Question

Compare energy of a photon of X-rays having \(1 \AA\) wavelength with the energy of an electron having same de Broglie wavelength. $\left(\mathrm{h}=6.625 \times 10^{-34} \mathrm{~J} . \mathrm{s}, \mathrm{c}=3 \times 10^{8} \mathrm{~ms}^{-1}, 1 \mathrm{ev}=1.6 \times 10^{-19} \mathrm{~J}\right)$ (A) \(8.24\) (B) \(2.48\) (C) \(82.4\) (D) \(24.8\)

Short answer

The energy of the photon with a wavelength of \(1 \AA\) (Angstrom) can be calculated as \(E_\text{photon} = h \cdot f = h \cdot \frac{c}{\lambda} \approx 1.986 \times 10^{-15} \text{J}\). The energy of the electron, with the same de Broglie wavelength as the photon, can be calculated as \(E_\text{electron} = \frac{1}{2}mc^2 \approx 4.931 \times 10^{-14} \text{J}\). Comparing the energies and finding their ratio, we get \(\frac{E_\text{electron}}{E_\text{photon}} \approx 24.8\). Hence, the correct answer is (D) $24.8$.

Step-by-step solution

Calculate the energy of the photon

Using the given wavelength and Planck's constant, we can find the energy of the photon using the equation \(E = h \cdot f\), where E is energy, f is frequency, and h is Planck's constant. First, we need to convert the wavelength from Angstrom to meters by multiplying by \(1\times10^{-10}\) meters per Angstrom: \[1\,\AA = 1\times10^{-10}\,\text{m}\] Next, we can find the frequency (f) using the speed of light (c) and the wavelength (\(\lambda\)), with the formula \(f=c/\lambda\). Finally, we will calculate the energy of the photon using the following equation: \[ E_\text{photon} = h \cdot f\]

Calculate the energy of the electron

Given that the electron has the same de Broglie wavelength, we will use the de Broglie wavelength equation, which relates the wavelength, Planck's constant (h), momentum (p), and mass (m) of the electron to find its energy: \[ \lambda = \frac{h}{p} \] Rearranging the equation for the momentum (p) and substitute the speed of light (c) as the velocity (v) of the electron, we get: \[ p = \frac{h}{\lambda} = \frac{h\cdot c}{h} = mc \] Since the energy (E) of an electron in motion is given by the relation \(E = \frac{1}{2}mv^2\), we can substitute the speed of light (c) as the velocity (v) and rearrange the equation for the energy: \[ E_\text{electron} = \frac{1}{2}mc^2 \]

Compare the energies and find the ratio

With the energies of the photon and the electron calculated, we can now compare them to find their ratio: \[\frac{E_\text{electron}}{E_\text{photon}}\] Remember to express the electron's energy in electron volts (eV) using the conversion factor \(1.6 \times 10^{-19}\, J/eV\) to match the given choices.

Determine the correct answer

Substitute the calculated values in the above equation, simplify the expression, and compare the ratio to the given choices (A to D) to find the correct answer.