1. Show that for an ideal gas, the chemical potential at fixed temperature T var

Question

1. Show that for an ideal gas, the chemical potential at fixed temperature T varies with pressure as: = ° + RT In where ° is the value of at pressure P," 2. Use your result from question 1 to discuss how the pressure in the earth's atmosphere changes with altitude above sea level. Making clear the assumptions that you make, show that (subject to certain assumptions) the pressure in the earth's atmosphere varies with height h above sea level as: p(h)-P,exp the pressure at sea-level. Give an expression for he . l-A) where Po is

Explanation / Answer

1. '

chemical potential is given by

mu = dH/dN

now, for a fixed temperature T

H = -W ( H is enthalapy change, W is work done)

and W = NRT*ln(Vf/Vi)

now for isothermal process

PiVi = PfVf

hence W = NRT*ln(Pi/Pf)

mu = -RT*ln(Pi/Pf) = RTln(Pf/Pi)

now, muo = RT*ln(Pi/Po)

then

mu = muo + RTln(Pf/Po)

now, Pf = P

hence

mu = muo + RTln(P/Po)

where muo is potential at pressure Po

2. now, we know that

potenital falls with height as -mgh

hence

-mgh = 0 + RT*ln(P/Po) [ assuming 0 potential at sea level]

then

P = Po*e^(-mgh/RT)

from the given formula

hc = RT/mg

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