For an object to escape from the surface of the Earth it must be moving at a spe
ID: 1344965 • Letter: F
Question
For an object to escape from the surface of the Earth it must be moving at a speed of about 11 km/s. Given that the mass of a hydrogen molecule is 3.3×10-27 kg, we want to determine the approximate Fahrenheit temperature at which much of the hydrogen in the atmosphere would escape.
a. What is the mathematical relationship between the average KE of the molecules and the absolute temperature of the gas?
b. What is the KEesc of a hydrogen molecule moving at the escape speed?
c. What is the absolute temperature of the gas when its average KE equals KEesc?
d. What is the equivalent Celcius temperature of the gas?
e. What is the equivalent Fahrenheit temperature of the gas?
f. Why has the Earth’s atmosphere lost most of its hydrogen in comparison to its other gases?
Explanation / Answer
a.
The average kinetic energy of the molecule in terms of temeprature is,
Kavg = (3/2)kT
b.
The average kinetic energy is,
Kesc = 0.5mv2 = 0.5*3.3×10-27*11x103* 11x103 = 2.0e-19 J
c.
If the KEesc is equal to the average kinetic energy, we get
(3/2)kT = 0.5mv2
Hence, the temperature is,
T = mv2 / 3k = 3.3×10-27*11x103* 11x103 / 3 * 1.38x10-23 = 9644.9 K
d.
the equivalent Celcius temperature of the gas:
T = 9644.9 K = 9644.9 - 273 = 9371.9 oC
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