Chapter 5, Problem 101A

Interpretation Introduction

Interpretation:

The frequency and the wavelength of an X-ray photon having $3.01\times {10}^{-18}\text{J}$ of energy needs to be determined.

Concept introduction:

X-ray is a high energy and high frequency electromagnetic radiation. It has wavelength ranging from $\text{0}\text{.01}\text{to}\text{10}\text{nm}$ and frequencies from $3\times {10}^{19}\text{to}3\times {10}^{16}\text{Hz}$. X-rays are found to reside between gamma rays and ultraviolet radiation on the electromagnetic spectrum.

Answer to Problem 101A

Frequency and wavelength of the X-ray photon is $4.54\times {10}^{15}{\text{s}}^{-1}$ and $6.60\times {10}^{-8}\text{m}$ respectively.

Explanation of Solution

Frequency can be calculated by using the equation given below where E represents the energy of the photon i.e. $3.01\times {10}^{-18}\text{J}$ and h represents the Planck’s constant i.e. $6.626\times {10}^{-34}\text{J}\text{.s}$.

$\begin{array}{c}{\text{E}}_{\text{photon}}=\text{h}v\\ v=\frac{{\text{E}}_{\text{photon}}}{\text{h}}\\ =\frac{3.01\times {10}^{-18}\text{J}}{6.626\times {10}^{-34}\text{J}\text{.s}}\\ =4.54\times {10}^{15}{\text{s}}^{-1}\end{array}$

Hence, the frequency of the X-ray photon is $4.54\times {10}^{15}{\text{s}}^{-1}$.

Wavelength of the X-ray photon can be determined by dividing the speed of the light by frequency. Speed of the light is $3.00\times {10}^8\text{m/s}$ and frequency is $4.54\times {10}^{15}{\text{s}}^{-1}$.

$\begin{array}{c}\text{λ}=\frac{c}{v}\\ =\frac{3.00\times {10}^8\text{m/s}}{4.54\times {10}^{15}{\text{s}}^{-1}}\\ =6.60\times {10}^{-8}\text{m}\end{array}$

Hence, the frequency of the X-ray photon is $6.60\times {10}^{-8}\text{m}$.

Conclusion

Thus, the frequency and wavelength of the X-ray photon is $4.54\times {10}^{15}{\text{s}}^{-1}$ and $6.60\times {10}^{-8}\text{m}$ respectively.