How do different types of mixtures affect phase equilibria?

[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?

Please help me out.Thanks in advance.

 

[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.
I try to answer your questions:

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

I hope this can help you!

 

[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 learned 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.

PLease explain this.

 

[Davide86]

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 learned 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.

PLease explain this.

"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).

 

[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.

 

[LeonhardEuler]

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.

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.

 

[PhaseShifter]

Also, a salt solution is considered a separate phase. Why is that?

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

 

[Urmi Roy]

From what I understood in LeonhardEuler's post, a phase is a collection of all those thermodynamic states that can be defined definitely 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 each other 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 into the discussion--is there no justification of it?

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

 

[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).

 

[LeonhardEuler]

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.

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".

 

[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.

 

[PhaseShifter]

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?

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.

 

[PhaseShifter]

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?

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.

 

[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?

 

[LeonhardEuler]

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

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.

 

[LeonhardEuler]

" 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.

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.

"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?

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.

 

[PhaseShifter]

What are the requirements to be metastable?

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.

 

[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?

 

[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 each other'. 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 each other'? 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 each other's existence.

 

[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.

 

[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.

 

[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 acquires similar propertues as its gaseous phae,so basically the liquid becomes a gas,it seems.
Please explain further if possible.

 

[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.

 

[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 acquired,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 acquired 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!

 

[PhaseShifter]

vapours are basically individual drops of liquid floating in air,which have been separated due to different times of getting heated

The vapor is a gas, not a liquid.

 

[Urmi Roy]

Besides this point am I basically right?

Again,I can also say that a gas is basically super-agitated liquid molecules.

 

[PhaseShifter]

I'm reading what you say, but it's not clarifying anything, except that you seem to be saying there are two or more liquid phases, which is incorrect.

If there are droplets, they are all one liquid phase, but the vapor is a separate phase altogether.

 

[PhaseShifter]

What do you mean "liquid molecules"?

You seem to be saying the gas is composed of tiny bits of liquid, which is not true at all.

 

[Urmi Roy]

Sorry to frustrate you,actually I'm just an ignorant beginner and it seems that I have quite a few conceptual mistakes.
Actually I was just trying to say that at the critical temperature the liquid molecules,acquire enough kinetic energy so as to not affectively attract each other anymore -just as in a gas the molecules hardly attract each other.
That's why below the critical temp., if we bring two liquid molecules close to each other they will still attract each other and form a droplet but above that temp.,we can no longer form do that.


About the vapour phase concept-I really had an idea that the vapour phase consists of highly energetic liwuid molecules--please tell me where I'm wrong.

 

[Borek]

Actually I was just trying to say that at the critical temperature the liquid molecules,acquire enough kinetic energy so as to not affectively attract each other anymore -just as in a gas the molecules hardly attract each other.

Kinetic energy doesn't change attraction. As a first approximation you can assume that attraction between molecules doesn't change with temperature (this is not true in general). Imagine two molecules that are 'glued' together by their attraction. They keep close. Now imagine you increase their kinetic energy. They still try to keep close (attraction has not changed) but they are faster thus able to move farther away.

Attraction changes with distance, so when the average distance between molecules grow, average attraction goes down. But I would not call it effect of kinetic energy.

--

 

[Urmi Roy]

Okay,so could I still apply my reasoning considering that at heated states,liquid molecules tend to move away from each other because they have a greater chance of moving away from each other?
Please explain the concept of critical temperature in the way I tried to (in vain)- by considering what happens at the molecular level at different temperatures- I think that would really help me understand this faster.
Thanks for your help.

 

[Urmi Roy]

Perhaps I'm bothering everyone too much with my doubts on 'critical temperature'.

Perhaps I could get someone to help me on another topic in phase equilibrium, Eutectic systems.

I'm trying to bring some good textbooks for myself,so please put up with me for another week or so.

Well,this is my problem...

I have not found any webpage that explains the 'chemical' reason as to why when we mix two solids,they lower each other's freezing points at all,why that lowering of freezing point is dependant on their relative amounts,and why there is such a temperature called as an eutectic point.

Also there are certain characteristics of the eutectic point that I don't understand intuitively...

1. Why does an eutectic mixture have to consist of two solids that are miscible in all proportions in the liquid phase but which do not react chemically?

2. Does anything unusual happen in the vapor state of the metals?

3. Why is the composition of the eutectic mixture fixed? Why can't the same thing happen with different compositions of the metals?

4. Why, at the eutectic point does the mixture behave like one pure substance?

Pleeeease help!

 

[PhaseShifter]

I have not found any webpage that explains the 'chemical' reason as to why when we mix two solids,they lower each other's freezing points at all,why that lowering of freezing point is dependant on their relative amounts,and why there is such a temperature called as an eutectic point.

Also there are certain characteristics of the eutectic point that I don't understand intuitively...

1. Why does an eutectic mixture have to consist of two solids that are miscible in all proportions in the liquid phase but which do not react chemically?

2. Does anything unusual happen in the vapor state of the metals?

3. Why is the composition of the eutectic mixture fixed? Why can't the same thing happen with different compositions of the metals?

4. Why, at the eutectic point does the mixture behave like one pure substance?

Pleeeease help!

First, Above the eutectic temperature you have a simple case of freezing point depression. This lowers the freezing point of a given metal when the same metal's concentration in the melt is low.

Second, you can think of the eutectic point as a sort of triple point, only you have two solid phases and one liquid phase instead of a liquid, solid, and gas. Above the eutectic temperature, you can have solid A in equilibrium with the melt, or solid B in equilibrium with the melt, but not both. Below the eutectic point, you have solid A in equilibrium with solid B, but no melt. Ath the eutectic point, all three phases are in equilibrium.

Third--solid solutions will behave differently than the melt due to the crystaline nature of the solid phases. Where the transition from solid to liquid brings an increase in entropy, the transition between solid phases really doesn't make much difference as far as entropy is concerned. However, increased lattice strain will give a high enthalpy of mixing, so the miscibility gap must get wider as temperatures are decreased.

So, to answer your questions one by one:

1. Nothing special happens above the eutectic temperature, just the same sort of freezing point depression you see with salt and ice.
2. Not really. It's just a point where three coexistence curves converge.
3. Actually, if you apply Gibbs' phase rule, you'll see you have one degree of freedom. This means the eutectic is not fixed. However, since neither solids nor liquids are very compressible you will need to use extreme pressure to see the eutectic temperature and composition shift by any significant amount, unless at least one of the solids has a very high change in volume on melting.
4. It really doesn't behave like a pure substance, since more than one solid phase can exist in equilibrium with the melt at that point.

 

[Urmi Roy]

Thanks, I'll ponder upon these points carefully before I proceed with this topic.

In the mean while I have a few scraps of questions left...after which I hope it'll be it for me.

Please explain these 2 points--

1."In systems that are too small -- even, say, a thousand atoms -- the distinction between phases disappears, since the appearance of non-analyticity in the free energy requires a huge, formally infinite, number of particles to be present."

Do I need to pay attention to this point?--though I didn't really understand it,I don't think it's too important.

2."Different parts of a system may exist in different phases, in which case the phases are usually separated by boundary surfaces."

Lastly,it says in a webpage,"The phase rule indicates that for a single component system at most three phases (usually gas, liquid and solid) can co-exist in equilibrium. "

Can we reason this out? I mean what exactly happens at the molecular level, at the temperature and pressure of the triple point which allows all the three states to stay in equilibrium? Actually,the notion of all three states in equilibrium seems quite impossible,so it would be nice I came to know how this is made possible and what's really happening to all those molecules.

 

[Urmi Roy]

Hi,after a lot of searching,I found an explanation as to why the phase rule exists and what exactly it means,in terms of chemical potentials.

"The basis for the rule (Atkins and de Paula,[2] justification 6.1) is that equilibrium between phases places a constraint on the intensive variables. More rigorously, since the phases are in thermodynamic equilibrium with each other, the chemical potentials of the phases must be equal. The number of equality relationships determines the number of degrees of freedom. For example, if the chemical potentials of a liquid and of its vapour depend on temperature (T) and pressure (p), the equality of chemical potentials will mean that each of those variables will be dependent on the other. Mathematically, the equation μliq(T, p) = μvap(T, p), where μ = chemical potential, defines temperature as a function of pressure or vice versa. "

How can we use this paragraph to explain the equation : F = C − P + 2 ,
and explain how Gibbs reasoned this out?

 

[PhaseShifter]

The first is related to statistics more than to any particular physical mechanism. Think of how relative standard deviation is related to sample size and consider what that means when considering a system of 1000 atoms vs a system of 100 000 000 000 000 000 000 000 atoms.

The seconds means exactly what it says--if one part of your system is in one phase, and another part is in a different phase, then you generally expect a surface or interface to separate the two phases.

What's so mysterious about having three phases in equilibrium? It simply means that {\Delta~G_fusion}={\Delta~G_vaporization}=0

 

[PhaseShifter]

How can we use this paragraph to explain the equation : F = C − P + 2 ,
and explain how Gibbs reasoned this out?

Keep in mind that's not exactly correct. Chemical equiibrium poses additional constraints, so it's actually F=C-P-R+2.

Let's start out considering the simplest system, which would have one component and only one phase.

The chemical potential is a function of temperature and pressure, regardless of what the equation of state is for that phase.

Now consider what happens when you increase the number of phases: The chemical potential of the substance has to be the same at equilibrium, so we have the constraint {\mu_a}(T,P)={\mu_b}(T,P), or \Delta\mu_{ab}(T,P)={\mu_a}(T,P)-{\mu_b}(T,P)=0. We can choose either T or P as an independent variable, but we can only choose one of them--the new constraint makes the other a dependent variable by default.

Now if we make a third phase

Likewise adding a third phase gives a second delta:
{\mu_a}(T,P)={\mu_c}(T,P) gives us \Delta\mu_{ac}(T,P)={\mu_a}(T,P)-{\mu_c}(T,P)=0. This gives us a different curve on a T,P graph, which will intersect the first curve at a single point if all three phases coexist at once. (or won't intersect at all if the three phases can't coexist)

At this point you're probably wondering why I don't use a third delta:
{\mu_b}(T,P)={\mu_c}(T,P) gives us \Delta\mu_{bc}(T,P)={\mu_b}(T,P)-{\mu_c}(T,P)=0. But notice that \Delta\mu_{bc}(T,P)={\mu_b}(T,P)-{\mu_a}(T,P)-{\mu_c}(T,P)+{\mu_a}(T,P)=\Delta\mu_{ac}(T,P)-\Delta\mu_{ab}(T,P)--so this third delta is simply a combination of the other two.
Mathematically it will always be zero when both of the others are.

Now let's go back to the one-phase system, but add a second component.
Now we have the chemical potential of each substance as a function of three independent variables: temperature, pressure, and mole fraction of component #1. We could also use the mole fraction of component #2, but {X_1}+{X_2}=1 so only one mole fraction is an independent variable.

 

[Urmi Roy]

Basically,does this mean that since chemical potential is a function of temperature and pressure,we can make any two components have the same chemical potential by fixing them at certain tempertatures and pressures (which must be specific for the pair,no?) and taking another anyone of them,if we equate its chemical potential at a certain temperature and pressure with yet another substance,and if that value of chem. potential is the same as the previous pair's,we get all three of them in equilibrium..right?

According to this,any 2 substances may have the same chemical potential at many values of temperature and pressure (like along the solid-liquid equilibrium curve of water),so we may even have many points at which all three substances stay in equlibrium--all we need to do is match the value of chemical potential of all the components.,after all,we can match up the chemical potentiala os the substances merely by adjusting their temperatures and pressures.

Also,it is not exactly clear why a 2 phase system has a degree of greedom of 2,and how the number degrees of freedom vary with increasing number of phases.

 

[PhaseShifter]

Having two components with the same chemical potential is possible, but not particularly useful or even meaningful.

What is important is having the chemical potential of any given component equal in each phase--the system will shift towards the state of lowest free energy, so a component in a phase where it has a high chemical potential will move to a phase where it has a lower chemical potential. In each phase, the chemical potential of a given component will be a different function of temperature, pressure, and composition.

Also, if you add the chemical potentials of all components multiplied by their stoichiometric coefficients for a given reaction, you get \Delta~G for that particular reaction.
Since \Delta~G={{\partial G}\over{\partial\xi}}, the system will again minimize the Gibbs free energy by reacting until \Delta~G=0. Because of this, a chemical reaction places an additional constraint just like an extra phase does.

Any time you add a new component to the system, you need an additional degree of freedom to describe the composition of the system.
Any time you add an additional phase or reaction, you need to use one of those degrees of freedom to describe "distance from (phase or chemical) equilibrium", which of course is no longer a degree of freedom under equilibrium conditions, since by definition it must be equal to zero.

 

[Urmi Roy]

Any time you add an additional phase or reaction, you need to use one of those degrees of freedom to describe "distance from (phase or chemical) equilibrium", which of course is no longer a degree of freedom under equilibrium conditions, since by definition it must be equal to zero.

I think I get it now. Whenever I add a new phase to the system,initially,it has its own temperature and pressure and hence chemical potential.However,it has to gradually adjust its chemical potential to attain \Delta~G=0. Since temperature and pressure of a system are dependant-a change in one will prompt the other to change,the newly added phase must modify any of these factors, thus changing its chemical potential to attain minimum chemical potential condition.Right?

 

[Urmi Roy]

"At alloy compositions and temperatures between the start of solidification and the point at which it becomes fully solid (the eutectic temperature) a mushy mix of either alpha( a mixture of mainly A metal with some less amount of B) or beta (mixture containing mainly B with some small amount of A)will exist as solid lumps with a liquid mixture of A and B. These partially solid regions are marked on the phase diagram."

Alpha and beta are also mixtures, but weren't we discussing that only one metal ,say A,(lead in our original discussion) separates out leaving liquid B (tin) in the mixture?

Do we define the compositions of the alpha or beta states at the eutectic point,or do we consider alpha as pure A,and beta as pure B?

 

[PhaseShifter]

Whenever I add a new phase to the system,initially,it has its own temperature and pressure and hence chemical potential.However,it has to gradually adjust its chemical potential to attain \Delta~G=0. Since temperature and pressure of a system are dependant-a change in one will prompt the other to change,the newly added phase must modify any of these factors, thus changing its chemical potential to attain minimum chemical potential condition.Right?

Right.

"At alloy compositions and temperatures between the start of solidification and the point at which it becomes fully solid (the eutectic temperature) a mushy mix of either alpha( a mixture of mainly A metal with some less amount of B) or beta (mixture containing mainly B with some small amount of A)will exist as solid lumps with a liquid mixture of A and B. These partially solid regions are marked on the phase diagram."

Alpha and beta are also mixtures, but weren't we discussing that only one metal ,say A,(lead in our original discussion) separates out leaving liquid B (tin) in the mixture?

This is correct above the eutectic temperature, The alpha phase can exist in an equilibrium mixture with the melt and the beta phase can also coexist in a mixture with the melt--but the alpha phase and beta phase can't coexist at equilibrium unless the temperature is lowered below the eutectic temperature.

Below the eutectic temperature, the mixture spontaneously separates into alpha and beta phases, but the melt is no longer thermodynamically stable.

The alpha and beta phases are themselves mixtures of two components, but they are each a homogeneous solid solution of one metal in the other. In the case of lead and tin, you can think of the two solid phases as "lead contaminated by tin" and "tin contaminated with lead".

 

[Urmi Roy]

Cool,it seems I'm getting somewhere at last!

I know its getting kind of long,but I can't help squeezing some things in!

1.When we are cooling the mixture,how can small amounts of lead separate out at a time,since we are lowering the temperature of the entire mixture --the entire lead should freeze all at once,shouldn't it?

2.Why does the tin suddenly freeze all at once at the eutectic point,which is not even the proper temperature for it to do so?

3. Why does lead’s rate of cooling decrease when it has started freezing?

 

[PhaseShifter]

1.When we are cooling the mixture,how can small amounts of lead separate out at a time,since we are lowering the temperature of the entire mixture --the entire lead should freeze all at once,shouldn't it?

When lead crystallizes out of solution, the composition of the remaining melt changes. This means the temperature has to be lowered still further to freeze out more lead, which will shift the melt composition even more...

2.Why does the tin suddenly freeze all at once at the eutectic point,which is not even the proper temperature for it to do so?

But it is the proper temperature for the tin to freeze out. This is a sort of triple point where solid lead, solid tin, and melt can coexist. Below this temperature the melt is no longer thermodynamically stable at any composition. At higher tin concentrations, there will be no solid lead above the eitectic temperature because that is not thermodynamically stable. At higher lead concentrations there will be no solid tin above the eutectic temperature because that phase is not thermodynamically stable.

3. Why does lead’s rate of cooling decrease when it has started freezing?

This is mainly due to the latent heat of fusion.

 

[Urmi Roy]

When lead crystallizes out of solution, the composition of the remaining melt changes. This means the temperature has to be lowered still further to freeze out more lead, which will shift the melt composition even more...

This means that say at a particular lowered temperature, the lead composition is getting prepared to freeze--there may be some parts of it which due to certain factors (unnevenness of heating etc.) may start loosing heat first---as soon as that part has separated out,the composition has changed. This lowers the freezing point and ther same process is repeated--am I okay?

But it is the proper temperature for the tin to freeze out. This is a sort of triple point where solid lead, solid tin, and melt can coexist. ...thermodynamically stable..

To tell you the truth,this still isn't very clear.
I think what it means is that its only at the eutectic point that freezing of tin is thermodynamically favoured...

but there seems to be some blurred aspects of it...1.inspite of the gradual lowering of temperature of the mixture,the tin does not cool at all...2.the eutectic temperature is lower than even the freezing point of tin,so does it in a way influence the freezing point of tin too?...3after the eutectic composition is reached,can we say that the alloy has become homogeneous and the grains of lead and tin are fully incorporated in a single stable structure,accounting for the fact that at this composition,the cooling curve resembles that of a pure,homogeneous substance.

This is mainly due to the latent heat of fusion.

If I'm not wrong,this represents the fact that the heat loss during cooling is partially offset due to the heat evolved during the solidification process.

 

[PhaseShifter]

To tell you the truth,this still isn't very clear.
I think what it means is that its only at the eutectic point that freezing of tin is thermodynamically favoured...

Freezing of tin can be thermodynamically favored above the eutectic temperature as long as the tin concentration is higher than at the eutectic composition. Below the eutectic temperature, both solid lead and solid tin are more stable than the melt, so the melt does not exist.

but there seems to be some blurred aspects of it...1.inspite of the gradual lowering of temperature of the mixture,the tin does not cool at all...

I'm not sure what you're getting at here...at equilibrium the entire system will be the same temperature. The phase diagram only shows the system at equilibrium, so the tin will be no warmer or cooler than any other part of the mixture.

2.the eutectic temperature is lower than even the freezing point of tin,so does it in a way influence the freezing point of tin too?...

As one substance crystallizes out of the melt, the composition of the melt shifts away from the composition of the solid. This gives you one curve for lead crystallizing out of the melt, and another curve for tin crystallizing out of the melt. The eutectic point is where the two curves intersect, and both the lead and tin can coexist with the melt. If you lower the temperature any further, the melt ceases to be stable leaving you with only solid tin and solid lead.

The situation is not unique to metals...a salt solution has a lower freezing point than either pure salt or pure water.

3after the eutectic composition is reached,can we say that the alloy has become homogeneous and the grains of lead and tin are fully incorporated in a single stable structure,accounting for the fact that at this composition,the cooling curve resembles that of a pure,homogeneous substance.

If there are grains of lead and grains of tin, the mixture is not homogeneous.

If I'm not wrong,this represents the fact that the heat loss during cooling is partially offset due to the heat evolved during the solidification process.

This part is correct. When cooling below the eutectic point, the melt will freeze completely, and both the lead and the tin have latent heats of fusion.

 

[Urmi Roy]

As one substance crystallizes out of the melt, the composition of the melt shifts away from the composition of the solid. This gives you one curve for lead crystallizing out of the melt, and another curve for tin crystallizing out of the melt. The eutectic point is where the two curves intersect, and both the lead and tin can coexist with the melt. If you lower the temperature any further, the melt ceases to be stable leaving you with only solid tin and solid lead..

Right,so basically, what happens is that the prescence of either of the metals lowers the freezing point (F.P) of the other. Since the lowering of freezing point depends upon the amount of impurity, the lead gets it F.p lowered to 183 deg. at 38% of tin and tin has its F.p lowered to 183 deg at 62 % of lead. This is dependant on the nature of the metals--how much of the other it requires to lower the temp. Since by 183 deg. tin has solidified,there is no more liquid tin to depress the F.p of lead anymore,(similar argument can be applied to lead),so at that temperature,both the metals have solidified.
Is that okay?

If there are grains of lead and grains of tin, the mixture is not homogeneous..

From what I understand now, below the eutectic point,since both the metals are frozen,they cool individually,so the cooling curve for the mixture resembles that of a pure element.

 

[PhaseShifter]

That seems to be correct.

 

[Urmi Roy]

That's a great relief.

I think I've got my foundations pretty strong,thanks to Phaseshifter,though I'll need some books to read.

I really appreciate the wonderful and more importantly,the continuous help that Phaseshifter has provided.

Thankyou very very much. :)

 

[Urmi Roy]

Hi,I'm sorry to be back with all my questions after almost 2 months,but inspite of all my books and websites, I can't find any complete description of phase diagrams of solid solutions and of peritectic mixtures.
I have my exams next week,so please try and help me within that period...as soon as my exams begin I'll be completely tied up with my other subjects also.

Here are all the doubts I have in regard to solid solution phase diagrams...I have only one more set of questions for peritectic systems...

(In reference to http://csmres.jmu.edu/geollab/Fichter/IgnRx/SolidSol.html )

We consider the phase diagram of Anorthrite and Albite solid solution...I'll refer to albite as Na and anorthrite to Ca, as these are the major metals involved (They've done the same thing on the website also)..

1. Why does calcium freeze so drastically at the beginning (of the freezing process of the liquid melt)?

2. While initially the rate of crystallization of calcium is greater, why does sodium incorporation (into the forming crystal) start increasing as the crystallisation continues?

3.Why will the first crystal formed due to solidification of calcium consist of any sodium at all -sodium has a much lower freezing point?

4.Is the liquidus curve in the albite-anorthrite curve in any way similar to that of the simple two component eutectic mixture (Like lead and tin)?

5. Why are there two curves-solidus and liquidus--(in the diagrams for the simple eutectic systems,the liquid phase and forming cryatal phase were represented on the same diagram)?

6.What does it mean to move down a vertical line crossing the liquidus line, solid + liquid region and the solidus line?