This notes sheet looks at the properties of substances, and how these change as the substances change between different equilibrium states.

Pure Substances

A pure substance is one that is chemically homogenous (it has the same chemical composition everywhere in the substance). Examples include

  • hydrogen
  • pure water
  • a mixture of water and steam
  • air (all the gaseous substances are distributed evenly, everywhere)

Examples do not include

  • Mixtures of oil and water (oil is not soluble in water, so the two will always be separate)
  • Mixtures of liquid and gaseous air (different parts of air condense at different temperatures, so the mixture is not homogeneous)

Phase Changes

While there are three main phases a substance can be in (solid, liquid and gas), there are also a number of sub-phases within these:

thermodynamic phases, phases of substances, properties of substances, chemical phases

As you can see, the liquid phase is broken into two bands:

  • subcooled (compressed) liquids are when the liquid is not about to evaporate. For example, water at 20°C
  • saturated liquids are when the liquid is about to evaporate. For example, water at 100°C

The gas phase is also split into two bands:

  • saturated vapours are vapours that are about to condense. This region overlaps with the saturated liquids phase, so at 100°C, water exists as a mixture of liquid about to vaporise and vapour about to liquify
  • superheated vapours are gases that are not about to condense. For example, steam at 300°C

The boiling point is also referred to as the saturation temperature and pressure. This is the given temperature and/or pressure that the liquid-vapour mixture is seen. It is important to note that the temperature remains constant during a phase change.

There are two other important point between phases: the critical and the triple point.

P-T diagram, pressure temperature diagram, pressure temperature graph, triple point, critical point, properties of substances
  • At the triple point, gas, solid and liquid phases can coexist.
  • Above the critical point, liquid and vapour can no longer be distinguished. It is the high-pressure form of the liquid-vapour mixture phase, known as the supercritical fluid phase.

Latent Heat

The energy absorbed or released during a phase change between solid and liquid is called latent heat of fusion; that absorbed or released between the liquid and vapour states is called latent heat of vaporisation.

Behaviour of Gasses

Equations of state (EoS) are laws that apply at any point in a given state. These are helpful to describe and model the behaviour of substances, especially of ideal gasses. Making three assumptions about gasses simplifies the models greatly:

  1. Momentum is conserved when gas molecules collide with the container wall
  2. Gas molecules have negligible volume
  3. Any attractive forces between gas molecules is negligible

These assumptions are known as the kinetic theory of gasses, and give us the ideal gas EoS:

PV = n \bar{R} T

  • n is the number of moles
  • \bar{R} is the molar gas constant, 8.31 kJ.kmol^{-1}.K^{-1}

A more common form of the equation uses the specific gas constant, R, and the mass, m, of gas:

PV = mRT

Dividing both sides by the mass gives the specific volume form:

Pv = RT

Since R is constant, we can write this as:

\frac{P_1 v_1}{T_1} = \frac{P_2 v_2}{T_2}

It follows from this that specific internal energy, u, is only a function of temperature, and not pressure:

u = f(T)

When Kinetic Theory Doesn’t Apply

If we cannot assume that molecular volume and inter-molecular attraction are negligible, then the EoS has to be re-written in van der Waals form:

(P+ \frac{a}{v_2})(v-b) = RT

  • a is the intermolecular attraction
  • b is the molecular volume

Perfect Gases

If the process occurs over a very small change in temperature, we can model the ideal gas EoS as being linear:

u_2 - u_1 = C_v (T_2 - T_1)

Cv here is the specific heat at a constant volume.


For a constant pressure process, work transfer is given as W = P x ΔV (See notes sheet on energy, heat & work). Applying this to the first law, Q – W = ΔU gives:

Q = (U_2 + PV_2) - (U_1 + PV_1)

Therefore, heat transfer is given as the change in (U + PV). Since U, P and V are all properties, (U + PV) must also be a property: enthalpy, H:

H = U + PV

Specific enthalpy is therefore given by:

h = u + Pv

Applying the EoS for ideal gasses, we can write specific enthalpy like this, too:

h = u + RT

For ideal gases, the change in internal energy is given by the integral of Cv, the specific heat at constant volume, with respect to temperature:

u_2 - u_1 = \int^{T_2}_{T_1} C_p dT

For ideal gasses:

C_v = \frac{du}{dT} \quad C_p = \frac{dh}{dT}

For a perfect gas, Cp and Cv are constant so the changes in enthalpy and internal energy can be modelled as:

h_2 - h_1 = C_p (T_2 - T_1)

u_2 - u_1 = C_v (T_2 - T_1)

From h = u + RT, we can write h - u = RT. Dividing both sides by T gives \frac{dh}{dT} and \frac{du}{dT} on the left-hand side, and just R on the right. Therefore:

C_p - C_v = R

Another important quantity is the ratio of the two specific heats, γ:

\frac{C_p}{C_v} = \gamma

  • For a perfect gas, TV^{\gamma - 1} and PV^{\gamma} are constant.

Vapours & Liquids

P-V diagram for steam, vapour-dome, vapour dome diagram, steam diagram, wet steam graph

The behaviours of vapours and liquids can be represented on a P-v diagram, as shown above. The isotherms are increasing towards to the top right, so the bottom left line represents the lowest temperature, and the top right line represents the highest temperature.

The grey shaded region of wet vapour is sometimes called the vapour-dome.

Vapour dome, wet steam graph, wet steam diagram, steam process, isobaric steam process

The straight horizontal line here represents an isobaric (constant pressure) process. The graphs are useful as they give a lot of information:

  1. The substance is a subcooled liquid, being heated up
  2. The substance has heated up and is now a saturated liquid
  3. The temperature is the same and the substance is made up of a mixture of saturated liquid and saturated vapour
  4. The temperature is the same and the substance is a saturated vapour
  5. The substance has heated up more and is now a superheated vapour
Vapour dome, vapour dome graph, vapour dome diagram, wet steam chart, wet steam diagram, thermodynamics

Properties on the liquid side of the vapour-dome are noted using subscript f (from the German word ‘flüssig’, meaning liquid), while properties on the vapour side are noted with subscript g (from the German word ‘Gas’, meaning… gas):

  • v_f, u_f and h_f lie on the saturated liquid line
  • v_g, u_g and h_g lie on the saturated vapour line

A change in any property between the saturated liquid and saturated vapour states is noted with the subscript fg:

h_{fg} = h_g - h_f

Dryness Fraction

Inside the vapour-dome (the wet vapour state), the isobars and isotherms are on top of one another. This means that for wet vapours, the two are dependent, and so the vapour can be fully defined with just v and either P or T.

Sometimes, we want to know how much of a wet vapour is saturated liquid and how much is saturated vapour. To do this, we use linear interpolation along the horizontal line of temperature and pressure. This is called the dryness fraction, x:

x = \frac{m_{vap}}{m_{vap} + m_{liq}} = \frac{m_{vap}}{m}

  • For a saturated liquid, x = 0
  • For a wet vapour, 0 < x < 1
  • For a saturated gas, x = 1

Often it is more helpful to use volume instead of mass:

x = \frac{V-V_f}{V_g-V_f}

Dryness fraction, vapour dome, wet steam fraction, thermodynamics, properties of substances

From this, we can derive useful equations to calculate the internal energy, enthalpy and volume at a given point in the wet vapour phase:

u = u_f + xu_{fg}

h =h_f + xh_{fg}

v = v_f + xv_{fg}

  • A pure substance is one that is chemically homogeneous
  • Subcooled liquids are not about the evaporate, saturated liquids are about to evaporate
  • Saturated vapours are about to condense, superheated vapours are not
  • Wet vapour is the mixture of saturated liquids and vapours
  • The equation of state, PV = nRT applies anywhere in a given state (here R is the molar gas constant)
  • This can be rewritten as PV = mRT, and Pv = RT (here R is the specific gas constant)
  • Internal energy is only a function of temperature
  • Enthalpy is given as h = u + Pv
  • For perfect gases, \Delta h = Cp \Delta T
  • For perfect gases, \Delta u =Cv \Delta T
  • \gamma  = \frac{Cp }{Cv}
  • Dryness fraction is the proportion of wet vapour that is saturated liquid: x= \frac{V-V_f}{V_g - V_f}