A free atom of iron emits `K_alpha` X-rays of energy 6.4 keV. Calculate the recoil kinetic energy of the atom. Mass of and iron atom `= 9.3 xx 10 ^(-26) kg`.

Answer

To solve the problem of calculating the recoil kinetic energy of a free iron atom emitting K_alpha X-rays of energy 6.4 keV, we can follow these steps:

### Step 1: Convert the energy of the photon from keV to joules
The energy of the emitted X-ray photon \( E_p \) is given as 6.4 keV. To convert this to joules, we use the conversion factor \( 1 \text{ eV} = 1.6 \times 10^{-19} \text{ J} \).

\[ E_p = 6.4 \text{ keV} = 6.4 \times 10^3 \text{ eV} = 6.4 \times 10^3 \times 1.6 \times 10^{-19} \text{ J} \]

Calculating this gives:

\[ E_p = 6.4 \times 1.6 \times 10^{-16} = 1.024 \times 10^{-15} \text{ J} \]

### Step 2: Calculate the momentum of the photon
The momentum \( p \) of a photon can be calculated using the formula:

\[ p = \frac{E_p}{c} \]

where \( c \) is the speed of light, approximately \( 3 \times 10^8 \text{ m/s} \).

Substituting the values:

\[ p = \frac{1.024 \times 10^{-15} \text{ J}}{3 \times 10^8 \text{ m/s}} = 3.4133 \times 10^{-24} \text{ kg m/s} \]

### Step 3: Apply conservation of momentum
According to the principle of conservation of momentum, the momentum of the photon is equal to the momentum of the recoiling iron atom:

\[ p_{\text{photon}} = p_{\text{atom}} \]

### Step 4: Calculate the recoil kinetic energy of the iron atom
The recoil kinetic energy \( K \) of the iron atom can be calculated using the formula:

\[ K = \frac{p^2}{2m} \]

where \( m \) is the mass of the iron atom, given as \( 9.3 \times 10^{-26} \text{ kg} \).

Substituting the values:

\[ K = \frac{(3.4133 \times 10^{-24})^2}{2 \times 9.3 \times 10^{-26}} \]

Calculating \( (3.4133 \times 10^{-24})^2 \):

\[ (3.4133 \times 10^{-24})^2 = 1.165 \times 10^{-47} \text{ kg}^2 \text{ m}^2/\text{s}^2 \]

Now substituting this back into the kinetic energy formula:

\[ K = \frac{1.165 \times 10^{-47}}{2 \times 9.3 \times 10^{-26}} = \frac{1.165 \times 10^{-47}}{1.86 \times 10^{-25}} \approx 6.25 \times 10^{-23} \text{ J} \]

### Step 5: Convert the kinetic energy to electron volts
To convert joules to electron volts, we divide by \( 1.6 \times 10^{-19} \):

\[ K = \frac{6.25 \times 10^{-23}}{1.6 \times 10^{-19}} \approx 0.390625 \text{ eV} \approx 0.391 \text{ keV} \]

### Final Answer
The recoil kinetic energy of the iron atom is approximately **0.391 keV**.

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