Question

The energy of a silver atom with a vacancy in K shell is 25.31 keV, in L shell is 3.56 keV and in M shell is 0.530 keV higher than the energy of the atom with no vacancy. Find the frequency of K_α, K_β and L_α X-rays of silver.

Answer 1

Given:
Energy of electron in the K shell, E_k = 25.31 keV
Energy of electron in the L shell, E_L = 3.56 keV
Energy of electron in the M shell, E_M = 0.530 keV

Let f be the frequency of K$\alpha$ X-ray and f₀ be the frequency of K_β X-ray.
Let f₁ be the frequency of L_α X-rays of silver.

∴ K_α = E_K − E_L = hf
Here, h = Planck constant
f = frequency of K$\alpha$ X-ray

$$\begin{gathered} f=\frac{E_{\mathrm{K}}-E_{\mathrm{L}}}{h} \\ f=\frac{(25.31-3.56)}{6.63\times {10}^{-34}}\times 1.6\times {10}^{-19}\times {10}^3 \\ f=\frac{21.75\times {10}^3\times {10}^{15}}{6.67} \\ f=5.25\times {10}^{18}\mathrm{Hz} \\ {\mathrm{K}}_{\mathrm{\beta }}=E_{\mathrm{K}}-E_{\mathrm{M}}=h{f}_0 \\ \Rightarrow f_0=\frac{E_{\mathrm{K}}-E_{\mathrm{M}}}{h} \\ \Rightarrow f_0=\frac{(25.31-0.53)}{6.67\times {10}^{-34}}\times {10}^3\times 1.6\times {10}^{-19} \\ \Rightarrow f_0=5.985\times {10}^{18}\mathrm{Hz} \end{gathered}$$

$$\begin{gathered} K_L=E_L-E_M=h{f}_1 \\ {f}_1=\frac{E_L-E_M}{h} \\ {f}_1=\frac{3.56-0.530}{6.63\times {10}^{-34}}\times {10}^3\times 1.6\times {10}^{-19} \\ {f}_1=7.32\times {10}^{17}\mathrm{Hz} \end{gathered}$$