Question

(a) The probability distribution for the height h of molecules of mass m in our atmosphere, assuming a constant temperature, is given by

P(h) ∝ e^-mgh/k_BT

Normalize the probability distribution and derive an expression for the average height of molecules in the atmosphere.

(b) Estimate the height of Oxygen molecules in the atmosphere, and of Nitrogen molecules.

You may use the result $\int_{0 }^{\infty}x^{n}e^{-x}dx = n!$

Answer 1

Pih The pocobability distribution for the height h of molecules of mass is in our atomosphere, is given by, dane-mgh/ket nigh/RET >> Pah) = A, e Now, the normalization Condition is given by, ibch) dh = 1 A jorezmgh/kat dhe Let; amgh/kot ex. → dh a bheann de 2 mg zing 29 PCW = 12 mg. e-mgh/ket a. The average height of molecules of in atmosphere to W E por ho pch).dh a ne2mgh/kat ah 2x

Let, singh x. kəТ .. dh a kot dx 3 ( x cºsx =1 - < = 2 Tag

b. We have, ko 1.3807 y 18-23 J.S.K' g = 9.8 m/s? tour oxygen molecules, me 32% 1.661x10 Now for a fixed temperature , T 5, the height of oxygen molecules in atmosphere wi he 71.3807x10-23x1 3 12 | 2x32 x 1.661810–24x9.8 . 0.00153.7312 m. be for Nitrogen molecules, ma 28 X16661X10249 B. The height of oxygens molecules in atmosphere will be iho 1.3807 x 10-23 XT 228% 1.661x1072489.8 0.015/XT372 m are used .