Question

Show that the stability criterion requires that:

a) S 0 P,N T,N c)

Answer 1

these are called concave function if

d²U/dX²  < 0

G = U - TS + PV

a) thus

02G or2

here specific heat is positive hence the derivative is negative, thus a concave function.

b) under constant pressure,

GVKT

since the compressibility is a positive quantity, the derivative is negative.

c)

by eq,

$(-C_P/T)(-V K_T) - (-S \times V )^2$

since the first term is a positive quantity, the whole term is positive.