Chapter 5: Simple Mixtures

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Welcome back to the Deep Dive.

Today, we're getting into a really core part of physical chemistry,

simple mixtures.

Yeah, absolutely.

Our plan is to break down focus five from Adkins physical chemistry, that's the simple mixtures chapter, and give you a clear kind of step -by -step guide to the main ideas.

Right, because this is where we move past just talking about pure substances, which is, you know, a bit idealized.

Things get more real.

Exactly.

We need to understand how adding something else, a second component, changes everything thermodynamically.

And the concept that ties it all together is the chemical potential.

We use the symbol mu maneuver.

And the big picture, like always in thermo, is equilibrium, which means that chemical potential for any substance has to be the same everywhere in every phase it's in.

That's the rule.

But before we get the equilibrium, we need a way to actually talk about the components in the mixture.

Which brings us to partial molar quantities.

Yes, this is fundamental.

It kind of shatters that simple idea that you can just add properties up.

Like volume.

Like volume, yeah.

Take one mole of water, add it to a big batch of pure water.

The total volume goes up by about 18 cubic centimeters, as you'd expect.

Right.

Makes intuitive sense.

But take that same mole of water and pour it into pure ethanol instead.

Okay.

Now the total volume only increases by about 14 cubic centimeters.

Wow, that's quite a bit less.

It's a huge difference, and it tells you straight away that the contribution a molecule makes, its partial molar volume, we call it VGL zeol.

It totally depends on what's around it.

So in ethanol, the water molecules pack differently.

Yeah, the hydrogen bonding network is all different.

They're forced into different arrangements.

So VGLs isn't just the volume of the molecule itself.

It's effective volume in that specific environment.

So any partial molar quantity, whether it's volume or enthalpy or Gibbs energy, it's really about the effective contribution in the mix.

Precisely.

And the most important one driving processes for Spontaneity is the partial molar Gibbs energy.

Which is the chemical potential.

Mooji bond.

That's it.

No joyer.

It is the central concept.

It literally tells you how much the total Gibbs energy of the system changes when you add a tiny bit more of component J, keeping everything else, pressure, temperature, other amounts, constant.

And the total Gibbs energy is just the sum of each component's amount times its chemical potential.

Yep.

Joll -jollers and a moo plus NB moo for a binary mixture.

So new Georgia basically dictates whether stuff happens spontaneously or not.

Okay, but here's where it gets really interconnected, right?

The components aren't acting alone.

Not at all.

They're linked.

Changing one affects the other.

And that's captured by the Gibbs -Duhem equation.

Thermodynamic tether, you called it.

Ha, yeah, I like that.

For two components, it's $1 math from moo plus NB math from moo.

It means if you do something that changes the chemical potential of A, then the chemical potential of B, new stuma must change in a corresponding way to keep that equation balanced.

So if U goes up, new birth has to go down.

Essentially, yes.

Weighted by their amounts.

You can't change one in isolation.

It ties the whole system together.

Okay, before we weight into, you know, real liquid mixtures, which can get complicated, let's set the baseline.

Mixing perfect gases.

Right, the ideal case.

This is our model for what a perfect solution would look like thermodynamically.

No interactions between molecules.

Exactly.

So when you mix perfect gases,

the enthalpy of mixing is zero, delta tex mix H dollar.

There's no heat released or absorbed because they don't care about each other.

So why do they mix spontaneously then?

Purely entropy.

Delta tex mix is always negative because the entropy of mixing, delta tex mix, is always positive.

It's just statistics spreading out into the larger volume, creating more disorder.

That's the sole driving force in the ideal case.

Got it.

So that ideal mixing model sets the stage for understanding liquid solutions.

It does.

Now we move to liquids and we use two key limiting laws to describe their behavior, especially near the extremes of composition.

Raoult's law and Henry's law.

Yep.

Raoult's law comes first.

We typically apply it to the solvent, to the component that's present in large excess.

Let's call it A.

Raoult's law says the partial vapor pressure of the solvent above the solution is just its mole fraction times the vapor pressure of the pure solvent.

So pH of all those XAPA.

And the reason the vapor pressure drops when you add a solute is?

Entropy again.

The solute increases the entropy of the liquid phase, making it more stable.

This lowers the tendency of the solvent molecules to escape into the vapor phase.

Lower escaping tendency means lower vapor pressure.

Makes sense.

So Raoult's works well when the solvent molecules are mostly surrounded by other solvent molecules, like when six a lot is close to one.

Exactly.

But what about the solute component B, especially when it's really dilute, when six ballers is close to zero?

It's an environment that's totally different then.

Completely different.

It's surrounded almost entirely by solvent A molecules, not other solute B molecules.

So its behavior can't possibly follow Raoult's law, which assumes a pure B environment as the reference.

So he needs its own law.

It does.

And that's Henry's law.

For the dilute solute, its partial vapor pressure is proportional to its mole fraction, six dollars.

But the proportionality constant isn't PBS.

It's an empirical constant, KB2.

So PB equals KBA.

And KBD depends on both the solute and the solvent.

Yes.

It reflects the specific interactions between solute B and solvent A in that dilute environment.

It's experimentally determined for each pair.

Henry's law is the limiting behavior for the infinite dilution.

Okay, so we have these two limiting laws.

But most solutions aren't perfectly ideal.

They're somewhere in between.

How do we handle the math for chemical potential then?

We like that simple Mu -Wu plus RT -LN -Su -E form for ideal cases.

We cheat, basically.

We introduce the concept of activity, A .J.

Deller.

Activity is like the effective mole fraction or concentration.

So we force the equation to look ideal.

We do.

We write Ul -Mo -A plus RT -LN -A -A -A.

It keeps the nice mathematical structure.

All the messiness, all the deviation from ideal behavior gets shoved into this correction factor called the activity coefficient gamma -J.

Okay.

And activity is related to mole fraction how?

Simple relation.

A -J equals gamma -J X -J -A.

The activity coefficient, gamma, tells you how much the real behavior deviates from the ideal behavior predicted by the mole fraction alone.

If gamma -J equals 8 -1, the solution behaves ideally with respect to that component.

If it's not one, it's non -ideal.

And getting this right depends on defining the standard state correctly.

Crucially important, yes.

Because gamma approaches one under different conditions for the solvent versus the solute.

Right.

You mentioned two standard states.

We did.

For the solvent, like component A, we use the Roe -Woltz law standard state.

Here, the activity coefficient gamma goes to one as the mole fraction $6 goes to one, the pure solvent.

Makes sense.

But for the solute component B, we use the Henry's law standard state.

For the solute, the activity coefficient gamma B goes to one as the mole fraction $6 goes to zero, infinite dilution.

Ah, because that's where Henry's law applies perfectly.

Exactly.

It's a bit confusing at first, this dual standard, but it's the rigorous way to handle real solutions across the whole composition range using activity coefficients.

Okay, so activity handles the general case.

What about models for why solutions might deviate, like slightly non -ideal cases?

A useful first step beyond ideality is the concept of a regular solution.

Regular solution.

What makes it regular?

It's regular in the sense that we assume the molecules still mix randomly.

So the entropy of mixing is assumed to be the same as for an ideal solution.

Delta text mix is still positive and calculated the same way.

Okay, so the randomness is still perfect.

What's not ideal then?

The energy.

The interactions.

In a regular solution, the interaction energy between an A molecule and a B molecule is different from the A -A and B -B interactions.

This means the enthalpy of mixing is not zero.

We often call this non -zero enthalpy the excess enthalpy, A -H -G -A -L -E -R.

So non -ideality in a regular solution comes purely from energy differences, not from, say, molecules clumping together or forming structures.

That's the core idea of the model, yes.

It captures energetic non -ideality while keeping the entropy simple.

We often model this excess enthalpy using a parameter, H -I, like H -C -Y, H -C -A, phi, like 1 -H -U.

And the sine of these goal tells you if mixing releases heat or absorbs it.

Right.

Negative Z means exothermic mixing.

A -B interactions are favorable.

Positive C means endothermic.

A -B interactions are less favorable.

And this parameter Z can then be plugged into things like the Margules equations to actually predict the activity coefficients.

Gamma -N and Gamma -B.

It gives us a basic physical model for non -ideality.

Got it.

Let's shift gear slightly to colligative properties.

These always felt a bit different.

They are interesting because they depend only on the number of solute particles, not what they chemically are, at least in the ideal limit.

Like freezing point depression, boiling point elevation.

Vapor pressure lowering, osmotic pressure.

Yeah.

And they all stem from the exact same cause.

Which is?

The solute lowers the chemical potential of the solvent, period.

Adding anything non -volatile to a solvent makes the solvent molecules happier, thermodynamically speaking, to stay in the liquid phase because of the increased entropy.

So the solvent's escaping tendency decreases.

Exactly.

Which means you need to lower the temperature more to make it freeze, freezing point depression, delta TFE, or raise the temperature more to make it boil, boiling point elevation, delta TB1.

Both are usually proportional to the molality of the solute.

Delta TS, KFBB, and delta TB equals KFBB.

Right.

And you often see that the freezing point depression, delta TFE, is significantly larger than the boiling point elevation, delta TB, for the same concentration.

Yeah.

Why is that?

Good question.

It comes down to visualizing the chemical potential plotted against temperature.

Remember, the enamel decreases as, increases for all phases.

The line for the solid phase on that plot has a very steep negative slope.

The liquid phase line has a less steep negative slope, and the gas phase line has the shallowest negative slope.

Got it.

Slopes get shallower going solid to liquid gas.

Exactly.

Now, when you add a solute, you lower the chemical potential of the liquid phase only, assuming non -volatile solute.

So that liquid line drops straight down on the plot.

Because the solid line is much steeper than the gas line, when the liquid line drops, its intersection point with the solid line shifts much further to the left, lower temperature, then its intersection point with the gas line shifts to the right, higher temperature.

Ah.

So the geometry of those intersecting lines dictates the magnitude of the shift.

Clever.

It's a nose visual explanation for why Caberol is usually larger than Caberol.

And the most sensitive colligative property is osmotic pressure.

Definitely.

Osmosis is that spontaneous flow of solvent across a membrane permeable only to solvent, moving from pure solvent side to the solution side, trying to dilute the solution and equalize the solvent's chemical potential.

And osmotic pressure is the pressure you need to apply to stop that flow.

Precisely.

For dilute solutions, the Van Hoff equation works well.

Pi equals BRT day, where B is the molar concentration of the can generate measurable pressures, which makes it great for measuring molar masses of big molecules.

Absolutely critical for things like polymers or proteins.

A tiny molar concentration of a huge molecule still gives a decent osmotic pressure, whereas the freezing point depression might be too small to measure accurately.

We even use virial expansions of the osmotic pressure equation to account for non -ideality at higher concentrations.

Okay.

Shifting perspective now from properties to mapping out stability across different conditions, we need phase diagrams.

Essential tools.

Yeah.

Especially for binary systems, two components.

Let's start with liquid vapor systems.

Typically temperature versus composition diagrams, right?

At constant pressure.

Usually, yes.

They show you at any given temperature what composition the liquid phase has and what composition the vapor phase in equilibrium with it has.

There are two curves.

One for the liquid, bubble point curve, and one for the vapor, dew point curve.

And if your overall system composition and temperature puts you between those two curves, then you have both liquid and vapor present.

And to figure out how much of each phase you have, you need the lever rule.

The indispensable tool for distillation calculations.

Absolutely.

You draw a horizontal line, a tie line, across the two -phase region at your temperature.

The line connects the liquid composition on one curve to the vapor composition on the other.

And the amounts are inversely proportional to the lengths of the segments.

Exactly.

$1 LLLVL is NVLLV free.

The amount of liquid times the distance on the lever to the vapor composition equals the amount of vapor times the distance to the liquid composition.

Simple geometry, but crucial for figuring out yields and separation processes.

But sometimes distillation hits a wall because of azeotropes.

Ah, yes.

Azeotropes.

These are mixtures that boil at a constant temperature without changing composition.

So the vapor has the same composition as a liquid.

Precisely.

On the phase diagram, this happens at either a maximum or a minimum in the boiling temperature curves.

Since distillation works by separating components based on the difference between liquid and vapor compositions, if they become identical at the azeotrope.

You can't separate them any further by simple distillation.

Correct.

It represents a practical limit to purification for many common mixtures, like ethanol and water.

Okay, what about binary solid liquid systems?

What's the key feature there?

The eutectic point.

This is the specific mixture composition that has the lowest possible melting point of any mixture of those two components.

Lower than either pure component.

Often, yes.

And the crucial thing is that this eutectic mixture melts or freezes sharply at a single temperature, the eutectic temperature like a pure substance would.

When it freezes, it forms a very fine -grained mixture of the two solid phases.

Like solder.

Solder is a perfect example, chosen for its low sharp melting point, which is near a eutectic composition.

It's also why throwing salt on icy roads works.

You're creating a mixture with a much lower freezing point, hopefully below the ambient temperature.

Are there more complex solid liquid behavior?

Oh yes.

You can have systems where the components react to form a stable solid compound, which then might melt normally, or you can get incongruent melting.

Incongruent.

That's where a solid compound, when heated, doesn't melt into a liquid of the same composition.

Instead, it decomposes into a different solid phase plus a liquid phase.

Things like the sodium -potassium system do this.

It just shows the variety and complexity possible even in binary systems.

And briefly, for three components, ternary systems.

Then we need triangular phase diagrams.

Since 6x allr plus xb plus xc equals len 1, you can plot the composition on an equilateral triangle.

These are used a lot, for example, to map out regions where three liquids might be fully miscible, or might separate into two or even three liquid phases.

Any key points on those diagrams?

One important feature is the plate point.

If you have a region where two liquid phases are in equilibrium, the plate point is a specific composition and temperature where those two phases become identical and merge into a single phase.

It's a kind of critical point for liquid equilibrium.

Okay, one last major topic.

The special and rather difficult case of ionic activities.

Why are ions so much harder to deal with than neutral molecules?

It really boils down to the nature of the forces.

Columbic forces, the electrostatic attractions and repulsions between ions, are very strong and very long -range.

Unlike the short -range forces between most neutral molecules.

Exactly.

Those long -range forces mean that ions interact significantly, even when they're quite far apart, leading to huge deviations from ideal behavior, even in solutions we'd normally consider quite dilute.

And there's a measurement problem too, right?

A fundamental one.

You can't just measure the activity of, say, the sodium ions in a salt solution by themselves.

You always have counter ions, like chloride, present to maintain charge neutrality.

You can't experimentally isolate the contribution of just one type of ion.

So how do we handle activity coefficients for ions?

We define a mean activity coefficient, gamma, gamma plus minus.

It's a kind of geometric mean of the individual ionic activity coefficients that we can't measure.

It essentially assumes the non -ideality is shared in a specific mathematical way between the cations and anions.

It's a practical workaround.

So gamma is what we actually determine experimentally.

Yes.

And to try and predict or understand gamma, especially in dilute solutions, we have the W.

Huckel limiting law.

What's the physical picture behind W.

Huckel?

It's based on the idea of the ionic atmosphere.

Think of any specific ion in the solution, say a positive ion.

On average, due to electrostatic attraction, it will be surrounded by a diffuse cloud or atmosphere that contains slightly more negative ions than positive ions.

So a net negative charge cloud around a positive ion.

Averaged over time, yes.

And this surrounding atmosphere of opposite charge exerts an attractive force on the central ion, stabilizing it.

Stabilizing means lowering its energy.

Lowering its energy, which means lowering its chemical potential compared to if it were truly isolated or behaving ideally.

This lowering of chemical potential is what's reflected in the activity coefficient being less than one.

And the Debbie Huckel law quantifies this.

It does for very dilute solutions.

It gives an equation showing that the

activity coefficient is proportional to the square root of the ionic strength and also depends on the charges of the ions.

But it's crucial to remember it's a limiting law.

Meaning it only works perfectly at?

At extremely low concentrations, approaching infinite dilution.

As concentrations increase, the model breaks down because the ionic atmosphere start to overlap and other interactions become important.

But it provides the correct limiting behavior and a fundamental physical insight.

Wow.

So looking back, we started with volume not being simply additive.

Right, the partial molar volume concept.

We moved through ideal limits, like Raoult's and Henry's laws.

Developed activity to handle reality.

Saw how chemical potential decrease drives colligative properties.

Mapped equilibrium with phase diagrams and the liver rule.

And ended with the intense non -ideality of ions due to long -range forces.

It really feels like the whole journey of simple mixtures is about understanding interdependence and moving away from simple additivity.

Absolutely.

The chemical potential truly is the central player.

Understanding how it changes with composition, temperature and pressure, and how the components are linked via Gibbs -Duhem gives us the power to predict and control real -world processes.

Think about distillation columns, designing alloys, understanding how cells maintain osmotic balance.

It all comes back to these principles.

So for everyone listening, maybe a final thought that you want, connecting back to Gibbs -Duhem.

If you dissolve something like salt in water, we know the chemical potential of the water plummets, drastically changing its freezing point.

Given that Gibbs -Duhem tether, how must the chemical potential, the activity, maybe even the partial molar volume of the salt itself be changing as you add more and more of it, forcing that huge change onto the water?

How does the system maintain that thermodynamic balance?

That interdependence is key.

Thinking about how one component forces changes in the other really deepens your understanding.

Thanks for tuning into this deep dive, everyone.