# Homework Help: Phases and phase equilibria

1. Sep 6, 2009

### Urmi Roy

I am doing phase equilibria in college and I don't have very good books (I am in the first semester and I haven't got the chance to get good books yet) to study from and neither am I being able to find any websites that will explain to me the basic concepts of the chapter(wikipedia goes straight on to the details without a good explain on the basics).
Please help me out by explaining these problems that I have and if possible suggest a good website where I can get good reading material.

1. Is a mixture of different phases always heterogeneous?

2.Are the components of a system the same as the phases comprising the system?

3.Why do we say that the degrees of freedom of a system is the measure of the minimum number of independant variables defining the system and not the maximum?

4. Is the degree of freedom of a system comprising 3 phases always one? How do we detremine the degree of freedom of a system by looking at it?

2. Sep 6, 2009

### Davide86

Hi! I suggest you to read the book "Introduction to statistichal mechanics" Bowley, Sanchez; there are a couple of chapters regarding different phases balance and phase transitions.

1 I guess so, but I don't know wheter exist some exotic systems that can behave differently

2 No, for example a single metal (lead e.g.) can have two different phases: normal state and superconductor state

3 Because you can use variables defined as combinations of the degrees of freedom to describe the system. In this case you would have more variables than degrees of freedom

4 You have to analyze the system. I mean, if you have a gas in isothermic contact with a heat reserviour, you know that you can vary pressure, number of particles, volume and chemical potential. So you can determine the numbers of the degrees of freedom

3. Sep 7, 2009

### Urmi Roy

Actually,the problem is that I don't have a clear conception of what a "phase" is. It is not as simple as what we learnt in school,as being the different states of matter.

From what I read in the page http://www.chemistrydaily.com/chemistry/Phase_(matter) it seems that any homogeneous matter can be considered as a "phase" since it has consistent chemical and physical properties throughout it. The page also says that in a phase,the thermodynamic variables are consistent throughout. However,there is an example that a gas at 0 degrees celcius and at say 200 degrees celcius are in the same phase but in different thermodynamic state. If they are in the same phase,their thermodynamic property of temperature should be equal.

4. Sep 7, 2009

### Davide86

"The thermodynamic variables are consistent throughout" doesn't mean that they always have the same value. I mean, consider a cup of water: you can change its volume, pressure and temperature (within certain range of values) without changing the phase (no boiling or freezing), so you can change the value of the thermodynamic variables.

The consistency is about the regularity of the functions of the thermodynamic variables: if a phase transition occurs the entropy (and other functions like specific heat, free energy, hentalpy and so on) shows discontinuity in some points, or other singularity (e.g. discontinuity of the derivative).

5. Sep 7, 2009

### Urmi Roy

So basically what the issue is that if there is any abrupt change in the thermodynamic variables,which we can detect by plotting that particular value against time and locating any discontinuity in the graph-right?

Also, a salt solution is considered a separate phase. Why is that? After dissolution,the salt does not become an integral part of the water it is mixed in--its a mixture after all.
In your previous post,I noticed that you are considering the possible phases as solid,liquid and gas,whereas,a salt solution,is also being considered a separate phase in some sources.This seems to be very confusing!

In the page that I mentioned,they are saying that even magnetism is an indicator of phase change--bu magnetism is not a thermodynamic property.PLease explain this.

6. Sep 7, 2009

### LeonhardEuler

The idea is that when you specify enough variables to describe the state of the system, you completely describe the state of the system at equilibrium. Properties like magnetic susceptibility are thermodynamic properties. Since they vary with temperature, pressure, volume, composition, entropy, etc ..., then given enough of these variables, the magnetic susceptibility is specified. Conversely, given the magnetic susceptibility, the state of the system can be specified with one less of the other variables.

(Aside: An exception would of course occur if one of these properties varied in a non-monotone way with the others. For instance, the density of water at 1 atm first rises with increasing temperature from 0 °C to 4 °C, then falls with increasing T thereafter. So if I told you there was one mole of pure water at 1 atm with a density of 999.992 kg/m3, the temperature could either be 3 °C or 5°C, and you couldn't tell. I have never heard this question adequately addressed.)

So if the magnetic susceptabilty does change discontinously, that does indicate a phase change.

As for the salt solution, it might make sense in context. If there is a membrane between pure water and a salt solution, then the salt solution can be considered another phase in this context because there is a spatially discontinuous change in concentration.

For your fourth question in your original post, it depends on the number of components. The phase rule is usually given as
F = C -P + 2
So for a pure substance like water, this would give zero degrees of freedom. This means there is only one point (called the triple point) at which solid, liquid and gas can co-exist in equilibrium, so no variables like temperature and pressure are needed to specify the system, since it will always be 273.16 K and 0.0060373057 atm.

However, a full statement of the phase rule needs to include the number of relevant reversible modes of work. For all systems you probably study in introductory thermodynamics, this will be 1 (PdV work). However, if, for instance, you are studying the thermodynamics of rubber, and are interested in the stretching process along a particular axis, this will then jump to 2 modes because of the stretching mode. Then, for instace, you can use n, P, T, and length to specify the system (and this is actually done when studying this problem). Other possibilities include the presence of electric and magnetic fields, which would need to be specified if you are not assuming them to be constant or neglecting them.

7. Sep 7, 2009

### PhaseShifter

A separate phase from what other phase(s)? It's meaningless to call it "separate" without anything that it's separate from.

Last edited: Sep 7, 2009
8. Sep 8, 2009

### Urmi Roy

From what I understood in LeonhardEuler's post, a phase is a collection of all those thermodynamic states that can be defined definately by all the thermodynamic parameters. What I mean is that suppose within a certain range of temperature,all the thermodynamic parameters vary monotonically with each other and there is no abrupt change in any of them. Phase denotes the range of values of the thermodynamic parameters within which they are all monotonically related to eachother and there is no abrupt change in any of them.So,ultimately,a phase is equivalent to a 'state of matter' as we studied in lower classes.

Now, about the salt solution,as LeonhardEuler said,it can be specified as a separate phase (apart from solid,liquid and gaseous states)from, say,pure water within the same range of temperatures because it has a separate set of thermodynamic variables at all those temperatures than the pure water.This implies that any homogeneous substance can be considered as a phase. Now, a salt solution is considered a separate phase,because it is a homogeneous mixture,but a mixture of iron pellets and alumunium pieces is not a separate phase as we cannot assign specific values of thermodynamic parameters at all points within its bulk at all temperatures. This means that the main criteion is homogeneity.

Please tell me if this is alright! I hope I've got it right.

The case of water seems to be an exception,since at 4 degrees celcius, there is an abrupt change in the thermodynamic parameter 'density' but inspite of that water regulates itself within the limits of its water 'phase' but returning to the same set of values of thermodynamic parameters.Somehow this doesn't seem to fit in to the discussion--is there no justification of it?

Lastly, the formula that LeonhardEuler suggested for calculating the number of degrees of freedom is definately useful,but is there no way we can calculate the number of degrees of freedom without taking recourse to the formula - just by logic?

9. Sep 8, 2009

### PhaseShifter

It is possible to gradually lower the concentration of the salt and obtain pure water without going through any abrupt changes, so the salt solution is the same phase as pure (liquid) water.

Also, the change in pure water at 4ºC is not abrupt by any means--it's a relative maximum in the plot of density vs temperature, not a discontinuity in any thermodynamic variables.

In the case of a salt solution, you can increase the weight fraction of salt continuously from pure water to a saturated solution, but then there's a sudden jump from the saturated solution to a solid consisting of 100% salt.

In the case of iron pellets and aluminum pellets, you have have a heterogeneous mixture containing two phases of distinctly different composition--one is iron, the other is aluminum. They have distinctly different densities, heat capacities, etc. All the aluminum pellets have identical thermodynamic properties (and thus form a single phase). The iron pellets likewise share the same thermodynamic properties as one another (and form a single phase), but these are distinctly different from the properties the aluminum (so the iron phase is not the aluminum phase).

Last edited: Sep 8, 2009
10. Sep 8, 2009

### LeonhardEuler

What I meant was that the salt solution could be considered a separate phase in certain specific contexts, like when it is separated from pure water by a membrane. The problem that is confusing you is that there are two distinct ways of using the word phase.

In general, a phase can be any distinct homogeneous region in a heterogeneous system. So if a salt solution is separated from water by a membrane, it is a different phase In that particular setup.

People also talk about phase diagrams, and the different phases of matter that a substance can be in under different conditions. Here, people are talking about the different phases that a substance will spontaneously separate into upon a change in conditions. So salt water is not a separate phase of water under this meaning of the word. It will be considered a separate phase if it is somehow separated from pure water by some other means, but that is not what is meant when you talk about the "phases of water".

11. Sep 9, 2009

### Urmi Roy

Right, I understand what you people are saying. Perhaps if I think of it like this that phase changes occur whenever there is a gross change in intermolecular forces,we get a new phase. Suppose we have water,which has a certain magnitude of average inter-molecular attraction,as we heat it,the forces of attraction decrease gradually,but the changes are very small,untill we reach 100 degrees celcius,where there is a gross change in intermolecular attraction. I know that it seems a little vague but this is what the basic theory seems to be.

Also,as in the second last post,the term phase generally refers to solid,liquid and gas(and maybe plasma,supercooled liquid etc.) but according to the situation,we can redefine the phases and be more specific,like we did when a salt solution is separated by a membrane from water.I found PhaseShifter's explanation a bit hard to understand,but I could perhaps put it in the way that I just said.

Again,when we talk about components in a system,what exactly do we mean? Water is considered to be a one component system,but what about at its triple point-it consists of ice,water and water vapour-is this still a one component system?

I read in that webpage that I referred to in my original post that a phase change occurs when the free energy of the system becomes non-analytic. What does that mean?

Thanks for you cooperation,please bear with me while I try to grasp the concept properly.

12. Sep 9, 2009

### PhaseShifter

At the triple point you have three different phases in equilibrium (solid, liquid, and gas) but the system is made from one component--water. If you dissolved salt in the water or added a nitrogen atmosphere, you would be adding a second component.

Here you can also see the phase rule in effect--adding a second component increases the number of variables needed to describe the system. The triple point of pure water is a well defined point, since you have three phases in equilibrium and one component. that gives 2-3(phases)+1(component)=0 degrees of freedom needed to describe the system.

In the example where salt is added to the system, it causes freezing point depression, and would lower the temperature and pressure of the triple point. According to the phase rule 2-3(phases)+2(components)=1 degree of freedom, so the triple point has become a "triple curve" where the pressure and temperature of the triple point are functions of the amount of salt added.

13. Sep 9, 2009

### PhaseShifter

The short version is "an analytic function is a function that can be described as a Taylor series." During a phase change the free energy (or one of its derivatives) will go through a discontinuity.

For example, the molar volume of water abruptly changes when it boils, evaporates, or freezes.
$$V={({{\partial G}\over{\partial P}})}_{T}$$
Since V is not continuous at the boiling point, G is non-differentiable with respect to P.

14. Sep 10, 2009

### Urmi Roy

Thanks PhaseShifter for the excellent help. Now that I have got that,please let me ask my next question (sorry to bother you with this,but I really need to understand this properly).

I read that "In practice, each type of phase is distinguished by a handful of relevant thermodynamic properties. For example, the distinguishing feature of a solid is its rigidity; unlike a liquid or a gas, a solid does not easily change its shape. "

Now, compressibility,rigidity are not thermodynamic properties so how can they be considered?

" Many of the properties of solids, liquids, and gases are not distinct; for instance, it is not useful to compare their magnetic properties. " -what does this mean.

"Metastable states may sometimes be considered as phases, although strictly speaking they aren't because they are unstable. For example, each polymorph of a given substance is usually only stable over a specific range of conditions. "-
Are diamond and graphite different phases of carbon or are they metastable states ? What are the requirements to be metastable?

15. Sep 10, 2009

### LeonhardEuler

Be careful of making statements like this. A thermodynamic property of a substance is basically any property that can be defined at equilibrium. Compressibility is actually a very commonly considered thermodynamic property, and there are multiple types of compressibility that are often analyzed in thermodynamics (isothermal and adiabatic compressibilty). Rigidity is somewhat ambiguous, but the word is often used as a synonym for stiffness, which again, is also a thermodynamic property.

Just because you don't run into certain variables in introductory thermo doesn't mean they are not thermodynamic properties.

16. Sep 10, 2009

### LeonhardEuler

What they are trying to say depends on the context, but without seeing that, what it looks like they are trying to say is that for most materials the magnetic properties do not change considerably after a phase change. This is obviously not true in general since there definitely is a big change in magnetic properties when going from, say, iron liquid to iron solid, but most often this will not be the case.

Diamond and graphite are in fact different phases of carbon, but diamond is metastable at normal atmospheric conditions. This means that if you have a diamond at room temperature and pressure, and you let it reach equilibrium, it will eventually turn to graphite (diamonds are not forever). However, this process happens extremely slowly, so you will never see this happen in your life time. So in truth there is only one phase of carbon at these conditions, but for some practical purposes it can make sense to think of a diamond phase at these conditions because it will only change extremely slowly.

17. Sep 10, 2009

### PhaseShifter

A metastable state does not have the lowest possible free energy, but is separated from the lowest energy state by a barrier of higher energy.

In the example above, at STP diamond has a higher free energy than graphite, but it takes a large amount of activation energy to break the bonds holding the diamond lattice together.

Last edited: Sep 10, 2009
18. Sep 11, 2009

### Urmi Roy

I'm sorry about being a litlle irresponsible in my remarks about rigidity and compressibility not being thermodynamic parameters--I have understood my mistake now.

Also, about metastable states,I understand that they are basically phases that exist within a narrower limit of values of thermodynamic parameters (like heat content--if I keep providing heat energy to a piece of diamond,beyond a certain limit,it will start converting into graphite.)

However, how are these 'narrower limits' determined?

19. Sep 11, 2009

### Urmi Roy

Please let me put in another question on a topic that has been puzzling me for quite some time.

The critical temperature for a body is the temperature beyond which its liquid and vapour states 'merge into eachother'. However there is no such temperature at which its solid and liquid,or even solid and gaseous states do the same.
Firstly, what is exactly meant by states 'merging into eachother'? Secondly,why are there no such corresponding temperatures for the solid state?Lastly, is there any relation or perhaps similarity between the critical temperature and thetriple point,as in the triple point also,the 3 phases of water merge into eachother's existance.

Last edited: Sep 11, 2009
20. Sep 11, 2009

### PhaseShifter

Those are two very different situations--at the triple point, you have three different phases in equilibrium.

The critical point is quite different.
It's easy to understand liquid water as a separate phase from the vapor--they have vastly different densities, there's very clearly a surface separating the two phases when you look at them, the liquid is more visccous than the gas, etc.

However, those differences shrink when you raise the temperature--the liquid expands and the gas becomes more dense as the vapor pressure increases. At the critical point the densities of the two phases become equal, and other differences vanish as well. For instance--surface tension drops to zero at the critical point, and heat of vaporization vanishes. The critical point isn't simply a transition from one state to another, it's a part of the phase diagram where the distinction between two phases vanishes, and they become a single phase.

Critical points allow a thermodynamic path where a system can start as one phase and end up as another phase, without actually going through a phase transition. (i.e. heat water up above the critical temperature at a fixed volume, expand it to 2000 times the original volume,then cool it back down at a fixed volume, and it's gone from liquid to vapor without evaporating or boiling.)

The reasons there isn't a liquid-solid critical point point are pretty simple when you think about it--the solid has molecules arranged in an orderly lattice, while the liquid has a random structure. because of that, there will always be a significant entropy difference between the two phases.

21. Sep 11, 2009

### PhaseShifter

Also, the phase diagram for water will look very different if you plot T vs V instead of T vs P.

In the T vs P graph, you see phase transitions whenever the conditions cross over a curve representing a phase transition, i.e crossing over the line from liquid to gas.

In a T vs V plot, you must remember at low temperatures a huge difference in density exists between the gas and liquid when they are in equilibrium. This is because the intermediate densities would form a thermodynamically unstable state, and a system prepared at that density will spontaneously separate into two different states in order to lower its free energy. So rather than a curve that abruptly ends at the critical point, you see an inverted U-shaped curved with the critical point at the maximum--liquid water is on the left of the curve (high density), the gas phase is on the right of the curve (low density), and the area underneath the curve is an unstable region where the water separates into a gas (with molar volume corresponding to the right-hand branch of the curve) and a liquid (with molar volume corresponding to the left-hand branch of the curve).

Sometimes you will see a phase diagram where there is a second, narrower U-shaped curve inside of this unstable region. The outer curve is called the binodal curve, and indicates where the transition to another phase occurs with no change in free energy, and defines the phases which coexist with one another at equilibrium. The inner curve is called the spinodal curve, and it separates the metastable states from the states that are truly unstable in every sense--essentially, it is the limit on how far the system can move into the unstable region without spontaneously separating into two phases.

22. Sep 12, 2009

### Urmi Roy

PhaseShifter, I'll need some time to really understand about the critical temperature,as I'm having a difficulty in seeing the difference between the same and the boiling point of a liquid,besides, the liquid doesn't seem to remain a liquid at this tempertature--it aquires similar propertues as its gaseous phae,so basically the liquid becomes a gas,it seems.

23. Sep 13, 2009

### PhaseShifter

At the boiling point, you can have two phases in equilibrium (liquid and gas).

These two phases have very different properties (density, heat capacity, compressibility, etc.) For example, at 100ºC, the liquid is approximately 1200 times more dense than the gas. At lower temperatures, the differences are more extreme.

These differences get smaller as you approach the critical point. For example, the liquid expands (and its density decreases), and the vapor pressure increases (increasing the density of the gas).

At the critical point, all differences between the two phases cease to exist. the densities, heat capacities, etc. become the same. Also, the heat of vaporization goes to zero at the critical point. Since the discontinuities in thermodynamic variables no longer exist, you no longer have two separate phases.

24. Sep 13, 2009

### Urmi Roy

PhaseShifter,do you think I can put it this way that---suppose we have a liquid in a container-which may be closed or open-- and we start heating it--the liquid molecules at the surface get the heat and have an opportunity to drift off into the air,so due to the kinetic energy aquired,the molecules can,sort of fly off into the air--depending on their kinetic energy,the new vapour state (vapours are basically individual drops of liquid floating in air,which have been separated due to different times of getting heated) has a certain pressure.However,inspite of their kinetic energy,if we bring two such drops close by,they will again form a liquid drop together,due to the still prevalent intermolecular forces,which can supercede the effect of the incresed energy of the molecules.

At a certain point of heating,the pressure becomes equal to the atmospheric pressure,but the state is still liquid.

After more heating,the kinetic energy aquired will be so high,that even if we bring two liquid vapours close together,they will not form a liquid drop again,since the intermolecular forces cannot supercede the effect of increased energy of the molecules. This is our critical temperature.

Please take some time to read through my rubbish,it'll really help me if you clarify any misconception I may still be having!

25. Sep 14, 2009

### PhaseShifter

The vapor is a gas, not a liquid.