The characteristics X-rays wavelength is related to atomic number by the relation `sqrt(nu)=a(Z-b)`
When Z is the atomic number, a and b are Mosley's constants. If `lambda_(1)=2.886Å` and `lambda_(2)=2.365Å` corresponding to `Z_(1)=55 and Z_(2)=60` respectively, the value of Z corresponding to `lambda=2.660Å` is

Options

A. 55
B. 57
C. 74
D. 507

Answer

To solve the problem, we will follow the steps outlined in the video transcript and derive the value of the atomic number \( Z \) corresponding to the wavelength \( \lambda = 2.660 \, \text{Å} \).

### Step 1: Understand the relationship
The relationship given is:
\[ \sqrt{\nu} = a(Z - b) \]
where \( \nu \) is the frequency, \( Z \) is the atomic number, and \( a \) and \( b \) are Mosley's constants. The frequency \( \nu \) can be expressed in terms of wavelength \( \lambda \) as:
\[ \nu = \frac{c}{\lambda} \]
where \( c \) is the speed of light.

### Step 2: Set up equations for the given data
For the first case with \( \lambda_1 = 2.886 \, \text{Å} \) and \( Z_1 = 55 \):
\[ \sqrt{\frac{c}{2.886}} = a(55 - b) \quad \text{(Equation 1)} \]

For the second case with \( \lambda_2 = 2.365 \, \text{Å} \) and \( Z_2 = 60 \):
\[ \sqrt{\frac{c}{2.365}} = a(60 - b) \quad \text{(Equation 2)} \]

### Step 3: Divide the two equations
Dividing Equation 1 by Equation 2 gives:
\[ \frac{\sqrt{\frac{c}{2.365}}}{\sqrt{\frac{c}{2.886}}} = \frac{a(55 - b)}{a(60 - b)} \]
This simplifies to:
\[ \sqrt{\frac{2.886}{2.365}} = \frac{55 - b}{60 - b} \]

### Step 4: Calculate the left side
Calculating the left side:
\[ \sqrt{\frac{2.886}{2.365}} \approx 0.905 \]
Thus, we have:
\[ 0.905 = \frac{55 - b}{60 - b} \]

### Step 5: Cross-multiply and solve for \( b \)
Cross-multiplying gives:
\[ 0.905(60 - b) = 55 - b \]
Expanding and rearranging:
\[ 54.3 - 0.905b = 55 - b \]
\[ b - 0.905b = 55 - 54.3 \]
\[ 0.095b = 0.7 \]
\[ b = \frac{0.7}{0.095} \approx 7.36 \]

### Step 6: Substitute \( b \) back into Equation 1
Now substituting \( b \) back into Equation 1:
\[ \sqrt{\frac{c}{2.886}} = a(55 - 7.36) \]
This simplifies to:
\[ \sqrt{\frac{c}{2.886}} = a(47.64) \]

### Step 7: Calculate \( a \)
Using \( c = 3 \times 10^8 \, \text{m/s} \):
\[ \sqrt{\frac{3 \times 10^8}{2.886 \times 10^{-10}}} = a(47.64) \]
Calculating the left side:
\[ \sqrt{1.041 \times 10^{18}} \approx 1.01 \times 10^9 \]
Thus:
\[ 1.01 \times 10^9 = a(47.64) \]
Solving for \( a \):
\[ a = \frac{1.01 \times 10^9}{47.64} \approx 2.12 \times 10^7 \]

### Step 8: Find \( Z \) for \( \lambda = 2.660 \, \text{Å} \)
Now we need to find \( Z \) for \( \lambda = 2.660 \, \text{Å} \):
\[ \sqrt{\frac{c}{2.660}} = a(Z - b) \]
Substituting known values:
\[ \sqrt{\frac{3 \times 10^8}{2.660 \times 10^{-10}}} = 2.12 \times 10^7(Z - 7.36) \]
Calculating the left side:
\[ \sqrt{1.13 \times 10^{18}} \approx 1.06 \times 10^9 \]
Thus:
\[ 1.06 \times 10^9 = 2.12 \times 10^7(Z - 7.36) \]
Solving for \( Z \):
\[ Z - 7.36 = \frac{1.06 \times 10^9}{2.12 \times 10^7} \approx 50.04 \]
\[ Z \approx 57.40 \]
Rounding to the nearest whole number gives:
\[ Z \approx 57 \]

### Final Answer
The value of \( Z \) corresponding to \( \lambda = 2.660 \, \text{Å} \) is approximately **57**.