The energy of X-ray photon of wavelength 1.65 Å is
`(h=6.6 xx 10^(-34)J-sec,c=3xx10^(8)ms^(-1),1eV=1.6xx10^(-19)J)`

Options

A. 3.5 keV
B. 5.5 keV
C. 7.5 keV
D. 9.5 keV

Answer

To find the energy of an X-ray photon with a wavelength of 1.65 Å, we can use the formula for the energy of a photon:

\[ E = \frac{h \cdot c}{\lambda} \]

Where:
- \(E\) is the energy of the photon,
- \(h\) is Planck's constant (\(6.6 \times 10^{-34} \, \text{J s}\)),
- \(c\) is the speed of light (\(3 \times 10^{8} \, \text{m/s}\)),
- \(\lambda\) is the wavelength of the photon.

### Step 1: Convert the wavelength from Ångströms to meters
The given wavelength is \(1.65 \, \text{Å}\). We need to convert this to meters:

\[ 1 \, \text{Å} = 1 \times 10^{-10} \, \text{m} \]

Thus,

\[ \lambda = 1.65 \, \text{Å} = 1.65 \times 10^{-10} \, \text{m} \]

### Step 2: Substitute the values into the energy formula
Now, we can substitute the values of \(h\), \(c\), and \(\lambda\) into the energy formula:

\[ E = \frac{(6.6 \times 10^{-34} \, \text{J s}) \cdot (3 \times 10^{8} \, \text{m/s})}{1.65 \times 10^{-10} \, \text{m}} \]

### Step 3: Calculate the energy in Joules
Calculating the numerator:

\[ E = \frac{(6.6 \times 3) \times 10^{-34 + 8}}{1.65 \times 10^{-10}} \]

Calculating \(6.6 \times 3 = 19.8\):

\[ E = \frac{19.8 \times 10^{-26}}{1.65 \times 10^{-10}} \]

Now, dividing:

\[ E = 19.8 \div 1.65 \times 10^{-26 + 10} = 12 \times 10^{-16} \, \text{J} \]

Thus,

\[ E \approx 1.2 \times 10^{-15} \, \text{J} \]

### Step 4: Convert energy from Joules to electron volts
To convert Joules to electron volts, we use the conversion factor \(1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J}\):

\[ E \text{ (in eV)} = \frac{1.2 \times 10^{-15} \, \text{J}}{1.6 \times 10^{-19} \, \text{J/eV}} \]

Calculating this gives:

\[ E \text{ (in eV)} = \frac{1.2}{1.6} \times 10^{4} \approx 0.75 \times 10^{4} \, \text{eV} = 7500 \, \text{eV} = 7.5 \, \text{keV} \]

### Final Answer
The energy of the X-ray photon is approximately **7.5 keV**.

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