Assume the Earth\'s atmosphere has a uniform temperature of 20.0 degree C and un

Question

Assume the Earth's atmosphere has a uniform temperature of 20.0 degree C and uniform composition, with an effective molar mass of 28.9g/mol. (a) Show that the number density of molecules depends on height y above sea level according to n_V = n_0 e^-m_0 gy/k_B^T, where n_0 is the number density at sea level (where y = 0). This result is called the law of atmospheres. (b) Commercial jetliners typically cruise at an altitude of 11.0km. Find the ratio of the atmospheric density there to the density at sea level. (c) The average height of a molecule in the Earth's atmosphere is given by y_avg = integral^infinity_0 yn_V (y) dy/integral^infinity_0 n_V (y) dy = integral^infinity_0 ye^-m_0 gy/k_B^T dy/integral^infinity_0 e^-m_0 gy/k_B^T dy. Prove that this average height is equal to k_B^T/m_0 g. (d) Evaluate the average height, assuming the temperature is 10.0 degree C and the molecular mass is 28.9 u, both uniform throughout the atmosphere.

Explanation / Answer

a )

we have equation

nV ( E ) = no exp ( - E / KB T )

and

n ( h ) = no exp ( - ghm / KBT )

now

n ( h ) / no = exp ( - ghm / KBT )

b )

n ( h ) / no = exp ( - ghm / KBT )

n ( h ) / no = exp ( - 9.8 X 0.0289 X 11000 / 8.314 X 293.15 )

n ( h ) / no = 0.2781

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