Moseley's law for `K_(alpha)` photon is given by `sqrt(v)=a(Z-b)` where `a` is a universal constant and `b` is a screening constant. Moseley's logic helps us assume `b=1` for `K_(alpha)` photon. If there is a percentage error `=10^(-2)` in the measurement of `b` due to actual orbital configuration of an atom, then the relative error in the measurement of `v` if `Z=51` is `p xx 10^(-q)`. Find the value of `(p q)`

Answer

To solve the problem step by step, we will follow the logic outlined in the transcript while providing a clearer mathematical structure.

### Step 1: Understand Moseley's Law
Moseley's law for \( K_{\alpha} \) photon is given by:
\[ \sqrt{v} = a(Z - b) \]
where \( a \) is a universal constant, \( Z \) is the atomic number, and \( b \) is the screening constant.

### Step 2: Differentiate Moseley's Law
To find the relationship between the errors in \( b \) and \( v \), we differentiate both sides with respect to \( b \):
\[ \frac{d(\sqrt{v})}{db} = a \cdot \frac{dZ}{db} - a \cdot \frac{db}{db} \]
Since \( Z \) is constant in this case, we can simplify this to:
\[ \frac{dv}{d b} = 2\sqrt{v} \cdot \frac{db}{d b} \]
This leads us to:
\[ dv = 2\sqrt{v} \cdot db \]

### Step 3: Express \( v \) in terms of \( Z \) and \( b \)
From the original equation, we can express \( v \) as:
\[ v = (a(Z - b))^2 \]
Now, substituting \( b = 1 \) and \( Z = 51 \):
\[ v = (a(51 - 1))^2 = (50a)^2 \]

### Step 4: Calculate the relative error in \( v \)
The relative error in \( v \) can be expressed as:
\[ \frac{dv}{v} = \frac{2db}{Z - b} \]
Substituting \( Z = 51 \) and \( b = 1 \):
\[ \frac{dv}{v} = \frac{2db}{51 - 1} = \frac{2db}{50} \]

### Step 5: Substitute the percentage error in \( b \)
Given that the percentage error in \( b \) is \( 10^{-2} \):
\[ \frac{db}{b} \times 100 = 10^{-2} \implies \frac{db}{b} = 10^{-4} \]
Assuming \( b = 1 \):
\[ db = 10^{-4} \]

### Step 6: Substitute \( db \) into the relative error equation
Now substituting \( db \) into the relative error equation:
\[ \frac{dv}{v} = \frac{2 \cdot 10^{-4}}{50} = \frac{2 \cdot 10^{-4}}{50} = \frac{4 \cdot 10^{-6}}{1} = 4 \cdot 10^{-6} \]

### Step 7: Express the relative error in the required form
The relative error can be expressed as:
\[ \frac{dv}{v} = 4 \times 10^{-6} \]
This can be written in the form \( p \times 10^{-q} \) where \( p = 4 \) and \( q = 6 \).

### Final Step: Calculate \( p \times q \)
Now, we need to find the value of \( p \times q \):
\[ p \times q = 4 \times 6 = 24 \]

### Conclusion
The final answer is:
\[ \boxed{24} \]