20) Thermal equilibrium
A one-component fluid has the fundamental equation
3 = CV‘/2(NU)‘/4, (1)
where C is a positive constant of suitable units and particle numbers are measured in
mole. To study thermal equilibrium, consider a closed composite system consisting of
two chambers, A and B. Originally the chambers are separated by a fixed, impermeable,
adiabatic wall and have the sa.me volume, 1 m3. One mole of the substance is placed
in chamber A, while two moles of the substance are placed in chamber B. The total
energy of the system is 600 J.
Na) Find an expression for the entropy of the composite system in terms of the total
energy, U = U4 + U3, and the fraction of energy in chamber A, y = UA/U-
(b) Use your favorite plotting program to plot the entropy (up to a constant) of the
composite system as a function of y; add a figure caption to your figure.
(c) The internal wall is now made diathermal and the system is allowed to come to
equilibrium. Calculate the equilibrium energy for each chamber. l‘
(d) Compare your answer with the result you obtain from the location of the maxi-
mum in your graph.
(20) Equilibrium under particle exchange V
To study equilibrium under particle exchange, consider a two-component fluid in a
closed composite system consisting of two chambers separated by a fixed, diathermal
wall that is permeable to the first component but impermeable to the second.
(a) Show that, in equilibrium, the following relations hold for the temperature T and
the chemical potential in of the first component
T(1) ._. 71(2), (2)
ul” = #32’. (3)
where the superscript (i), 2′ E {1, 2}, indicates the chamber.
(b) The chambers are filled with a two-component fluid with the fundamental equa-
tion
U3/2v N1 N2
l – – N« R1 -, 4
5 NA + NRln (N5/2R3/293/200 N112 n N ,._ n N
With N = N1+ N2.
Wht’I”(‘ Ii H.314 J/(nml K) is thv ith-nl ans mnstmit and P9. 0. and A are positive
(~unst.unt.s with .~untnhl«- units. Thv purtirlv numlwrs nrv rm-nsurod in mul. the
(‘ltt’U.’,_Y in Jutth-, nnd thv vuhinw in In”. ()rip,iImll_v. tho :~‘_’stvni is prepared with
thv fnlluwilip, pnmItwt«’I‘.s’
N{” =.- 0.5 lltui. Ni” (175 mul. ~’“’ = 5.0 x 1n*‘m1’. T*“ = 300 K. ts)
Ni?) *= 1.0 moi. N-in (Hit) Inul. Vi” = 5.1) x 10 3 m3. Tm = ‘2-50 K-
Aftor t-.quililn’ium is 1-st.nlilisiu.~d. what urv thv vnlm-s of the particle numbers. the
l.(‘Ill[)l’t‘tlUll‘(‘, and thv pn-ssiiro in thv vluunhvrs’.’
For the mitrnpy hmctions in vqimtiuns (1). mid (4)
i Sketch, by hand or using a plotting prugnun. S as ll function of L’ for constant
particle nutnlu-r(s) and vnlnlnc-.
Show that S is I! hnlllugvltt-utts first unlc-r tum-tmn nf the vxtvnsivo snrialilvsr
Citvck if S imt.i.-stir-.-c pustulntv IV
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