Find the wavelength of the `K_(alpha)` line in copper `(Z=29)`, if the wave length of the `K_(alpha)` line in iron `(Z=26)` is known to be equal to `193 "pm"` (Take `b=1`)

Answer

To find the wavelength of the \( K_{\alpha} \) line in copper (Z=29), given that the wavelength of the \( K_{\alpha} \) line in iron (Z=26) is 193 pm, we can use the formula that relates the frequency (or wavelength) to the atomic number (Z) of the element.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The relationship between the wavelength (\( \lambda \)) and the atomic number (Z) can be expressed as:
\[ \sqrt{\nu} = A \cdot (Z - B) \]
where \( \nu \) is the frequency, \( A \) is a constant, and \( B \) is a correction factor. Here, \( B = 1 \).

2. **Expressing Frequency in Terms of Wavelength**:
We know that frequency (\( \nu \)) is related to wavelength (\( \lambda \)) by:
\[ \nu = \frac{C}{\lambda} \]
where \( C \) is the speed of light.

3. **Setting Up the Equations**:
For copper (Z=29):
\[ \sqrt{\frac{C}{\lambda_{Cu}}} = A \cdot (29 - 1) = A \cdot 28 \]
Squaring both sides gives:
\[ \frac{C}{\lambda_{Cu}} = A^2 \cdot 28^2 \quad \text{(Equation 1)} \]

For iron (Z=26):
\[ \sqrt{\frac{C}{\lambda_{Fe}}} = A \cdot (26 - 1) = A \cdot 25 \]
Squaring both sides gives:
\[ \frac{C}{\lambda_{Fe}} = A^2 \cdot 25^2 \quad \text{(Equation 2)} \]

4. **Dividing the Two Equations**:
Dividing Equation 1 by Equation 2:
\[ \frac{\frac{C}{\lambda_{Cu}}}{\frac{C}{\lambda_{Fe}}} = \frac{A^2 \cdot 28^2}{A^2 \cdot 25^2} \]
Simplifying gives:
\[ \frac{\lambda_{Fe}}{\lambda_{Cu}} = \frac{28^2}{25^2} \]

5. **Substituting the Known Wavelength**:
We know \( \lambda_{Fe} = 193 \, \text{pm} \):
\[ \lambda_{Cu} = \lambda_{Fe} \cdot \frac{25^2}{28^2} \]
Substituting the value:
\[ \lambda_{Cu} = 193 \cdot \frac{25^2}{28^2} \]

6. **Calculating the Wavelength**:
Calculating \( 25^2 = 625 \) and \( 28^2 = 784 \):
\[ \lambda_{Cu} = 193 \cdot \frac{625}{784} \]
Now, calculating:
\[ \lambda_{Cu} \approx 193 \cdot 0.797 = 154.2 \, \text{pm} \]

### Final Answer:
The wavelength of the \( K_{\alpha} \) line in copper is approximately \( 154 \, \text{pm} \).