At what temperature will the root-mean-square velocity of helium molecules reach the first cosmic velocity

At what temperature will the root-mean-square velocity of helium molecules reach the first cosmic velocity (7.9 km / s)? The mass of a helium atom is 6.4 · 10 ^ -27 kg. Take Boltzmann’s constant equal to k = 1.38 · 10 ^ -23J / K.

The relationship between the mean square velocity vsk of molecules and the temperature T:

vsk = (3kT / m0) ^ (1/2), where k is the Boltzmann constant, m0 is the mass of the molecule.

We square both parts:

vsk ^ 2 = 3kT / m0;

T = (m0 * vsk ^ 2) / 3k =

= (6.4 * 10 ^ (- 27) kg * (7900) ^ 2 (m / s) ^ 2) / (3 * 1.38 * 10 ^ (- 23) J / K =

= (6.4 * 10 ^ (- 4) kg * 62.41 * 10 ^ 6 (m / s) ^ 2) / (4.14 (kg * m ^ 2) / (K * s ^ 2)) = 9 647.923 K.

Answer: 9 647.923 K.