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### The Ideal Gas Law

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Course Overview 0:16
• Thermodynamics & Classical Thermodynamics
• Structure of the Course
• The Ideal Gas Law 3:06
• Ideal Gas Law: PV=nRT
• Units of Pressure
• Manipulating Units
• Atmosphere : atm
• Millimeter of Mercury: mm Hg
• SI Unit of Volume
• SI Unit of Temperature
• Value of R (Gas Constant): Pv = nRT
• Extensive and Intensive Variables (Properties) 15:23
• Intensive Property
• Extensive Property
• Example: Extensive and Intensive Variables
• Ideal Gas Law 19:24
• Ideal Gas Law with Intensive Variables
• Graphing Equations 23:51
• Hold T Constant & Graph P vs. V
• Hold P Constant & Graph V vs. T
• Hold V Constant & Graph P vs. T
• Isochores or Isometrics
• More on the V vs. T Graph
• More on the P vs. V Graph
• Ideal Gas Law at Low Pressure & High Temperature
• Ideal Gas Law at High Pressure & Low Temperature

### Transcription: The Ideal Gas Law

Hello and welcome to www.educator.com.0000

Welcome to the first lesson of Physical Chemistry.0002

Before I actually launch into the Physical Chemistry, I want to talk a little bit about the course as a whole so0006

that you have an idea of what is that you are in for and what is there to expect.0011

Physical chemistry is taught in two ways.0018

We teach classical thermodynamics first and then we could quantum mechanics and then we go back and teach statistical thermodynamics.0023

In other words, we go back and we use quantum mechanics to explain what we learned in thermodynamics.0032

The other way that is taught, it seems to be becoming more and more popular these days, its quantum mechanics first0039

and then statistical and classical thermodynamics are done simultaneously.0046

This particular course does classical thermodynamics first, so I go with the first method.0052

Classical thermodynamics first then it goes on to quantum mechanics, spectroscopy next, and then finally, statistical thermodynamics.0057

Many choices have to be made in AP Chemical course about which topics to cover.0067

The more topics one covers, the less time one has to cultivate it for understanding of the fundamentals.0072

In other words, time is finite if I spend more time on this, it is more topics, but it gives me less time to assimilate what it is that I have learned before.0079

I have always been one who believes that it is more fruitful to read three novels closely and deeply than to have read 20 novels,0091

and for the sake of impressing be able to enumerate 20 novels.0098

You get a lot more out of doing less but doing it very well, that is always the case.0102

If you ever run across a situation where you have to make a choice, I promise you, if you make the choice of doing fewer things0109

but doing them well and deeply, it will be a lot better than having done multiple things.0115

It is just the way the mind works, it is just the way discipline works, is just a way to true deep learning which actually works.0121

I have chosen what I consider to be the most important topics for a strong foundation.0129

If, after further reflection, I feel that what I have left out deserves to be covered or if there is a clear demand for topic that I have left out,0134

I will absolutely be happy to add that topic of the course, however many topics that might be.0142

If what I have presented in the course is understood to a reasonable degree then any topic that I have left out,0148

but your particular course does cover, it should be reasonably easy to follow.0155

That should be very easy to follow.0159

I have presented as a solid, good, foundational course.0161

This course is a very important part of your scientific literacy.0168

It is a first exposure to pulling back the curtain and exposing what is actually going on.0171

I hope that you find it as beautiful as I do.0178

Best wishes and let us jump right on in.0180

Mostly, what we will be talking about is gases.0191

We will talk about liquid and solids and we will see a couple of them in the problems but gas is going to be what we are most concerned with.0195

Let us begin with the ideal gas law.0203

The ideal gas law, you know it as PV = NRT, the pressure given a gas, the pressure × volume = the number of mol × the gas constant × the temperature in Kelvin.0207

That is it, nothing more than that.0218

It is an equation state.0232

It is an equation state and you are going to hear that term a lot as an equation of state.0235

What that means is the four variables, pressure, volume, temperature, and the number of mol.0241

Describe completely the state of our gases in at a given moment.0259

Describe the state of the gas.0265

Let us talk about units, units are going to be very important in this course.0275

In fact, probably, the single most annoying thing that you would have when you do with your problems is remembering to deal with the units and covering the units.0279

It just makes all this crazy but there it is.0289

Let us talk about units, the SI unit of pressure is a Pascal, symbolized as Pa.0294

Pressure is defined as force per unit area.0318

When I have the amount of force, if I divide it by the particular area over which I’m applying the force, I get the pressure, pressure = f/a.0325

Even force is a Newton and the unit of area is a square meter so 1 Pa = 1 N/sq m.0336

From here, we can go ahead and manipulate the units to serve our purposes.0353

From here, we can manipulate units as necessary to move between equivalent unit expressions.0363

If you ever lose your way to the problem.0398

Let us think about the units and that will help you solve the problem in many of the cases, particularly in classical thermodynamics.0402

It will let you move between equivalent unit expressions.0409

Let us go ahead and do not stick to the page here.0414

We have 1 N/ sq m, in Newton is a kg m/s² /m².0417

We have 1kg m² s, there we go.0434

1 N/ m², which is a Pascal, is equal to 1 kg m/s² /m².0449

If I multiply by m/ m, I end up with 1 kg m²/ s² / m³ and that is equal to 1 J/ m³.0463

1 N/ m² which is 1 Pa is also equal to 1 kg /ms² = 1 J / m³.0482

That is it, I’m just manipulating units.0491

I have 1 atmosphere of pressure is equal to 1.01325 × 10⁵ Pa, this exact.0497

I have 1 atmosphere is equal to 760 tors, that is also exact.0519

The mm of mercury Hg, is actually bigger than the tors by a factor of 14 parts or 10⁸ which is clearly insignificant.0529

We take 1 mm of mercury equal to 1 tors.0560

The SI unit of volume, as I have said is a unit of pressure, now we will do it as a unit of volume is actually the cubic meter not the cubic centimeter.0574

In practice, we use cubic centimeters and we use liters.0589

1 liter is 1 cubic decimeter = 1000 cubic centimeters = 10⁻³ cubic meters.0604

1 liter is 10⁻³ cubic meters, this is exact.0619

In the next page, the SI unit of temperature is the Kelvin.0630

The value of r, the gas constant, pv= nrt, r = pv/ nt, p0 v0/ n0 t0.0654

If I take the pressure to be 1 atmosphere.0678

Let us do it this way, I will just write out what does we still have here.0689

P0, if we take 1 atmosphere as our base which we said as 1.01325 × 10⁵ Pa.0699

If I take my initial volume or basic volume to be 22.41383 L, and if I take N to be 1 mol which is going to be 22.41383 × 10⁻³ m³/ mol.0712

We are trying to stick with SI units, temperature = 273.15 Kelvin.0745

Therefore, if I put all of these in values for p0 v0 n0 t0, sticking with m³/mol Pa, I end up with r= 8.31441 Pa m³/ mol Kelvin.0754

Recall that 1 Pa = 1 J/ m³, so 1 Pa m³ = 1 J.0782

Pa m is a Joules, therefore, r = 8.31441 Js/mol Kelvin.0801

That is where all that stuff comes from, J/ mol Kelvin.0811

If I wanted it in liters/ atmosphere/ mol Kelvin, I get 0.08206.0817

I will give the number here 0.058, that is liter atmosphere/ mol Kelvin.0824

If you want an equivalent expression for J at liter atmosphere, just use this conversion factor right here.0835

If you want an equivalent between J and liter atmosphere, and you will, then 8.31441 J/ mol Kelvin × 1 mol Kelvin is 0.08206 liter atmosphere mol Kelvin.0844

It will cancel mol Kelvin and I’m left with 101.3 j/ liter atmosphere.0894

That is the conversion factor.0905

Keep a very close eye on units and conversions.0913

It is going to be very important in this course.0915

Let us talk about, I will go to the left here.0920

Let us talk about extensive and intensive variables.0925

Extensive and intensive variables or properties.0931

Let us start off with the ideal gas law, pv = nrt, intensive property.0946

The two intensive properties in this are pressure and temperature.0958

The reason they are intensive is that they do not depend on the amount of the substance present.0965

An intensive property and intensive variable is the one that does not depend on the amount of it present.0984

An extensive properties here are volume and the number of moles, they do depend on amount.0991

Let us see, mass is another extensive property because the amount of mass depends on how much you have.1013

You have 3g or 4g, it is going to make it a difference.1026

Mass is an extensive property.1028

If I take 1 mass of iron, if I take 1 g of iron and 10 g of iron both sitting on a table, the temperature is going to be the same.1031

If I measure the temperature of the 1 g or 10 g, it does not matter how much s there.1039

The temperature is intensive, it has nothing to do with how much iron is there but the mass does, volume does, number of moles does.1044

That is the difference between the two.1053

Let us see, this is very important now.1057

The ratio of 2 extensive variables always gives an intensive variable.1065

For example, our example is going to be, if I have mass which is an extensive variable and if I have volume which is an extensive variable,1101

if I take the ratio of the mass to the volume I get the density.1118

The density is an intensive variable.1128

1 g of iron, 10 g of iron, the density of iron stays.1131

1 g of iron has a certain volume, 10 g of iron has a certain volume, that is our extensive but if I take the ratio of 2 extensive, I will end up with intensive variable.1138

That is very deep, very profound, and very important, density intensive.1148

I repeat, the ratio of 2 extensive variables always gives an intensive variable.1156

We would often do that.1160

Let us go ahead and take a look at pv = nrt.1165

We have pv = nrt and we are going to divide both sides but n.1170

I end up with pv / n = rt.1184

I'm going to rewrite this as, basically what I am going to do is take this.1188

I’m going to take whatever volume, which is an extensive variable, I’m going to divided by n, which is also an extensive variable.1193

They are both extensive, the ratio of 2 extensive is an intensive.1203

I’m going to rewrite this as pv ̅ = rt, where V ̅ Is just equal to v/n.1206

I have taken the volume and divide it by n.1216

It is called the molar volume, volume per mol.1220

The ideal gas law written that way consists of all intensive variables.1226

We can discuss gases and their properties without worrying about whether there is 1 or 50 mol.1261

Now the amount does not matter.1267

When amounts do not matter, that is where we begin to uncover and elucidate underlying truths.1270

Now, we can discuss gases and their properties without worrying about whether we have 5 mol or 50 mol.1277

It is not going to make any difference.1312

Any fundamental property of the system should never depend on how much is there.1315

As I said, we wish to elucidate general truths in any specific case we might deal with the amount.1348

But when we talk about generalities, we never talk about the amount.1368

We should have to talk about the amount.1370

We should plot it in every situation across the board.1372

In general, volume is not the only thing that you will see with a line over it.1382

In general, any variable with a line over it is a molar quantity.1388

In other words, it is been divided by the number of moles that are present.1403

Molar quantity meaning it has been divided by n, the number of moles.1407

Let us talk about being able to graph equations.1431

Graphing equations, we will clearly graph an equation is just insanely important in science.1435

This is a relation among 3 variables, pressure, a molar volume, and temperature.1457

It is a relation among 3 variables.1468

If any 2 or none, the 3rd is automatically known.1477

Let us solve for each variable separately.1499

We are going to solve for p, v, and we are going to solve for t.1501

Let us start by solving for p.1506

P= rt/( v) ̅ , I’m going to write it as rt × 1 / V ̅.1509

This is 1/V ̅, these are hyperbolas.1521

I’m going to hold temperature constant and I’m going to graph pressure vs. Volume.1529

Holding t constant is very important.1539

We will graph p vs. V ̅, in other words, p is on the y axis and V ̅ Is the x axis.1549

As the molar volume changes, it gets bigger.1569

What happens to the pressure or as the pressure gets bigger, what happens to the volume?1573

Here is what we get.1579

This is the pressure, it is going to be at atmospheres, the axis is going to be molar volume, it is going to be deci³/ mol.1588

Molar volume what you get when you graph this equation by holding t constant, you have to put this.1602

You will end up with hyperbolas.1619

This is the one for 100 Kelvin, this is one for 200 Kelvin, again, I get different graphs, lines, curves, one for each temperature that I’m holding constant.1623

I am holding temperature constant and I’m changing v and let us see what happens to p.1641

This might be 400 Kelvin.1647

Clearly, the whole thing, there is an infinite number of these.1650

For a given v or volume, higher temperature means a higher pressure.1662

You already know this from General Chemistry that a higher t means a higher p, that is it what is going on.1691

Let us go forward again.1701

Every point of the PD graph or the PD plane, every single point of that two dimensional plane represents a particular state of the gas.1710

In other words, it represents a particular pressure, a particular volume, and a particular temperature.1735

It represents a particular state of the gas by holding the temperature constant, by holding the t constant.1742

I can strain the states to follow the curve.1773

Here is my p, here is my v, and I have a bunch of different temperatures.1788

Now as I change pressure, as I change volume, the change is going to follow that curve.1795

It is just going to bounce around, it is going to follow that curve for different temperatures.1805

It is going to be another curve for another temperature.1809

When I make changes, it goes up and down along the curve.1813

These curves for this particular graph holding temperature constant, these curves are called isotherms.1819

Isotherms just mean equal heat.1830

That is all it means, equal heat or in other words, holding temperature constant.1837

In thermodynamics, when you hear the term isothermal that means I'm keeping the temperature constant, that is all it means.1842

These curves are called isotherms when t is held constant.1848

Let us start again with pv = rt, this time let us go ahead and solve for volume.1868

V is equal to rt/p, I can write this as r/p × t.1877

That is interesting, this is linear.1890

I’m going to hold pressure constant.1894

Now let us hold pressure constant, this is constant and r is already a constant.1900

I’m going to graph v vs. T and express v as a function of t.1910

Here is the graph that I get.1929

This is my volume, this is my temperature in Kelvin, and this is linear.1934

And I hold p as a constant at different pressures, this is what looks like.1941

This one might be 3 atmospheres, let us say this one is 2 of atmospheres, let us say this one is 1 atmosphere.1952

What I have done is I have expressed this pv= nrt, now I have graphed the volume vs. Temperature.1965

At different pressures that I choose, the hold constant, I have these lines.1974

When I hold pressure constant, these lines are called isobars.1982

If I pick it as pressure 3 atmospheres, if I raise the temperature, the volume is going to travel along this line.1986

This is going to be everywhere.1994

At 2 atmospheres, if I change the volume, if I change the temperature the volume is going to travel along that line.1996

It will go backwards.2000

If at 1 atmosphere, it is going to travel along this line.2003

This is a state and a state for a given volume, for a given temperature, and for a given pressure.2006

These lines are called isobars.2016

When pressure is held constant, the lines that you get are called isobars.2023

When pressure is held constant and you have volume vs. Temperature those are called isobars.2027

As pressure rise, of course r with p, the slope decreases.2047

That is why from your perspective, as the pressure rises, the line gets closer.2063

It becomes more flat closer to the x axis.2069

Let us start again with pv ×rt, this time let us go ahead.2083

Let us try this, let us go p = r/ vt.2102

Let me shift one thing and make sure that.2124

P = rt, pressure = rt/ v.2133

What we did is we took our t and we held temperature constant.2143

I’m going to take r/ v × t, I’m going to take r/ v separate × t.2152

Again, we get a linear graph and now I’m going to graph pressure vs. Temperature but I'm going to hold the volume constant.2159

And what I get here is this a linear graph.2170

Pressure, temperature, different volumes, so again, we get something linear2178

This one might be 30 deci³/ mol, this one might be 20 deci³/ mol, this one might be 10 deci³/ mol.2191

Again, we are talking about molar volume.2210

Whenever I have these, whenever I have a pressure, temperature, graph for a particular volume,2215

if I change the temperature my pressure is going to move along this line.2221

These lines are called isochors or isometrics.2226

For a constant volume, for constant molar volume, we get things called isochors or isometrics.2234

Lines of constant volume, that is it.2257

Let us go ahead and draw this all out here.2262

I have got p = rt / V ̅.2265

I have got p and v = r/ p × t and I got p = r/ v × t.2280

This graph I ended up with hyperbolas.2302

This t was held constant and these were called isotherms.2307

Here we have v along this axis, this is p along that axis and this is going to be v along that axis.2321

This is v along this is axis and this is t along this axis.2335

I ended up getting these things, these were called isobars.2337

In this case, I held p as a constant.2343

Basically, the thing that you are taking with the gas constant here are t/ v, these are the variables, hold that thing called constant.2348

V and t are variables, hold the other variable constant.2355

Here it is p and t, we will hold the other variable constant.2358

We are going to hold this constant and again here we have pressure, here we have temperature, and we ended up with a bunch of v and this are called isochors.2361

Let me take a look at this one right here, this was molar volume vs. Temperature.2376

This equation here which was molar volume= r/ p × t.2399

This equation, when I look at these pictures, this equation and graph is easier to see, as temperature drops to 0 it is telling me that the volume is going to drop to 0.2409

This equation and graph, they imply that at t = 0, the molar volume = 0.2433

OK this does not happen to real gases and here is why.2447

Gases as t decreases, at some value of t the gas actually liquefies, it turns into liquid.2466

The gas liquefies so a further drop in temperature does not change the volume.2485

When something is liquid, the temperature in that stays at that volume.2499

A further drop in t does not change the volume.2505

It does not increase the volume.2511

In other words, this is an ideal gas would behave this way that real gases liquefy at low temperatures.2514

Therefore, the graph itself real gases, they do not go to 0.2519

OK now first equation that we did was this one where we have pressure and where we had the molar volume.2529

This equation and this graph which was p = rt/ v.2543

1/v or v was the x axis, these imply that as pressure increases, as I go up in pressure it implies that the volume actually go down to 0.2553

As p increases, the volume goes to 0 but again, that does not happen.2579

At a certain pressure, at a certain p, the gas liquefies.2589

Let us say you are just squeezing all the gas particles at some point they will just turn into liquid.2598

The gas liquefies.2603

No further increase in p reduces the molar volume.2606

This is called isothermal compression.2635

It is isothermal because you are moving along isotherm.2636

These are called isotherms because you are compressing it.2639

In other words, you will be increasing the pressure.2641

As you increasing the pressure, what is going to end up happening is the volume is going to get closer and closer to 0.2644

But at some point, it is going to liquefy and it is not going to go any further.2652

Hence, the ideal gas that is why it is called the ideal gas law.2656

It is never a real gas law, hence, an ideal gas.2660

At low pressures and high temperatures, the ideal gas law is actually very accurate.2670

The ideal gas law is quite accurate at describing gases behavior.2691

Deviations from ideal behavior occur at high pressures and or low temperatures.2721

In other words, as the molecules starts to get really close to each other.2750

We will talk about that more in subsequent sections.2755

Thank you for joining us here at www.educator.com, we will see you next time for our continuation of classical thermodynamics.2758