Answer
Observed scattering angle is $54^{0}$
Work Step by Step
Given that
wavelength of incident X-ray $\lambda_{0}=0.280nm$
wavelength of X-ray after scattering $\lambda=0.281nm$
Wavelength shift or Compton shift is given by
$\Delta \lambda= \lambda- \lambda_{0}=\lambda_{C} (1-cos\theta)$
where $\theta$ is scattering angle
for scattering by electron $\lambda_{C}=\frac{h}{m_{e}c}=2.43\times10^{-3}nm$
$\lambda_{0}=0.280nm$ ,$\lambda=0.281nm$
putting values in above equation we will get
$0.281nm- 0.280nm=2.43\times10^{-3}nm(1-cos\theta)$
or $(1-cos\theta)=\frac{ 0.001}{2.43\times10^{-3}}=0.4115$
or $cos\theta=1-0.4115=0.5885$
$\theta=cos^{-1}(0.5885)=53.94^{0}\approx54^{0}$
