Question

A substance whose state is pecified by P,V,T can exist in two distinct phases. At a given temp T the two phases can coexist if the pressure is P(T).

The following information is known about the two phases. At the temps and pressures where they can coexist in equilibrium,

i) There is no difference in the specific volume of the two phases

ii) there is no difference in the specific entropy of the two phases

iii) the specific heat cp and the volume expansion coefficient a(alpha) are diffrent for the two phases

a) Find dP(T)/dT as a fuction of T.

b) What is the qualitative shape of the transition region in the P-V diagram? In what way is it different from that of an ordinary gas-liquid transition? ( The phase transition we have described is a second-order transition.)

Answer 1

a) If two face is in equilibrium have different molar volume, there free energy will increase by different amount when pressure change at fixed temp T. Only way to maintain equilibrium of different pressure is to change temperature as well

For two phase in Equilibrium

G₁=G_s

dG₁=dG_s

for infinite change in T and P ( so that the system remain in equilibrium)

dG₁=V₁dP-S₁dT

dG_s=V_sdS-S_sdT

V₁dP-S₁dT=V_sdP-S_sdT

At Equilibrium =-=0

and =

the clapeyron equation.

(b) Quality is nothing other than the fraction of the mass in a vapor- liguid system that is in the vapure phase

for define quantity. consider the closed vessel which contain a mixture of vapure and liquid of a pure substance.

the mass of vapor = M_v nad total volume = V_v

Mass of Liquid = M_l and toal volume = V_l

Then the toal mass in the vessel is the sum of the masses of vapor and liquid.

M=M_l+M_V -(1)

Total volume of the vesele is sum of the volume of two phase

V=V_l+V_V -(2)

relation between specific volume v and taoal volume V

V=vM -(3)

Now consider liquid phase. the specific volume of the liquid when two phase are present, will alway be v_f

so V_l = v_f M_{l    -(4)}

same as above write vapure phase

V_v = v_gM_V    - (5)

substituting eg 3,4,and 5 in to eq- 2 nad dividing through M and then eleminating M using eq - 1

we get

- (6)

at this point quality x is defined

x=

by substituting eg 7 into eq 6

we get

V = x v_g + (1-x) v_f

we get relation between specific volume V and quality x in the two phase region

note* quality is only define for two phase regions

When x= 0

v= v_f

when x = 1

v = v_g

any point on hte saturated liquid curve has quality zero and any point on the saturated vapure curve has a quality of one.

Point inside teh two phase region will have quakty between zero and one.