z - 0.75 \approx 112.24 \]

Now, adding 0.75 to both sides:
\[ Z \approx 112.24 + 0.75 \]
\[ Z \approx 113.00 \]

### Step 7: Conclusion
The atomic number of element X is approximately 113.