Question

The escape speed from the Earth is 11.2 km/s, so that a gas molecule traveling away from Earth near the outer boundary of the Earth's atmosphere would, at this speed, be able to escape from the Earth's gravitational field and be lost to the atmosphere. The composition of our atmosphere is approximately 78% dinitrogen (N₂),21% dioxygen (O₂), and other trace molecules. At what temperature is the rms speed of (a) dioxygen molecules, and (b) helium atoms equal to 11.2 km/s ? (c) Can you explain why our atmosphere contains oxygen but not helium?

Answer 1

The escape velocity of 11.2 km/s is equivalent to 11200 m/s and 40320 km/h. Oxygen and helium would need to reach extremely high temperatures to attain the same speed. Oxygen remains in the atmosphere because it is heavier, whereas lightweight helium can escape into space. So the correct answer is option B.

Step-by-step explanation:

The escape velocity of any object from Earth is 11.2 km/s. To express this speed in meters per second (m/s), we multiply by 1000, getting 11200 m/s. To express it in kilometers per hour (km/h), we multiply by 3600 (the number of seconds in an hour), obtaining 40320 km/h.

The temperature at which dioxygen (O₂) molecules have a root-mean-square (rms) speed equal to Earth's escape velocity can be calculated using the formula for the kinetic theory of gases, which relates the rms speed (Vrms) to the temperature (T) and molar mass (M) of the gas:

Vrms = √(3kT/M)

where k is the Boltzmann constant (1.38 x 10^-23 J/K) and M is the molar mass of oxygen in kilograms per mole (0.032 kg/mol). Rearranging the formula to solve for temperature and substituting Vrms with 11.2 km/s (converted to 11200 m/s) will give the required temperature.