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**Unformatted text preview: **1. Consider a dilute gas of particles in the atmosphere. Near the earth's surface, the force on a particle of mass m may be taken as a constant, F =-mgJ, where J is a unit vector in the vertical direction. a.) By enforcing the number constraint on the ideal gas ((N) = LsfCEsIJ1.1 r), with fCEsI J1., r) = Exp (~) Exp (- ~ )), derive the expression /lint = r log (~) for the chemical ( ) 3/2 potential of an ideal gas. The quantum concentration is nQ = 2:: 2 (5 points) b.) Write an expression for the total chemical potential for a gas particle in terms of the internal chemical potential and external chemical potential. (5 points) c.) What conditions must be satisfied for the total chemical potential to be uniform with height? (5 points) d.) The density of particles at height y = 0 is given as n(O). Calculate the density n(y) of the dilute gas of atoms of mass m at temperature r as a function of the height y above the earth's surface. (10 points) (' ( ) /A / '7-- -~. /r- tv-=- 2..f.( 'i':J",....}{, 7 ) ..,... ~ /~ /l' ::. e e ~ , 'I J 7 'L L ~ .-.:... ~ or-&.. C~'ca:.( PJ~~.....-u ~.~ Irn(>~ eM. ~ r,.rf6r .... 'r Je.L ~ eAVO Ii. i-/-er.c.e. N ~ e:n- J: ':': =. e./.( ( 'f i. wh~ ~ I ir Jo- P (V ~-h 6A ~t"-C . 4 < 'h. .~ a. f (f1\j\4... rar-~c.lL I;' a. bcq( 0+ V"\~"'I~ V d"J... . L _ ~ + JJ _ ('no. f( M ... I 1:....

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