Chapter 2: The First Law

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Welcome back to the Deep Dive.

We've got the sources you sent on the basics of thermodynamics, specifically focused to from Atkins Physical Chemistry.

Our mission today, to really break down the first law of thermodynamics, we want to make sense of energy transformation.

You know, how chemical reactions and physical stuff give off heat or do work.

Exactly.

And this first law, it's fundamental.

It basically says the total energy in an isolated system stays constant, can't create it, can't destroy it.

Okay, let's unpack this.

Right.

So first things first in thermodynamics, we need to define our terms.

We talk about the system, that's the bit we're interested in, like the chemicals reacting in a flask.

And everything else in the surroundings.

That's where we usually measure things like temperature or pressure.

Precisely.

And how the system interacts with the surroundings is key.

We classify systems based on that.

You've got open systems, they swap both energy and matter, like an open beaker of boiling water.

Then closed systems, which swap energy, maybe heat, but not matter.

Think of a field bottle warming up in the sun.

And finally, the ideal case for the first law, the isolated system.

It exchanges absolutely nothing with the surroundings, neither energy nor matter.

It's like a perfect thermos flask, theoretically.

Got it.

So once we've defined our system, we need to talk about how energy actually moves.

The two main ways are work and heat.

That's right.

Work in the thermodynamic sense is about achieving motion against some kind of opposing force.

Think lifting a weight or a gas expanding and pushing a piston.

Energy then is basically the capacity to do that work.

And heat.

Heat is different.

It's energy transfer that happens specifically because there's a temperature difference between the system and its surroundings.

There's a neat molecular picture here too, isn't there?

Work involves organized motion molecules moving together, like electrons in a wire or gas pushing a piston uniformly.

Whereas heating involves disorderly motion.

It's about stirring up the random thermal jiggling of molecules, much less coordinated.

And the type of boundary matters too.

A diathermic wall lets heat pass through freely.

Like a metal container.

While an adiabatic wall is a perfect insulator,

no heat gets through.

Think of that ideal thermos again.

Which leads us to processes being exothermic releasing heat, like burning fuel.

Or endothermic absorbing heat from the surroundings, like ice melting or water evaporating.

Okay, so all this energy flowing in and out or being stored, it has to be accounted for.

That brings us to the system's total energy account.

Internal energy, which we call dollar.

Right.

Dollars is the grand total of all the kinetic energy, the motion energy, and all the potential energy, the energy stored in bonds and interactions of every single particle in the system.

It's an extensive property, meaning more stuff means more internal energy.

And here's a really critical point you highlighted.

Internal energy, dollar is a state function.

Yes, absolutely crucial.

Its value depends only on the current state of the system.

It's temperature, pressure, volume, composition.

It does not depend on how the system arrived at that state.

So wait, if I start with reactants and end with products, the change in internal energy, the delta U, is the same whether I did the reaction quickly, slowly through 10 intermediate steps or just one direct step.

Exactly that.

The path taken for the transformation is irrelevant to the overall change in U.

That's the power of a state function.

And this is what lets us write the first law so simply, mathematically,

delta U plus Q plus dollar.

The change in internal energy, delta U, equals the heat added to the system plus the work done on the system.

It shows that heat and work are, in a sense, equivalent ways to change the system's energy bank account.

They are the energy currencies.

Now you mentioned work.

A common type is expansion work, right?

When a gas pushes back the surroundings.

Yes, the work done by the system when it expands by a volume dollier against a constant external pressure is dollier's dollar.

The negative sign is important.

It means when the system does work, extends,

its internal energy decreases, assuming no heat flow.

And is there a way to get the maximum possible work out of an expansion?

There is.

Theoretically, it's called reversible expansion.

This happens when the external pressure is only infinitesimally lower than the internal pressure of the gas at every single tiny step of the expansion.

It's perfectly balanced, extracting the maximum energy.

This is like lowering the pressure outside incredibly slowly.

Precisely.

For a perfect gas expanding isothermally, constant temperature, the total work done in a reversible expansion from volume 5 psi b dollars comes out as del U was nRT Ln vFeI.

Okay, back to measuring delta U1.

You said state functions are powerful.

How does delta U plus Q help us measure things?

Well, think about doing a reaction where the volume is held absolutely constant, like in a strong sealed container, what chemists call a bomb calorimeter.

If the volume doesn't change, delta V is zero.

So the expansion work del TV does must also be zero.

Exactly.

If volume is constant, delta Vd a dollar, and no other kind of work like electrical work is being done, then one U a dollar.

The first law equation simplifies right down to delta U alkuvia.

Ah, so the change in internal energy is just the heat measured at constant volume.

We just measure the temperature change of the calorimeter.

That's it.

It gives us a direct way to measure delta U on reactions, particularly combustion.

And this leads directly to the concept of heat capacity at constant volume.

Heat capacity.

That's how much heat you need to raise the temperature, right?

Specifically, Cpdol is the change in internal energy with respect to temperature when the volume is held

Mathematically, Cvdol is school's partial CVA.

It tells you how much the internal energy climbs for each degree rise in temperature at constant volume.

Okay.

Constant volume is neat for bomb calorimeters, but honestly, most chemistry doesn't happen in a sealed bomb.

It happens in beakers, flasks, or even biological cells, usually open to the atmosphere, which means they are at constant pressure, not constant volume.

And as they react, they might expand or contract constantly doing work on the atmosphere or having work done on them.

So measuring dollars under those conditions doesn't give you delta U directly because the dollars isn't zero anymore.

The bookkeeping gets messy.

It does.

And that's precisely why we introduce a new state function to make life easier under these very common conditions.

This is enthalpy symbolized by a dollar.

Here's where it gets really interesting.

How is enthalpy defined?

It's defined simply as decal each plus PVV.

We take the internal energy and we add the product of the system's pressure and volume.

Why add that specific PVV term?

Because watch what happens when we look at the change in enthalpy, delta H at constant pressure.

A small change decal a day is date for plus PDV plus VDP due.

Using data is DQ plus DB and assuming only expansion work plus BXDV plus D day.

Whereas any additional work like electrical, we get data queue, PXXDV plus D devolved plus PDV plus VDPD.

Okay, that was a bit complicated.

If the process happens at constant pressure, degal T and defy, and the external pressure matches the system pressure, and there's no additional work, then the DDV and plus PDV day terms cancel out.

Leaving just DSPB day checks.

Exactly.

At constant pressure, with only PV work being done, the change in enthalpy is simply equal to the heat supplied.

Delta HQP did.

So that's the magic.

Measuring flow in an open beaker directly gives you the enthalpy change for the reaction.

That's incredibly useful.

It's why enthalpy is arguably the most important thermodynamic function for chemists and biologists.

Most things happen at constant atmospheric pressure.

Measuring heat gives you delta H dal.

And just like we had CVI for internal energy, we now have a heat capacity at constant pressure.

Right.

CVP is the change in enthalpy with respect to temperature the pressure is held constant.

So CPE equals partial H marshal TP.

Now, why is CPPH are generally bigger than CVBA?

I remember that from classes.

Think about heating a gas at constant pressure.

As you put heat in, the temperature rises and the gas expands, pushing back the surroundings.

It has to do work.

Ah, so some of the heat energy you supply goes into raising the temperature, increasing internal energy, but some of it is immediately used to do

Exactly.

To get the same temperature rise as you would at constant volume, where all heat goes to internal energy, you need to add more heat at constant pressure to also cover the work cost.

So a TAPI has to be greater than CVRA.

How much greater?

Is there a relation?

For a perfect gas, it's a very simple relationship.

CPCV equals nR doer, where n is the number of moles and R is the gas constant.

That NOLAR term basically represents the extra work done by the gas per degree of temperature rise.

That difference also explains why delta H and delta U can be significantly different for reactions involving gases, right?

Yes.

Remember, delta H equals U plus PVO.

So the change is delta H equals delta U plus delta PV.

For solids and liquids, the PV term is usually piney and doesn't change much.

But for gases, PVs can be substantial.

And using the perfect gas law, PV and RPV, we can approximate the change as

where delta NE is the change in the number of moles of gas in the reaction.

Precisely.

So delta H approaches delta U plus delta RT way.

If a reaction produces more gas molecules, delta NOLAR, the system expands, does work, and delta H is more positive or less negative than delta U.

If it consumes gas, delta NOLAR, the system contracts, work is done on it by the surroundings, and delta H is more negative or less positive than delta U.

Let's take that water formation example again.

$2 text, H2 text.

We start with three moles of gas with basically zero moles of gas, so delta NE makes three three.

Right.

So delta H, delta U, three RT, though.

Since RT is positive, delta H will be a larger negative number, more exothermic than delta U in this case, because the atmosphere collapsing in on the system effectively adds energy to it, making you not drop as much as H.

Oh, wait, I might have flipped the sign intuition there.

Let's rethink.

Delta H, delta U plus delta TV, delta ingot is negative, so delta H is negative, delta H, delta U plus.

This means delta H is less than delta U1.

For an exothermic reaction where delta U is negative, delta H will be even more negative.

The collapsing atmosphere does work on the system, contributing to the energy release measured as heat.

Yes, that makes sense.

Okay, good.

So delta H is the key for practical chemistry.

Now, how do we use it systematically?

That gets us into thermochemistry.

Right.

We need a standard baseline for comparison.

This is the standard state, the pure form of a substance at a pressure of one bar.

Temperature isn't part of the definition, but it's usually specified, often as 298 .15 K, 25 degrees C.

And using this, we define the standard enthalpy change.

In the little circle means standard.

That's the enthalpy change when reactants in their standard states transform into products in their standard states.

And because enthalpy is a state function.

Ah, path independence again.

This must be where Hess's law comes in.

You got it.

Hess's law says that the overall standard enthalpy change for a reaction is the same whether it happens in one step or multiple steps.

You can add and subtract the delta H and its values of known reactions to figure out the delta H and its for a reaction you don't know, as long as they combine to give the overall reaction.

Like calculating the enthalpy of sublimation, solid to gas, by adding the enthalpy of fusion, solid to liquid, and the enthalpy of vaporization, liquid to gas.

Perfect example.

Delta encaminous plus delta texite works because it also only cares about the start and end states, solid and gas.

This state function property is incredibly powerful.

It leads to something even more useful, right?

The standard enthalpies of formation.

The standard enthalpy of formation.

This is the standard reaction enthalpy for the formation of one mole of a compound from its elements in their reference states.

Reference state.

That's the most stable form of the element at one bar and the specified temperature.

Like graphite for carbon, O2 gas for oxygen, N2 gas for nitrogen.

Exactly.

And crucially, the standard enthalpy of formation of any element in its reference state is defined as zero.

It's our baseline.

So if we have tables of these delta schoolies values for all sorts of compounds, then we can calculate the standard enthalpy change for any reaction without ever doing the experiment.

We just sum up the delta tex values of all products, multiply by their stoichiometric coefficients, and subtract the sum of the delta tex values of all the reactants.

So it's delta tex volume, delta tex reactants, and the progesty, summed to telpinase.

That's amazing.

Built entirely on dollars being a state function.

It's the backbone of practical thermochemistry.

But what if we want to know the reaction enthalpy at a different temperature, not the standard 298k where most formation data is listed?

Ah, good question.

That's where Kirchhoff's law helps.

It tells us how delta tex changes with temperature.

How does it work conceptually?

Well, the enthalpy of the reactants changes with temperature according to their total heat capacity.

The enthalpy of the products also changes with temperature according to their total heat capacity.

If these heat capacities are different, then the gap between the reactant enthalpy and product enthalpy, which is the reaction enthalpy, must also change as temperature changes.

Makes sense.

If products get hotter, faster than reactants, the enthalpy difference will shift.

Precisely.

Kirchhoff's law formalizes this.

It says the change in reaction enthalpy when you go from temperature T dollar to T dollar two is the integral from two dollar to T dollars dollars of the difference in molar heat capacities between products and reactants.

So delta tex D2, delta tex NETT1 plus delta Cp on plus Cdp omas.

If the heat capacity difference is roughly constant, it simplifies, but the principle holds.

Exactly.

It allows us to adjust tabulated 298k beta to other temperatures relevant for industrial processes or specific conditions.

Okay, let's step back a bit for section 2D.

We keep saying state function.

The math behind this involves exact and inexact differentials, right?

Can you clarify that?

Sure.

Think of it like altitude versus distance traveled on a hike.

The change in altitude depends only on your start and end points.

That's like DTAM or D dollars, an exact differential.

It's integral delta U1 or delta H day is path independent.

But the actual distance you walk depends entirely on the path you choose winding or straight up.

Right.

That's like heat and work deli doll.

They're inexact differentials.

The total heat transferred or work done depends completely on the process path.

You can get the same delta U day via many different combinations of tall and dollar.

So the math respects this difference.

We can integrate gain easily between states, but integrating data or deli requires knowing the specific path.

Exactly.

And this structure lets us define some interesting properties using partial derivatives.

One is the internal pressure.

It's defined as partial VD.

So how internal energy changes when volume changes if we keep temperature constant?

What does that physically mean?

It essentially measures the strength of molecular forces within the substance.

For a perfect gas, molecules don't interact, so spreading them out by increasing volume doesn't change their potential energy.

Joule found experimentally that for a near perfect gas T at T equals dollars.

But for real gases or liquids, molecules do attract each other.

Right.

So if you pull them apart by expanding the volume at constant T, you have to increase their potential energy like stretching a spring.

This means dollar increases as fire increases.

So IOT is positive for most real substances where attractions dominate.

It's a measure of the stickiness of the molecules.

Interesting.

And there's another general relation between CBA and CVA evils, not just for perfect gases.

Yes, a more general one.

CPCV is alpha two TV kappa.

It involves the expansion coefficient, which measures how much volume changes with temperature and the isothermal compressibility, which measures how much volume changes with pressure.

That looks more complex, I guess it shows CPI is generally greater than CPO or CPI even for liquids and solids because things usually expand when heated.

Alpha dash.

Correct.

Unless alpha is zero, which happens for water around four degrees C, CPI will be greater than CV.

Now what's fascinating here is the Joule Thompson effect.

This sounds related to the internal pressure idea.

It is, but it's measured under different conditions.

The Joule Thompson effect looks at what happens to the temperature of a gas when it undergoes expansion where the enthalpy is kept constant isenthalpic expansion.

This happens when a gas expands through a throttle or porous plug in an insulated pipe.

So gas flows from high pressure to low pressure through some restriction and we measure the temperature change.

Exactly.

We measure the Joule Thompson coefficient defined as partial T partial pH doll,

the change in temperature per unit change in pressure at constant enthalpy.

And what happens?

Does it cool down or heat up?

For a perfect gas, nothing.

One in a unit.

Nothings.

No intermolecular forces.

No temperature change on isenthalpic expansion.

But for real gases, it depends.

Depends on what?

Primarily on temperature and pressure relative to the gas's inversion temperature.

Below the inversion temperature, attractive forces dominate.

As the gas expands, molecules move apart, overcoming these attractions.

This requires energy, which comes from their kinetic energy.

They slow down, meaning the gas cools.

Precisely.

Below TDAJ is positive expansion.

Pressure drop leads to cooling, temperature drop.

This is the effect used in refrigerators and to liquefy gases like nitrogen and oxygen in the lin process.

And above the inversion temperature.

Above T dollars, repulsive forces tend to dominate or kinetic energy effects are more significant.

Expansion actually leads to a slight heating of a gas, so more LARS negative.

Most gases are below their inversion temperature at room temperature, which is why compressed air cools as it escapes.

Hydrogen and helium are exceptions.

Their inversion temps are very low, so they warm up on expansion at room temp.

That's really practical.

Okay, final section 2e.

Adiabatic changes.

What defines these?

Adiabatic means no heat transfer.

$12 dollars.

The process happens in a thermally insulated system, or so fast that heat doesn't have time to flow.

If Keeney dollars, the first law becomes very simple.

Delta U.

Right.

Any change in internal energy is solely due to work done.

If a gas expands adiabatically, it's negative because work is done by the system, its internal energy must decrease.

And for a gas, lower internal energy means lower temperature.

So adiabatic expansion always causes cooling.

Yes.

The system pays for the expansion work by using up its own thermal energy.

The work done is directly related to the temperature drop.

For a perfect gas, CV delta T fall.

How does this look on a pressure volume graph compared to an isothermal expansion?

An adiabatic expansion curve called an adiabat is steeper than an isotherm starting from the same point.

Why steeper?

Because in an isothermal expansion, as the volume increases, the pressure drops only due to the increased volume.

Cropto 1bui.

The system draws heat from the surroundings to keep T constant.

But in an adiabatic expansion?

Pressure drops for two reasons.

First, the volume increases, just like before.

Second, the temperature falls as the gas does work.

Lower temperature further reduces the pressure, since pp propto.

So the pressure plunges more steeply for a given volume change.

Got it.

The relationship for a reversible adiabatic expansion of a perfect gas is PV gamma tex constant, where gamma is the heat capacity ratio, Cp Cvi's.

Since gamma is always greater than 1, this confirms the steeper dependence on V compared to the isothermal PV pex constant.

Exactly.

That gamma factor captures the extra pressure drop due to cooling.

Okay, let's try and wrap this up.

We covered a lot of ground.

We did.

The key takeaways really start with the first law, delta u plus w dou.

It's the basic energy conservation rule.

Internal energy, tout de la, is the total energy store, and it's a state function path independent.

But for practical constant pressure chemistry, enthalpy, del H delta u plus PV dou, is often more useful.

Because it's change, delta H equals the heat flow we can easily measure.

Delta H equals Qp dou.

And because $1 are state functions, we get incredibly powerful tools like Hess's law, allowing us to calculate reaction enthalpies from standard enthalpies of formation.

Khrushchev's law lets us adjust these for temperature.

We also saw how the math of state functions, exact differentials, contrast with heat and work, inexact, and how properties like internal pressure and the Joule -Thompson effect review the importance of intermolecular forces and connect to real -world applications like gas liquefaction.

And finally, adiabatic processes showed us the direct link between work done and temperature change, explaining why idovats are steeper than isotherms.

So what does this all mean?

We've basically laid out the fundamental rules for energy accounting in chemistry.

Understanding internal energy and especially enthalpy is absolutely vital.

Think about it calculating the energy released by fuels, the caloric content of food, the efficiency of engines or power plants, even understanding metabolic processes in our own bodies.

It all comes back to the principles, particularly enthalpy changes, discussed here.

It really is the foundation for understanding energy flow in almost every chemical and physical process.

We hope this deep dive helps solidify those concepts for you.

Absolutely.

Thanks for tuning in and exploring the first law with us.

We'll catch you on the next deep dive.