Answer
The frequency of the scattered photon is $2.4\times 10^{19}~Hz$
Work Step by Step
We can find the initial wavelength:
$\lambda_i = \frac{c}{f_i}$
$\lambda_i = \frac{3.0\times 10^8~m/s}{3.0\times 10^{19}~Hz}$
$\lambda_i = 1.00\times 10^{-11}~m$
$\lambda_i = 10.0~pm$
We can find the Compton shift in wavelength:
$\Delta \lambda = \frac{h}{mc}~(1-cos~\theta)$
$\Delta \lambda = \frac{6.626\times 10^{-34}~J~s}{(9.1\times 10^{-31}~kg)(3.0\times 10^8~m/s)}~(1-cos~90^{\circ})$
$\Delta \lambda = (2.427~pm)~(1)$
$\Delta \lambda = 2.427~pm$
We can find the wavelength of the scattered photon:
$\Delta \lambda = \lambda_f-\lambda_i$
$\lambda_f = \lambda_i+\Delta \lambda$
$\lambda_f = (10.0~pm)+ (2.427~pm)$
$\lambda_f = 12.427~pm$
We can find the frequency of the scattered photon:
$f_f = \frac{c}{\lambda_f}$
$f_f = \frac{3.0\times 10^8~m/s}{12.427\times 10^{-12}~m}$
$f_f = 2.4\times 10^{19}~Hz$
The frequency of the scattered photon is $2.4\times 10^{19}~Hz$
