In order to predict how systems will behave, we need to understand their properties at different equilibrium states.
A pure substance is one that is chemically homogenous (it has the same chemical composition everywhere in the substance). Examples include
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)
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:
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.
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.
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:
Momentum is conserved when gas molecules collide with the container wall
Gas molecules have negligible volume
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:
A more common form of the equation uses the specific gas constant, R, and the mass, m, of gas:
Dividing both sides by the mass gives the specific volume form:
Since R is constant, we can write this as:
It follows from this that specific internal energy, u, is only a function of temperature, and not pressure:
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:
If the process occurs over a very small change in temperature, we can model the ideal gas EoS as being linear:
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:
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:
Specific enthalpy is therefore given by:
Applying the EoS for ideal gasses, we can write specific enthalpy like this, too:
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:
Meanwhile the change in enthalpy is given by the integral of Cp, the specific heat at constant pressure, with respect to temperature:
For ideal gasses:
For a perfect gas, Cp and Cv are constant so the changes in enthalpy and internal energy can be modelled as:
From h = u + RT, we can write h – u = RT. Dividing both sides by T gives dh/dT and du/dT on the left-hand side, and just R on the right. Therefore:
Another important quantity is the ratio of the two specific heats, γ:
For a perfect gas:
Vapours & Liquids
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.
The straight horizontal line here represents an isobaric (constant pressure) process. The graphs are useful as they give a lot of information:
The substance is a subcooled liquid, being heated up
The substance has heated up and is now a saturated liquid
The temperature is the same and the substance is made up of a mixture of saturated liquid and saturated vapour
The temperature is the same and the substance is a saturated vapour
The substance has heated up more and is now a superheated vapour
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):
A change in any property between the saturated liquid and saturated vapour states is noted with the subscript fg:
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:
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:
From this, we can derive useful equations to calculate the internal energy, enthalpy and volume at a given point in the wet vapour phase:
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, Δh = Cp ΔT
For perfect gases, Δu =Cv ΔT
γ = Cp / Cv
Dryness fraction is the proportion of wet vapour that is saturated liquid: x = V - Vf / Vg - Vf