A`K_(alpha)` X-ray emitted from a sample has an energy of `7.46 ke V. Of which element is the sample made?

Options

A. Calucium `(Ca, Z = 20)`
B. Cobalt `(Co, Z = 27)`
C. Cadmium `(Cd, Z = 48)`
D. Nickel `(Ni, Z = 28)`

Explanation

To determine the element from which the K-alpha X-ray is emitted, we can follow these steps:

### Step 1: Understand the K-alpha X-ray Transition
K-alpha X-rays are produced when an electron transitions from the n=2 energy level to the n=1 energy level in an atom. The energy of the emitted X-ray can be calculated using the formula:
\[ E_{K_{\alpha}} = E_{I} - E_{F} \]
where \( E_{I} \) is the energy of the electron in the initial state (n=2) and \( E_{F} \) is the energy of the electron in the final state (n=1).

### Step 2: Calculate the Energies for n=1 and n=2
The energies for the hydrogen-like atom are given by the formula:
\[ E_n = -\frac{Z^2 \cdot 13.6 \, \text{eV}}{n^2} \]
For \( n=1 \):
\[ E_{1} = -\frac{Z^2 \cdot 13.6}{1^2} = -13.6Z^2 \, \text{eV} \]
For \( n=2 \):
\[ E_{2} = -\frac{Z^2 \cdot 13.6}{2^2} = -\frac{Z^2 \cdot 13.6}{4} = -3.4Z^2 \, \text{eV} \]

### Step 3: Substitute Energies into the K-alpha Energy Formula
Now, substituting \( E_{1} \) and \( E_{2} \) into the K-alpha energy equation:
\[ E_{K_{\alpha}} = E_{2} - E_{1} \]
\[ E_{K_{\alpha}} = (-3.4Z^2) - (-13.6Z^2) \]
\[ E_{K_{\alpha}} = 10.2Z^2 \, \text{eV} \]

### Step 4: Convert Given Energy to eV
The given energy of the K-alpha X-ray is \( 7.46 \, \text{keV} \). Converting this to electron volts:
\[ 7.46 \, \text{keV} = 7.46 \times 10^3 \, \text{eV} \]

### Step 5: Set Up the Equation
Now we can set up the equation:
\[ 7.46 \times 10^3 = 10.2Z^2 \]

### Step 6: Solve for Z
Rearranging the equation to solve for \( Z^2 \):
\[ Z^2 = \frac{7.46 \times 10^3}{10.2} \]
Calculating this gives:
\[ Z^2 \approx 730.39 \]
Taking the square root:
\[ Z \approx \sqrt{730.39} \approx 27.02 \]

### Step 7: Identify the Element
The atomic number \( Z \approx 28 \) corresponds to Nickel (Ni), which is the element from which the K-alpha X-ray is emitted.

### Final Answer
The sample is made of Nickel (Ni).

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