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. 63
B. 67
C. 74
D. 507

Answer

To solve the problem, we will follow these steps:

### 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 constants.

### Step 2: Convert wavelength to frequency
We know that frequency \( \nu \) can be expressed in terms of wavelength \( \lambda \) as:
\[ \nu = \frac{c}{\lambda} \]
where \( c \) is the speed of light. Therefore,
\[ \sqrt{\nu} = \sqrt{\frac{c}{\lambda}} = \frac{\sqrt{c}}{\sqrt{\lambda}} \]

### Step 3: Set up equations for given values
We have two sets of values:
1. For \( \lambda_1 = 2.886 \, \text{Å} \) and \( Z_1 = 55 \):
\[ \sqrt{\frac{c}{2.886}} = a(55 - b) \]
2. For \( \lambda_2 = 2.365 \, \text{Å} \) and \( Z_2 = 60 \):
\[ \sqrt{\frac{c}{2.365}} = a(60 - b) \]

### Step 4: Take the ratio of the two equations
Taking the ratio of the two equations:
\[ \frac{\sqrt{\frac{c}{2.886}}}{\sqrt{\frac{c}{2.365}}} = \frac{a(55 - b)}{a(60 - b)} \]
This simplifies to:
\[ \sqrt{\frac{2.365}{2.886}} = \frac{55 - b}{60 - b} \]

### Step 5: Calculate the left-hand side
Calculating the left-hand side:
\[ \sqrt{\frac{2.365}{2.886}} \approx 0.905 \]

### Step 6: Set up the equation
Now we have:
\[ 0.905 = \frac{55 - b}{60 - b} \]
Cross-multiplying gives:
\[ 0.905(60 - b) = 55 - b \]
Expanding this:
\[ 54.3 - 0.905b = 55 - b \]

### Step 7: Solve for \( b \)
Rearranging gives:
\[ b - 0.905b = 55 - 54.3 \]
\[ 0.095b = 0.7 \]
\[ b = \frac{0.7}{0.095} \approx 7.36 \]

### Step 8: Find the value of \( a \)
Now we can find \( a \) using one of the original equations. Using \( \lambda_1 \):
\[ \sqrt{\frac{c}{2.886}} = a(55 - 7.36) \]
Calculating \( 55 - 7.36 = 47.64 \):
\[ \sqrt{\frac{c}{2.886}} = a \times 47.64 \]
Substituting \( c = 3 \times 10^8 \, \text{m/s} \) and converting \( \lambda_1 \) to meters:
\[ \sqrt{\frac{3 \times 10^8}{2.886 \times 10^{-10}}} = a \times 47.64 \]
Calculating the left-hand side:
\[ \sqrt{1.041 \times 10^{18}} \approx 1.020 \times 10^9 \]
Thus:
\[ 1.020 \times 10^9 = a \times 47.64 \]
Solving for \( a \):
\[ a = \frac{1.020 \times 10^9}{47.64} \approx 2.14 \times 10^7 \]

### Step 9: Find \( Z \) for \( \lambda = 2.660 \, \text{Å} \)
Using the formula again:
\[ \sqrt{\frac{c}{2.660}} = a(Z - b) \]
Calculating \( \sqrt{\frac{c}{2.660}} \):
\[ \sqrt{\frac{3 \times 10^8}{2.660 \times 10^{-10}}} \approx \sqrt{1.127 \times 10^{18}} \approx 1.06 \times 10^9 \]
Substituting \( a \) and \( b \):
\[ 1.06 \times 10^9 = 2.14 \times 10^7 (Z - 7.36) \]
Solving for \( Z \):
\[ Z - 7.36 = \frac{1.06 \times 10^9}{2.14 \times 10^7} \approx 49.5 \]
Thus:
\[ Z \approx 49.5 + 7.36 \approx 56.86 \]
Rounding gives \( Z \approx 57 \).

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

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