Temperature & Kinetic Energy (DP IB Physics: SL): Revision Note

The surface temperature of the Sun is 5800 K and contains mainly hydrogen atoms.

Calculate the average speed of the hydrogen atoms, in km s⁻¹, near the surface of the Sun.

Answer:

Step 1: List the known quantities

• Temperature, T = 5800 K

• Mass of a hydrogen atom = mass of a proton, m_p = 1.673 × 10⁻²⁷ kg

• Boltzmann constant, k_B = 1.38 × 10⁻²³ J K⁻¹

Step 2: Equate the equations relating kinetic energy with temperature and speed

• Average kinetic energy of the molecules:  ${\overline{E}}_k=\frac{3}{2}{k}_{B}T$

• Kinetic energy:  $E_k=\frac{1}{2}m{v}^2$

$\frac{3}{2}{k}_{B}T=\frac{1}{2}{m}_p{v}^2$

Step 3: Rearrange for average speed and calculate

$\frac{3k_{B}T}{m_p}=v^2$

$v=\sqrt{\frac{3k_{B}T}{m_p}}=\sqrt{\frac{3\times \left(1.38\times {10}^{-23}\right)\times 5800}{1.673\times {10}^{-27}}}$

Average speed:  v = 11 980 m s⁻¹ = 12 km s⁻¹