Question

Consider a planet with an average surface temperature of 735 k. will it lose water molecules (molecular weight = 34) in its atmosphere to space if the planet's escape speed is 10.4 km/s?

Answer 1

Answer: No, the planet will not lose water molecules in its atmosphere to space

Explanation:

Given,

The temperature of the planet is (T) = 735 K

The molecular weight of the water molecule is (m) = 34 g = 0.034 kg

The escape speed of the planet (V) = 10.4 km/s = 10400 m/s

The Boling point of water is =  100 degrees Celcius = ( 273+100) = 373 K

Therefore, the planet's temperature (735 K) > Boling temperature of water (100 K)

Thus water molecules will act as gas molecules.

Hence, the kinetic energy of water molecules is = $\frac{3}{2}kT$  

where k = Boltzman Constant = 1.381 × $10^{-23}$ $m^{2} .kg.K^{-1} s^{-2}$

If v be the velocity of water molecules then,

    $\frac{1}{2}mv^2 = \frac{3}{2}kT$

⇒$v^{2}= \frac{3*1.381*10^{-23}*735 }{0.034}=8.956*10^{-19}\\ v=9.464*10^{-10} m/s$

As it is shown that the velocity of water molecule is very very less than the escape speed of that planet ($(v < < V)$.

Therefore, the planet will not lose water molecules in its atmosphere to space as velocity of water molecules less than the escape speed .

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