In a planetary atmosphere in hydrostatic equilibrium, the number density ’n’ of

ID: 1434471 • Letter: I

Question

In a planetary atmosphere in hydrostatic equilibrium, the number density ’n’ of a gas at a height h, n(h) is given by

n(h) = n(0) exp [- (h-h0)/H]

where H is called the ‘scale height’ (=kT/Mg)

k is the Boltzmann’s constant

T the temperature(2000 K),

M the atomic mass; M is 4 atomic mass units (2 protons & 2 neutrons) for helium

g is the gravity(= 0.8 x earth’s gravity, for Venus).

n (0) is the number density at a reference height of h0.

By what fraction does the density of helium changes between the altitudes of 500 km and 2000 km in the atmosphere of Venus?

Lets first converst 2000 and 500 km into meter for consistency and assume ho =0

Required fraction is given by:

n(2000000m)/ n (500000m)= exp (-2000000/H ) / exp (-500000/H)

n(2000000m)/ n (500000m)= exp (-2000000/H + 500000/H)

= exp (-1500000/H)

=exp (-1500000 x Mg/kT )

= exp[-1500000 * (4*1.6605 * 10-27 * 0.8*9.8)/(1.38064852 * 10-23 * 2000 )]

=exp(-2.8287)

= 0.059 Answer

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