Let `lambda_(alpha'), lambda_(beta),and lambda'_(alpha)` denote the wavelength of the X-ray of the `K_(alpha), K_(beta), and L_(alpha)` lines in the characteristic X-rays for a metal. Then.

Options

A. `lambda_(alpha)gt lambda_(alpha)gtlambda_(beta)`
B. `lambda_(alpha)gtlambda_(beta)lambda_(alpha)`
C. `1/lambda_(beta)=1/lambda_(alpha)+1/lambda_(alpha)`
D. `1/lambda_(beta)+1/lambda_(alpha)=1/lambda_(alpha)`

Answer

To solve the problem regarding the wavelengths of the K_alpha, K_beta, and L_alpha lines in characteristic X-rays for a metal, we can follow these steps:

### Step 1: Understand the Energy Levels
The transitions corresponding to the K_alpha, K_beta, and L_alpha lines can be related to the energy levels of the hydrogen atom (or hydrogen-like atoms). The relevant transitions are:
- K_alpha: Transition from n=2 to n=1
- K_beta: Transition from n=3 to n=1
- L_alpha: Transition from n=3 to n=2

### Step 2: Calculate the Energies
We can calculate the energies for these transitions using the formula for the energy levels of hydrogen-like atoms:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]

Calculating the energies:
- For n=1:
\[ E_1 = -13.6 \, \text{eV} \]
- For n=2:
\[ E_2 = -\frac{13.6}{2^2} = -3.4 \, \text{eV} \]
- For n=3:
\[ E_3 = -\frac{13.6}{3^2} \approx -1.51 \, \text{eV} \]

### Step 3: Calculate the Energy Differences
Now, we calculate the energies for each transition:
- **K_alpha**:
\[ E_{K_\alpha} = E_1 - E_2 = (-3.4) - (-13.6) = 10.2 \, \text{eV} \]

- **K_beta**:
\[ E_{K_\beta} = E_1 - E_3 = (-1.51) - (-13.6) = 12.09 \, \text{eV} \]

- **L_alpha**:
\[ E_{L_\alpha} = E_2 - E_3 = (-1.51) - (-3.4) = 1.89 \, \text{eV} \]

### Step 4: Relate Energy and Wavelength
The relationship between energy and wavelength is given by:
\[ E = \frac{hc}{\lambda} \]
Thus, we can express the wavelengths in terms of energy:
\[ \lambda = \frac{hc}{E} \]

### Step 5: Compare Wavelengths
Since energy is inversely proportional to wavelength, we can conclude:
- \( E_{K_\beta} > E_{K_\alpha} > E_{L_\alpha} \)
This implies:
- \( \lambda_{K_\beta} < \lambda_{K_\alpha} < \lambda_{L_\alpha} \)

### Step 6: Final Relationships
From the above relationships, we can summarize:
- \( \lambda_{L_\alpha} > \lambda_{K_\alpha} > \lambda_{K_\beta} \)

### Step 7: Evaluate Options
Now, we can evaluate the given options based on the relationships we derived. The correct relationship is:
\[ \frac{1}{\lambda_{K_\beta}} = \frac{1}{\lambda_{K_\alpha}} + \frac{1}{\lambda_{L_\alpha}} \]

### Conclusion
Thus, the correct answer is option C.

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