Characteristic X-rays of frequency `4.2xx10^18` Hz are produced when transitions from L-shell to K-shell take place in a cartain target meterial. Use Mosley's law to determine the atoic number of the target matderial. Given Rydberg constant `R =1.1xx10^7 m^(-1)`.

Explanation

To determine the atomic number of the target material using Moseley's law, we can follow these steps:

### Step 1: Understand Moseley's Law
Moseley's law states that the frequency of the X-rays emitted during electronic transitions can be expressed as:
\[ \frac{1}{\lambda} = R \cdot (Z - 1)^2 \cdot \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
Where:
- \( \lambda \) is the wavelength of the emitted X-rays,
- \( R \) is the Rydberg constant,
- \( Z \) is the atomic number,
- \( n_1 \) and \( n_2 \) are the principal quantum numbers of the initial and final energy levels.

### Step 2: Determine the Values
In this case:
- The frequency \( f = 4.2 \times 10^{18} \) Hz,
- The Rydberg constant \( R = 1.1 \times 10^7 \, m^{-1} \),
- The transition is from the L-shell (n=2) to the K-shell (n=1), so \( n_1 = 1 \) and \( n_2 = 2 \).

### Step 3: Convert Frequency to Wavelength
Using the relationship between frequency and wavelength:
\[ \lambda = \frac{c}{f} \]
where \( c \) (the speed of light) is approximately \( 3 \times 10^8 \, m/s \).

Calculating \( \lambda \):
\[ \lambda = \frac{3 \times 10^8 \, m/s}{4.2 \times 10^{18} \, Hz} = \frac{3}{4.2} \times 10^{-10} \, m = 0.714 \times 10^{-10} \, m = 7.14 \times 10^{-11} \, m \]

### Step 4: Substitute into Moseley's Law
Now we substitute \( \lambda \) into Moseley's law:
\[ \frac{1}{\lambda} = R \cdot (Z - 1)^2 \cdot \left( \frac{1}{1^2} - \frac{1}{2^2} \right) \]
Calculating \( \frac{1}{\lambda} \):
\[ \frac{1}{\lambda} = \frac{1}{7.14 \times 10^{-11}} \approx 1.4 \times 10^{10} \, m^{-1} \]

Now substituting into the equation:
\[ 1.4 \times 10^{10} = 1.1 \times 10^7 \cdot (Z - 1)^2 \cdot \left( 1 - \frac{1}{4} \right) \]
\[ 1.4 \times 10^{10} = 1.1 \times 10^7 \cdot (Z - 1)^2 \cdot \frac{3}{4} \]

### Step 5: Solve for \( Z \)
Rearranging the equation:
\[ (Z - 1)^2 = \frac{1.4 \times 10^{10}}{1.1 \times 10^7 \cdot \frac{3}{4}} \]
Calculating the right side:
\[ (Z - 1)^2 = \frac{1.4 \times 10^{10}}{8.25 \times 10^6} \approx 1.696 \times 10^3 \]

Taking the square root:
\[ Z - 1 \approx \sqrt{1.696 \times 10^3} \approx 41.2 \]
Thus,
\[ Z \approx 42.2 \]

Since \( Z \) must be an integer, we round it to:
\[ Z = 42 \]

### Final Answer
The atomic number of the target material is \( Z = 42 \).

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