Topic 14 - Vaporization

8/20/2019 Topic 14 - Vaporization

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School of Physics and Astronomy

Junior Honours Thermodynamics GJA 2015-2016

Lecture TOPIC 14 (Finn: 9.1, 9.2, 9.3)

Synopsis: Phases: the PVT surface and the equilibrium condition for two phases.

Equation of state for water, given any two of P, V, T gives us the third, and the associated phase(s).

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The equilibrium condition for two phases at fixed temperature and pressure

In thermal equilibrium, P and T are constant throughout the system, so G must be at a minimum.

Consider a system of two coexisting phases in equilibrium with specific (i.e. per unit mass) Gibbs free

energy g1 and g2 and the masses M 1 and M 2: the total Gibbs free energy for the system is G = g1 M 1 +g2 M 2.During a phase transition, the proportions of the two phases change, but T and P do not.

Conservation of mass: dM 1 +dM 2 = 0,

combined with dG = g1dM 1 +g2dM 2 = VdP−SdT = 0 (for T and P fixed) gives

g1 = g2

Conclusion: when two component phases of a system coexist at constant temperature and pressure, their specific

Gibbs functions are equal. If g for one phase is lower than the other then only the phase with the lower g is

present. Under a given external temperature, coexistence is only possible a one fixed external pressure.

The phase boundary is a line on P-T diagram

With fixed volume boundaries, equilibrium still requires P and T to be the same in each phase, and g1 = g2. Sothe pressure is defined as above, however the volume constraint V = v1 M 1 +v2 M 2 can be met by adjusting the

amounts of the two phases.

There is an area of phase coexistence on a V-T diagram (and on P-V)

1. Phase diagram The projection of the equation of state surface on to the PT is called a phase diagram

because it shows which phase is stable for given T,P, and the phase boundaries between pairs of phases.

2. Fixed volume

The projection on to the TV plane also shows the boundaries, but this time the coexistence at fixed

volume at the phase transitions is directly evident. Projection onto the PV plane may even allow two

solutions (e.g. water at ambient pressure has a density maximum, so there may be two temperatures atwhich P,V are the same. In this unusual case, specifying two conjugate variables P and V is not enough

to determine the state of the system.

3. Conjugate variables and positive quantities

P,V are conjugate (appear together in free energy expressions), if ∂ P/∂ V > 0 free energy can be spon-

taneously reduced by volume collapse, implying the system is not in equilibrium. So negative thermal

expansion is allowed, but negative bulk modulus is not. The TS equivalent of this statement (positive

heat capacity) is another statement of the Second Law.

4. Triple point

Ice, water and water vapour coexist at a unique temperature and pressure at the triple point. Useful for

defining a standard temperature point for thermometry.

5. Sublimation

It is the change of phase directly from solid-to-vapour. This happens for “dry ice”, which is solid CO 2,

under ambient pressure; the heat absorbed by the solid dry ice causes it to change directly into vapour

without passing via a liquid phase. The same occurs for ice, but at reduced pressure. Equally liquid CO2

can be formed at room temperature, but only at very high pressure.

6. Critical point

Point at which the liquid-vapour transition vanishes. Dense gas is indistinguishable from liquid, it is

usually called a “fluid”.

7. Phase transitions

Melting and boiling should be familiar. The stable phase is always the one with the lowest free energy

(Helmholtz or Gibbs, depending on boundary conditions).

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The Helmholtz function as an indicator of equilibrium in fixed volume

The key property of the Helmholtz function F=U-TS is

that it has a minimum value for a system in equilibrium

at constant T and V . This makes it ideally suited to

treating a mixture of two phases of a single substance,

such as ice and water. The equilibrium condition be-

comes

dF = PdV −SdT = 0

An isotherm on a plot of F vs V typically comprises

multiple sections, one for each phase.

Volume

H e l m h o l t z = U − T S

Isotherms

Liquid

Solid

F may be minimised by a mixture of phases. These can be linked by a common tangent, the slope of

which is the phase transition pressure. ∂ F

∂ V

T

= P(T )

The phases have different specific volumes, so relative amounts of the two phases must give the externally

specified volume V = N 1v1 + N 2v2 ; where N 1 and N 2 are the number of moles of each phase, which can

change spontateously subject to dN 1 = −dN 2 and v1,v2 the molar volumes. This is sometimes known

as the lever rule. Note that the Helmholtz free energies f 1 and f 2 are not equal at equilibrium, and the

pressure and temperature remain constant as the two phase mixture is compressed, provided it remains

at equilibrium.

Isotherms of a Van der Waals fluid

The Van der Waals, equation of state yields more interesting isotherms than for an Ideal gas. The equation

can be rewritten in the form:

Pv3− (Pb+ RT )v2 +av−ab = 0

where v = V /n, n being the number of moles of molecules in the sample.

In general, for given a and b (ie a given gas) for each P,T there can be three possible values of v. The

three “roots” of the equation coincide for a unique set of values of P, v and T defining a “critical point”,

with coordinates (Pc = a27b2 , vc = 3b, RT c = 8a

27b).

• The Van der Waals equation can be rewritten with a and b replaced with Pc,T c,vc

• For T > T c, the equation has only one root.

• At the Boyle temperature T B = a/( Rb) = 27T c/8, d (PV )/dP)T is zero . At higher T “volume

exclusion” effects (b) are important, and PV increases with pressure. At lower T “bonding” effects

(a) are important, and PV decreases with pressure. Most real gases have T b ≈ 2.75T c

• Considering a virial equation of state for gases, TOPIC 12,

Pv

RT = 1 + B2

1

v

+ B3

1

v

2

+ ...

Bi are all functions of T , and at the Boyle Temperature B2 = b− a RT

= 0.

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Liquid-vapour transition for the Van der Waals equation of state

Van der Waals equation can conveniently be written in terms of reduced parameters V /V c, T /T c, P/Pcwhere Pc,T c,V c are the values of P,T ,V at the critical point (Not to be confused with the triple point):

P/Pc + 3

(V /V c)2

3V V c−1

= 8 T

T cVdW’s Eqn of state

with V c = 3b, Pc = a

27b2, T c =

8a

27 Rb

This equation gives a mechanically unstable solution (ie negative compressibility, (dP/dV )T > 0) for

T < T c over a range of molar (specific) volumes. This range becomes larger as the temperature is lowered

further. The fluid therefore must break up into two phases with different densities which in equilibrium

(same P, T) must have equal Gibbs Free energies. Since the boundary condition is fixed total volume, the

system minimises its Helmholtz free energy.

This leads to the construction shown in the second figure, in which a horizontal line is drawn connecting

the denser phase (liquid) at the left and less dense phase (vapour) on the right. The ‘real’ isotherm movesalong this line. The position of the line is such as to enclose equal areas between it and the isotherm

traced by the Van der Waals equation of state. Remembering that both lines refer to the same isotherm

the equal area construction ensures that:

• The change in Gibbs free energy between the two points at its end is zero. (∂ g/∂ P)T = v therefore

gl−gv = ∆g = Tconst vdP = 0 performing the integral along the VdW isotherm.

• If we construct a heat engine running round the closed cycle moving along the horizontal line from

liquid to vapour and back along the hypothetical VdW isotherm or vice-versa no work can be done

according to the second law, since there is no difference in temperature.

Phase and the PVT surface

Real substances have similar PVT surfaces to the van der Waals fluid, with a critical point and critical

isotherm. However, because the van der Waals substance is by construction homogeneous it cannot show

the phase transitions to solid crystalline phases.

A real substance compressed slowly at constant temperature might pass through a succession of equilib-

rium states abcdef including a vapour-to-liquid transition (bc), following the “easy” compression of the

vapour. The denser liquid is less compressible but may be transform into successive solid phases. Triple

points occur where three phases coexist at a temperature T TP. For the case of a liquid-solid-gas triple

point for T < T TP no liquid phase is formed; the solid transforms directly to vapour (sublimation).

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Topic 14 - Vaporization

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