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Thermodynamics: Phases and their transition, Lecture notes of Thermodynamics

Phase diagram, Triple point of water, First order phase transformation, Clausius-Clapeyron equation, Second order phase transformation

Typology: Lecture notes

2018/2019

Uploaded on 08/07/2019

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Phases and their transition
We loosely understand that a substance can have different phases like
solid, liquid and gas or vapor. We typically employ state variables like
p,V,Tfor its description. In gas or vapor phase, for instance, we use
equation of state such as pV =RT or (p+a/V2)·(Vb) = RT etc. These
equations are valid over a restricted range of p,V,T.
We plot here water’s TVdiagram (isobars) and pVdiagram
(isotherms) over a wide range of p,T,Vfor different values of pand T
respectively. We get the following,
pf3
pf4
pf5
pf8
pf9

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Phases and their transition

We loosely understand that a substance can have different phases like

solid, liquid and gas or vapor. We typically employ state variables like

p, V , T for its description. In gas or vapor phase, for instance, we use

equation of state such as pV = RT or (p + a/V 2 ) · (V − b) = RT etc. These

equations are valid over a restricted range of p, V , T.

We plot here water’s T − V diagram (isobars) and p − V diagram

(isotherms) over a wide range of p, T , V for different values of p and T

respectively. We get the following,

Phase change processes of water : Keep pressure constant at 1 kPa, the

standard atmospheric pressure – the lower most line in T − V diagram.

I At low T and V , we get compressed or subcooled water, expanding

only slightly as T increases.

I When T = 100o^ C, water exists as a liquid that is about to vaporize

(saturated liquid).

I At T = 100o^ C, as more heat is transferred, the saturated water

starts vaporizing, saturated liquid-vapor mixture – boiling.

I T remains constant at 100o^ C until all the water is vaporized,

saturated vapor – a vapor that is about to condense.

I T of pure vapor continue to rise and we get superheated vapor.

As the pressure increases, the saturated liquid line and vapor line come

closer and the saturated liquid-vapor mixture phase gets shorter until the

point at which it vanishes i.e. the saturated liquid and vapor states

become identical – no distinction between liquid and vapor phase.

This specific point is called critical point.

Tc = 374. 14 o^ C, pc = 22.06MPa = 217.7 atm, Vc = 0.0032 m^3 /kg

Phase diagram

The saturated vapor / liquid line in both the T − V and p − V diagrams

suggests the temperature at which liquid changes to vapor phase changes

with pressure. Similarly, the pressure at phase change depends on

temperature.

Saturation temperature Tsat : temperature at which pure substance

changes phase at a given pressure.

Saturation pressure psat : pressure at which pure substance changes

phase at a given temperature.

I Start with ice at p = 100 KPa and add heat to it. Water

temperature will rise until T = Tsat ≈ 0 o^ C when ice begins to melt

and T remains constant till all the ice is melted.

I After all ice has melted, if we continued to add heat, water

temperature rises and we boil the water at T = Tsat ≈ 100 o^ C.

Again, T remains constant until all the water has boiled to vapor.

I With increasing p, the above process will repeat till p = pc.

I Have we started at p < 0 .61 KPa, ice goes sublimates to vapor.

First order phase transition : co-existing phases

1. Condition of co-existence of phases

μ 1 (T , p) = μ 2 (T , p)

2. First derivative of G is discontinuous across phase boundary.

3. Volume and entropy are discontinuous across phase boundary

V =

( ∂G

∂p

T

and S = −

( ∂G

∂T

p

4. CP of mixture of two phases during phase transition is infinite

CP = T

∂S

∂T

p

→ ∞, and so is β =

V

∂V

∂T

p

Since co-existing phases have different entropies, system must absorb or

release heat during phase transition – latent heat

L ≡ H = T ∆S at const. pressure

Clausius-Clapeyron equation

We apply Gibbs condition G 1 (T , p) = G 2 (T , p) to study properties of

boundary of first order transition across which there is a discontinuous

change of S and V. A first order phase change is accompanied by latent

heat L = T ∆S and volume change.

Consider points A(T , p) and B(T + dT , p + dp) on phase transition line

A : G 1 (T , p) = G 2 (T , p) and B : G 1 (T + dT , p + dp) = G 2 (T + dT , p + dp)

Taylor expanding about A and using Gibbs condition,

G 1 (T , p) +

∂G 1

∂T

p

dT +

∂G 1

∂p

T

dp +... = G 2 (T , p) +

∂G 2

∂T

p

dT +

∂G 2

∂p

T

dp + [( ∂G 1 ∂T

p

∂G 2

∂T

p

]

dT =

[(

∂G 2

∂p

T

∂G 1

∂p

T

]

dp

From Maxwell relations it follows that on phase boundary

dp dT

S 2 − S 1

V 2 − V 1

T (S 2 − S 1 )

T (V 2 − V 1 )

L

T (V 2 − V 1 )

− Clausius-Clapeyron eqn

Among other uses of Clapeyron eqn., the sign of dp/dT determines how

volume changes (expands or shrinks) under phase transition.