Question

Much of the gas near the Sun is atomic hydrogen. Its temperature would have to be (T) for the average velocity (v_avg) to equal the escape velocity from the Sun. What is that velocity?

a) (300 , km/s)
b) (500 , km/s)
c) (700 , km/s)
d) (1000 , km/s)

Answer 1

The escape velocity of atomic hydrogen near the Sun at a temperature of 1.5 × 107 K is determined by using the root-mean-square velocity formula for gaseous particles, which requires the Boltzmann constant and the mass of a hydrogen atom to calculate.

Step-by-step explanation:

The student's question is regarding the escape velocity of atomic hydrogen near the Sun at an extremely high temperature. Using the root-mean-square (rms) velocity for gaseous particles, we can determine that at a temperature of 1.5 × 107 K (15 million kelvins), the average velocity of atomic hydrogen would need to match the escape velocity of the Sun. The escape velocity is the speed that a particle must reach to escape the gravitational pull of a celestial body without further propulsion.

The formula we use to calculate the rms speed (Vrms) of a particle in a gas, where k is the Boltzmann constant, m is the particle mass, and T is the temperature, is:

Vrms = sqrt(3kT/m)

Since the temperature and mass of hydrogen are known quantities, we can use this formula to calculate the escape velocity. The actual calculation of this velocity requires additional information such as the mass of a hydrogen atom and the value of the Boltzmann constant, and it also involves square roots and large numbers, calculations typically done using a scientific calculator or suitable software.

We have not been provided with the atomic mass of hydrogen or the value of the Boltzmann constant in this context; however, this information is widely available and would typically be used to calculate the escape velocity. The options given (300, 500, 700, 1000 km/s) are suggestive of the scale of the velocities considered.