Chapter 3: The Second and Third Laws

Welcome to Last Minute Lecture.

This free chapter overview is designed to help students review and understand key concepts.

These summaries supplement not replaced the original textbook and may not be redistributed or resold.

For complete coverage, always consult the official text.

Welcome to the Deep Dive.

If you need a shortcut to understanding the universal rules that govern chemical and physical change,

well, you are definitely in the right place.

Today, we are taking a, let's say, fascinating and sometimes pretty mind -bending trip into the second and third laws of thermodynamics.

Yeah, the big ones.

Exactly.

We're pulling all the essential knowledge directly from the core chapter on this topic in Atkin's physical chemistry.

Our mission today, really, is to try and transform these pretty complex concepts,

spontaneity, entropy,

free energy, take them from those dense textbook sections and turn them into clear, cohesive story.

We want you to really grasp the fundamental why behind chemical processes.

And that why really starts with directionality, doesn't it?

Thermodynamics doesn't really care how fast something happens that's kinetics.

That's kinetics, yeah.

It cares about the ultimate destination.

It tells us the inherent natural tendency of a change.

We all kind of know diamond wants to turn into graphite.

Eventually, yes, even if it takes forever.

Exactly.

Even if it takes geological time.

And that idea, spontaneous change, that's the absolute core theme here.

The first law, remember, just told us energy is conserved.

Right.

Delta U equal Q plus W1.

Energy isn't created or destroyed.

Exactly.

Which means lots of changes are allowed.

They're permissible.

But we need the second law to pick out which of those actually happen naturally.

So it's like the universe's rule book on direction.

Precisely.

Which way does the change go?

Okay.

So if spontaneity isn't just about seeking lower total energy because the universe's energy is constant, what is the real driving force?

What determines that direction?

Yeah, that's the million dollar question, isn't it?

And it's not minimum energy.

It's maximum dispersal.

Dispersal.

Dispersal.

The driving force is this fundamental tendency for energy and matter to just spread out, to be dispersed into the environment.

Okay.

Like when an ice cube melts, the heat energy spreads out.

Exactly.

The thermal energy that was kind of localized in the surroundings disperses into the water molecules as they gain freedom.

Or when a gas expands, the matter itself disperses into a bigger volume.

Okay.

I see.

And the property that actually puts a number on this that quantifies this dispersal, that's entropy, right?

Capital S.

It's entropy, yes.

And this leads us right to the formal statement of the second law.

Which is?

A spontaneous change is always accompanied by an increase in the total entropy of the entire isolated system.

Isolated system meaning the system we're looking at plus its surroundings.

Exactly.

System plus surroundings, often just loosely called the universe.

For any spontaneous process, delta S total plus delta S or LB must be greater than zero.

Okay.

Delta S 27th.

That's the signpost for spontaneity.

That's the fundamental signpost.

Now Atkins gives us two key ways to define entropy.

There's a macroscopic one and then a molecular one.

They have to connect somehow.

They do, beautifully.

Let's start with the classical thermodynamic definition.

It relates the change in entropy, DTS, to heat transfer.

Okay.

Specifically, it's the heat transfer reversibly, dA shed divided by the absolute temperature, T.

So, add a test, dTrica first.

Why reversibly?

That always trips people up.

Yeah, it's crucial.

Entropy is a state function, remember.

The overall change in S only depends on the start and end points, not the path.

Right.

Like altitude.

Doesn't matter if you take the winding road or the straight path up the mountain.

Perfect analogy.

But to calculate that change correctly, to make sure we're getting the true state function value, we have to imagine using a specific type of path, a reversible one.

It's a calculation trick, essentially.

Yeah, okay.

Because ArrayBS is DQ RevU2, got it.

Uh -huh.

And the other definition, the molecular one.

Ah, that's the Boltzmann formula.

Zero dolly equals KLNW1.

This gives us that intuitive feel for what entropy really is.

K is Boltzmann's constant, and W.

W is the number of microstates.

It's literally the number of different ways the molecules in your system can be arranged, their positions, their energy levels.

So more ways to arrange things means more W.

Exactly.

More ways means more W, more randomness, more dispersal, and therefore higher entropy.

Logarithm of W increases, so S increases.

Which explains why gas expansion is spontaneous.

More volume means way, way more possible positions for the molecules.

W goes way up.

Exponentially up.

That's the statistical driving force.

And this core idea, delta -stod increasing, it ties everything together, doesn't it?

Like, why heat only flows from hot to cold?

Absolutely.

Heat flowing from cold to hot spontaneously would decrease the total entropy.

The universe forbids it.

Same reason you can't have a perfect engine converting all heat into work.

That would also violate delta -stod.

It would.

Both the Kelvin statement, no perfect heat engine, and the Clausius statement, heat flows hot to cold, are just consequences of the second law.

And mathematically, how do we nail this down?

You mentioned the Clausius inequality earlier.

Right.

The Clausius inequality looks simple.

D -E -Q -D -I, the equal sign, holds for a reversible process.

And the greater than sign.

That holds for any irreversible spontaneous process.

And if the system is isolated, D -E -Q -D -I excludes down dollars.

So the inequality becomes D -E -Q -D -I dollars.

Entropy can only increase or stay the same at equilibrium.

Which proves spontaneous processes must increase total entropy and are inherently irreversible.

Okay?

Oh, it's a new package.

Right.

So now we know what entropy is, conceptually.

But how do we actually calculate the change in entropy, delta -sensor, for real world things?

Like, say, expanding a gas isothermally.

Good question.

For an ideal gas expanding at constant temperature,

the change in system entropy is always delta -S -N -R -L -N -V -S -E -I -2.

It's positive because the volume increases.

And because S is a state function, that value is the same whether it spans slowly or poof into a vacuum.

Exactly the same delta -sensor.

But the surroundings are different.

Ah, right.

So if it expands into a vacuum -free expansion, totally spontaneous.

Then delta -sensor is zero.

Nothing happens in the surroundings.

Delta -sensor is zero.

So delta -sensor is just delta -sensor, which is positive, confirmed, spontaneous.

And if we do it reversibly, slowly?

Then heat must flow in from the surroundings to keep the temperature constant during expansion.

That heat flow causes delta -sensor to be negative, and it exactly cancels out the positive delta -sensor.

Making delta -sensor is P -negative.

Okay, that makes sense.

It hangs together nicely.

What about phase transitions?

Boiling water, melting ice at the transition temperature.

That calculation looks simpler in the book.

It is, usually.

At the constant transition temperature, delta -qubit chain, the delta -keller is just the enthalpy change.

Delta -sheller is divided by keller.

So delta -s, delta -h -tune.

Like for boiling water, delta -hg is the enthalpy of vaporization.

You put heat in.

Delta -h is positive, heat absorbed.

So delta -sals is positive, which makes sense.

Gas is way more disordered than liquid.

Huge entropy increase.

And that leads to Troughton's rule, right?

That kind of surprising observation that most liquids have roughly the same entropy of vaporization, around 85 J.

Kallemol.

Yeah, it's a fascinating rule of thumb.

It tells us that the dominant factor in vaporization entropy is just the massive increase in volume, the dispersal of matter into the gas phase.

And that's pretty similar for many liquids.

But not water.

Ah, water.

Always the exception.

Its entropy of vaporization is significantly higher.

Why?

Because liquid water is unusually ordered due to hydrogen bonding.

So when it boils, the increase in disorder is even greater than for, say, benzene.

Precisely.

You're starting from a more ordered state.

Okay.

And heating something up, without a phase change, just raising the temperature.

If we assume the heat capacity, Cp, Kall's, is constant over the range, the entropy change is delta -s Cp -l -ne -t -f -ti -sa -e -a -a -t -p -f -t -a -t -i.

Again, positive if you heat it up.

Makes sense.

More thermal energy means more ways to distribute it, higher W, higher S.

Exactly.

So far, these are all changes in entropy, delta -s -d -a.

But textbooks have tables of absolute standard entropies.

Where do those come from?

We need a zero point.

We do.

And that's where the third law comes in.

These third law entropies are found experimentally.

You cool the substance down, ideally close to absolute zero.

Zero Kelvin.

And then you carefully measure its heat capacity, CpATD, as you slowly heat it back up to the You have to account for any phase transitions along the way, too, adding their delta -s -s weight.

And the actual calculation involves an integral.

Yes, you're essentially calculating the area under the curve of CpT2 plotted against T, from zero K up to your final T, summing up all those tiny d -eva -T contributions.

But how do you measure CpT -t -eva -l's at absolute zero, or even really, really close?

You can't, really.

That's the trick.

For non -metallic solids near zero K, the heat capacity follows a predictable pattern, the Wt -cubed law.

CpT is proportional to T5 -3 -2.

Ah, so you measure down as low as you practically can, and then use that T relationship to extrapolate the curve mathematically down to T -zeros.

Exactly.

It allows us to complete the integration.

Which brings us to the statement of the third law itself.

The third law of thermodynamics,

sometimes called the Nernst heat theorem, it states that the entropy of all perfect crystalline substances is zero at absolute zero.

See eyeballs.

Perfect crystal, meaning perfectly ordered.

Perfect crystal, perfect alignment, no defects.

At T -abio, all particles are in their lowest possible energy state.

There's only one way to arrange them, one single microstate.

Ah, so W1.

And Boltzmann tells us C $ -P -ul -N -W1.

Since the long one is zero, S must be zero.

Precisely.

It connects the macroscopic law to the microscopic picture perfectly.

And crucially, this existence gives us our anchor point.

It allows us to determine those absolute standard molar entropies, some that you see in tables.

Which we can then use to calculate the standard entropy change for any reaction, delta -sominus, just by taking products minus reactants weighted by stoichiometry.

Yep.

Delta -sominus, septa -sominus, easy peasy once you have the table values.

Okay, quick pivot.

The second law, delta -stop -itsiliver, is fundamental.

But calculating the entropy change of the surroundings all the time is, frankly, a pain for chemists working with a system in a flask.

It really is.

We want criteria that focus just on the system.

That's much more practical.

And that's where the new energy functions come in, Helmholtz and Gibbs.

Exactly.

We define two new state functions designed for specific conditions.

First, Helmholtz energy, A, divided as UTSC.

This is useful if you're working at constant volume and temperature.

Spontaneity criterion for that is delta -air -dollar.

Correct.

But chemists more often work at constant pressure and temperature, open beakers, standard lab conditions.

So we need Gibbs energy.

We need Gibbs energy.

Yeah.

Defined as dia -dollar -eats -HTSC.

Remember, H is one dollar plus PVA.

So yaw -dollar -U plus PVTSC.

This is the workhorse for chemistry.

And the new rule for spontaneity under these common conditions, constant T and P.

Elegantly simple.

A process is spontaneous if the Gibbs energy of the system decreases.

Delta -GT must be negative.

Delta -G delta must be negative.

That's the key.

Forget the surroundings for a moment.

Just look at the system's G.

Exactly.

And look at the definition again.

Delta -G, delta -HT delta -S.

This equation is, well, it's pretty much the heart of chemical thermodynamics.

It's a balance, isn't it?

Or a competition.

It absolutely is.

It's a tug of war between two tendencies.

Delta -H -dollar represents the tendency towards minimum energy.

Exothermic reactions, negative delta -H -dollar are favored.

And T -delta -H -dollar represents the tendency towards maximum entropy or disorder.

Positive delta is favored, making D delta -side negative.

Precisely.

So a reaction can be spontaneous, delta -G -dollar, either because it's strongly exothermic, large negative delta -H -dollar, or because it creates a lot of disorder, large positive delta -J, especially at high temperatures where the T factor magnifies the entropy term.

So you could even have an endothermic reaction, delta -H -positive, be spontaneous.

Absolutely.

If the increase in system entropy, delta -T is large enough, and the temperature T is high enough, the negative delta -tailer term can overwhelm the positive delta -H, making delta -G -negative overall.

Think of dissolving some salts in water it gets cold, delta -H -D -R.

But it happens spontaneously because the ions dispersing create a lot of entropy.

Right, right.

Now this is where it gets really interesting for practical applications.

Delta -G isn't just a signpost, it tells us about work.

Yes.

This is crucial.

The change in Gibbs energy, delta -G -R, for a process at constant temperature and pressure represents the maximum amount of non -expansion work you can possibly get out of that process.

Non -expansion work?

What's that?

It's any work other than the work done by the system just expanding against the atmosphere.

That's PVO or work.

Think electrical work in a battery, or mechanical work done by a muscle, or biochemical work in a cell.

Useful work.

So when Atkins gives the example of glucose oxidation, delta -G is about note of 2865 that number.

That's the maximum energy available from metabolizing one mole of glucose to power.

Muscles, nerves, whatever.

Exactly.

That's the theoretical limit of energy that's truly free.

Hence, Gibbs free energy to do useful biological work.

The negative sign tells us it's spontaneous.

The magnitude tells us how much useful energy is released.

Wow.

Okay.

And just like with entropy, we have standard values for this.

Standard Gibbs energy is a formation.

We do.

Delta of geominus, dude.

They're tabulated just like enthalpies and entropies.

Defined as the Gibbs energy change when one mole of a compound is formed from its elements in their standard states.

By convention, delta of geominus for elements in their standard states is zero.

And for H plus ions in water.

Also set to zero by convention.

This allows us to calculate the standard Gibbs energy change for any reaction.

Delta of gominus.

Using the same products minus reactants formula we use for enthalpy and entropy.

Delta of gominus.

Text reactants.

Okay.

Brings it all together calculationally.

Now, final stretch.

We've got the first law, the second law.

Atkins brings them together into one equation.

Yes, the fundamental equation.

It combines the first law, d eolicals dq plus d dollar, and the second law definition of entropy.

D eolus dq revi, roge levit, for a reversible process in a closed system, doing only PV work.

You substitute pdq plus the ddso and del os desli pdv into the first law.

And you get del osos tds pdv.

That's it.

Del osos tds pdvd.

Okay.

And the beauty is, although we derived it thinking about a reversible change, because u, t, s, p, and v are all state functions,

this equation actually holds true for any infinitesimal change, reversible or irreversible, in a closed system doing no additional work.

It's incredibly powerful.

And from this and similar equations for h, a, and g, you can derive Maxwell relations.

Exactly.

The Maxwell relations are these really neat cross -derivative equalities, partial t, partial vs, partial p, partial s, vs.

They pop out because u, h, a, and g are state functions, meaning their mixed second partial derivatives are equal.

They let you relate properties that seem totally unconnected, useful, but maybe a bit deep for today.

Agreed.

Let's stick with g.

Since g is our main tool at constant t and p, how does it actually change with temperature and pressure?

Good question.

The fundamental equation for g is ddg equals vdp sdt2.

This tells us everything.

Okay, so at constant pressure, dp dollar, ds dtg equals sdts or partial g partial tp equals hdt t.

Right.

And since entropy s is always positive… g must always decrease as temperature increases at constant pressure.

Makes sense.

Higher t favors more disorder, lower g.

Exactly.

And the substance with the highest molar entropy, usually the gas, will have the steepest downward slope of g vs t.

That's why substances eventually boil and become gases at high enough temperatures.

The gas phase g drops below the liquid or solid g.

And there's the Gibbs -Helmholtz equation too that relates the change in delta g with temperature to delta h.

Yes, give it to delta gt partial db, delta ht22.

It's super useful for calculating how equilibrium constants change with temperature because delta g is related to k.

But again, maybe for another day.

Fair enough.

What about pressure dependence?

From dg equals vdp sdt2 at constant temperature, we get dg equals vtt partial v.

Correct.

The change in g with pressure depends directly on the volume, v.

So for solids and liquids, their molar volumes are tiny.

Tiny.

So their Gibbs energies are almost independent of pressure, unless you're talking about geological pressure.

But for gases, molar volumes are huge.

Huge.

And they depend on pressure, baked dollar of trop to 1 pi alpha for an ideal gas.

So the Gibbs energy of a gas is very sensitive to pressure.

Integrating ddl pi happy mouse for an ideal gas gives gmn s plus rt ln ln s.

A logarithmic dependence.

So squeezing a gas increases its Gibbs energy quite significantly.

Dramatically.

It makes sense you're reducing its dispersal, its entropy, counteracting the natural tendency.

Wow.

Okay.

We have covered a lot of ground there.

Let's try a quick recap.

Sounds good.

We started with spontaneity being driven by dispersal, this spreading out of energy and matter.

We quantified that with entropy, s, the key rule.

Total entropy must increase for any spontaneous change, delta stot schurro, that's the second law.

Correct.

Then we found our absolute zero point for entropy with the third law,

c dollars for a perfect crystal at zero kelvin.

This lets us get absolute entropy values.

Oh.

But calculating delta stot is cumbersome, so we introduce Gibbs energy g for systems at constant t and p.

Spontaneity means delta g -systias must be negative.

Yep.

JLGOR is HTSE and delta g -war is our signpost.

And incredibly, that delta g also tells us the maximum useful non -expansion work we can get from a process.

Maximum useful work.

Check.

And we wrapped up with the fundamental equation, Dela dollars is TDSPDV and saw how g changes with t and p.

A whirlwind to orbit, hopefully clear.

Absolutely.

Now, you mentioned something earlier, derived from the source material, that seemed almost paradoxical about delta g and work.

Ah, yes.

For some reactions, particularly exothermic ones, that also produce a lot of entropy, the magnitude of delta g -day, the maximum useful work you can get out, can actually be greater than the magnitude of delta u or delta h, the heat released.

Wait, how can you get more work out than the energy released by the system itself?

That sounds like breaking the first law.

It doesn't break it.

What happens is the process is so spontaneous, driven by both and entropy,

that the system can actually draw in heat energy from the surroundings and convert that absorbed heat, along with the energy for the reaction itself, the useful work.

The universe gives you a little energy rebate because the process is so favorable.

Adkins calls it nature providing a tax refund.

A thermodynamic tax refund.

I like that.

So, okay, final thought for our listeners then.

If these processes, like in fuel cells or even biological systems, can effectively borrow thermal energy from their surroundings to boost the useful work output, what does that imply?

What are the ultimate limits, maybe the design principles, for creating future machines or processes that could efficiently harvest energy, we normally just consider low -grade waste heat, by coupling it to a sufficiently spontaneous process?

That is a provocative question.

Tapping into that free environmental heat.

Yeah, something definitely for you all to think about.

Indeed.

Well, thank you for guiding us through that.

My pleasure.

Thanks for joining us for this deep dive into the fundamentals of directional change in the universe.