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Broadening of Spectral lines Due to Temperature

The temperature of a gas is due to random kinetic energy of its atoms and molecules. For a gas at temperature  
\[T\]
  consisting of atoms of mass  
\[m\]
  with root mean square speed equal to  
\[v_{RMS}\]
  we can write
\[Average Kinetic Energy= \frac{1}{2} m v^2_{RMS} = \frac{3}{2} kT \rightarrow v_{RMS} = \sqrt{\frac{3kT}{m}}\]

The surface temperature of our Sun is about 6000 degrees Kelvin and the surface consists mostly of ionized hyrogen - electrons and protons.
\[v_{RMS} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 6000}{1.67 \times 10^{-27}}} =12000 m/s \]

This will result in a broadening of the spectral lines
Considered as a redshift we have  
\[z=\pm \frac{12200}{3 \times 10^8} = \pm 4.07 \times 10^{-5}\]
.