3.D.2.c. Sublimation Curve

The coexistence curve between the solid and the gas phases is called the sublimation curve [see Fig.3.4].  The relevant Clausius-Clapeyron is

        ( dP dT ) coex = h g h s T( v g v s ) = Δ h sg TΔ v sg                       (3.18)

where Δ h sg  is the molar latent heat of sublimation.

Neglecting v s  and using the ideal gas approximation for v g , we have

Δ v sg v g = RT P

so that

        ( dP dT ) coex = PΔ h sg R T 2                                             (3.19)

 

Thus, along the sublimation curve,

        Δ h sg =R T 2 dT dP P =R dlnP d( 1 T )                  (3.20)

Exercise 3.3

Near triple point of NH3, we have

sublimation curve:           lnP=27.79 3726 T

vaporization curve:         lnP=24.10 3005 T

1.         Find T and P of triple point.

2.         Find latent heats of sublimation and vaporization.

Answer

1.     The sublimation and vaporization curves meet at the triple point.  Hence,

                27.79 3726 T t =24.10 3005 T t           Þ    T t =195.4K

                ln P t =27.79 3726 195.4 =6.13kPa

2.     For the sublimation curve,

                dP PdT = 3726 T 2            Þ            Δ h sg R T 2 P ( dP dT ) coex = 3726 R 31  kJ/mol

        For the vaporization curve,

                dP PdT = 3005 T 2             Þ            Δ h lg 3005 R 25  kJ/mol

Low Temperature Case

Using

        ds= ( s T ) P dT+ ( s P ) T dP

                = c P T dTv α P dP              where      ( s P ) T = ( v T ) P =v α P

the molar enthalpy can be written as

        dh=Tds+vdP

                = c P dT+v( 1T α P )dP                    (3.21)

Now, at low temperatures, the vapor pressure along the sublimation curve is very low.

Thus, we can neglect the pressure variation and write

        dh c P dT                 on sublimation curve

so that

        h= h 0 + T 0 T c P dT

where h= h 0  at T 0 .

The latent heat of sublimation is therefore

        Δ h sg = h g h s Δ h sg 0 + T 0 T ( c P g c P s )dT              (3.22)

Eq(3.20) can thus be integrated to give

        ln P P 0 = 1 R T 0 T Δ h sg T 2 dT

                = 1 R T 0 T 1 T ' 2 [ Δ h sg 0 + T 0 T' ( c P g c P s )dT'' ]dT'

                = Δ h sg 0 R ( 1 T 0 1 T )+ 1 R T 0 T 1 T ' 2 [ T 0 T' ( c P g c P s )dT'' ]dT'            (3.23)

which can be used to extrapolate experimental measurements.