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 1 answerLast reply by: Professor Dan FullertonTue Oct 28, 2014 6:19 PMPost by Siyan He on October 28, 2014For AP exam, do we need to memorize the constants like kB, k, Î± Î² for some kinds of substances 1 answerLast reply by: Professor Dan FullertonWed Jul 30, 2014 9:22 AMPost by Jamal Tischler on July 30, 2014Couldnt the temperature to be Kavg = 1 J/K * T ? It would have been the same thing in divisions. 1 answerLast reply by: Professor Dan FullertonSat Aug 24, 2013 7:26 PMPost by Janaki Ramam Vedula on August 24, 2013could you please solve the below question :-two identical rods of metal are wielded in series then 20 cal  of heat flows through them in 4 min.if the rods are connected in parallel the same amount of heat will flow in how many minutes? 1 answerLast reply by: Professor Dan FullertonMon May 13, 2013 6:39 AMPost by Nawaphan Jedjomnongkit on May 13, 2013From Ex 9: How do we know that the glass also have 1 L volume? So if we think in this way is that mean if I have a bottle of 1 L water, the bottle that hold 1 L of water inside also has volume of 1L ? So if I have a thick and thin bottle with the same ability to hold same amount of water, would they have different volume of glass while they hold same volume of water? Thank you.

### Temperature, Heat, & Thermal Expansion

• The internal energy of an object, known as its thermal energy, is related to the kinetic energy of all the particles comprising the object. Because there is a range of particle kinetic energies, the system is modeled as a distribution of kinetic energies and analyzed using Maxwell-Boltzmann statistics.
• The total thermal energy is the sum total of the kinetic energies of the constituent particles.
• The average kinetic energy of the particles is directly related to the temperature of the object.
• Even though two objects can have the same temperature (and therefore kinetic energy), they may have different internal energies.
• Heat is the transfer of thermal energy from one object to another object due to a difference in temperature.
• Conduction is a thermal energy transfer process in which particle collisions transfer momentum from on object to another. After many collisions, both systems of particles have the same temperature. On the microscopic scale, this is a probabilistic process.
• Heat transfers by conduction, convection, and radiation.
• When objects are heated, they tend to expand. When they are cooled, they tend to contract.

### Temperature, Heat, & Thermal Expansion

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
• Objectives 0:12
• Thermal Physics 0:42
• Explores the Internal Energy of Objects Due to the Motion of the Atoms and Molecules Comprising the Objects
• Explores the Transfer of This Energy From Object to Object
• Temperature 1:00
• Thermal Energy Is Related to the Kinetic Energy of All the Particles Comprising the Object
• The More Kinetic Energy of the Constituent Particles Have, The Greater the Object's Thermal Energy
• Temperature and Phases of Matter 1:44
• Solids
• Liquids
• Gases
• Average Kinetic Energy and Temperature 2:16
• Average Kinetic Energy
• Boltzmann's Constant
• Temperature Scales 3:06
• Converting Temperatures 4:37
• Heat 5:03
• Transfer of Thermal Energy
• Accomplished Through Collisions Which is Conduction
• Methods of Heat Transfer 5:52
• Conduction
• Convection
• Quantifying Heat Transfer in Conduction 6:37
• Rate of Heat Transfer is Measured in Watts
• Thermal Conductivity 7:12
• Example 1: Average Kinetic Energy 7:35
• Example 2: Body Temperature 8:22
• Example 3: Temperature of Space 9:30
• Example 4: Temperature of the Sun 10:44
• Example 5: Heat Transfer Through Window 11:38
• Example 6: Heat Transfer Across a Rod 12:40
• Thermal Expansion 14:18
• When Objects Are Heated, They Tend to Expand
• At Higher Temperatures, Objects Have Higher Average Kinetic Energies
• At Higher Levels of Vibration, The Particles Are Not Bound As Tightly to Each Other
• Linear Expansion 15:11
• Amount a Material Expands is Characterized by the Material's Coefficient of Expansion
• One-Dimensional Expansion -> Linear Coefficient of Expansion
• Volumetric Expansion 15:38
• Three-Dimensional Expansion -> Volumetric Coefficient of Expansion
• Volumetric Coefficient of Expansion is Roughly Three Times the Linear Coefficient of Expansion
• Coefficients of Thermal Expansion 16:24
• Example 7: Contracting Railroad Tie 16:59
• Example 8: Expansion of an Aluminum Rod 18:37
• Example 9: Water Spilling Out of a Glass 20:18
• Example 10: Average Kinetic Energy vs. Temperature 22:18
• Example 11: Expansion of a Ring 23:07

### Transcription: Temperature, Heat, & Thermal Expansion

Hi everyone and welcome back to Educator.com.0000

I am Dan Fullerton and I am thrilled to be opening up our unit today on thermophysics.0003

Our objectives are going to be to calculate the temperature of an object given its average kinetic energy, to describe the temperature of a system in terms of a distribution of molecular speeds, and describe thermal equilibrium as a probability process where energy is typically transferred from high to low energy particles.0011

We will also explain heat as the process of transferring energy between systems at different temperatures, and finally calculating the linear and volume metric expansion of a solid as a function of its temperature.0027

Thermophysics explores the internal energy of objects due to the motion of the atoms and molecules comprising the objects.0045

It explores the transfer of this energy from object to object, known as heat, a transfer mechanism for energy.0052

The internal energy of an object, known as its thermal energy is related to the kinetic energy of all the particles comprising the object.0063

The more kinetic energy the constituent particles have as they move in their vibrations as part of that object, the greater the objects thermal energy.0071

For most systems, the kinetic energy of the constituent particles is not the same, it is a distribution, therefore the system is modeled as a distribution of kinetic energies, typically using Maxwell Boltzmann's statistics.0080

As we talk about temperature and phases of matter, in solids the particles comprising the solids are held together very tightly, therefore their motion is limited to just vibrating back and forth in their given positions.0103

In liquids, the particles can move back and forth across each other, but the object itself does not have a defined shape.0116

In gases, the particles move throughout the entire volume available, but in all cases the total thermal energy is the sum of the kinetic energies of the constituent particles.0122

Average kinetic energy and temperature -- actual kinetic energies of individual particles may vary significantly and the average kinetic energy we can find by taking 3/2 times this constant Kb, known as Boltzmann's constant times the temperature and that temperature should be in Kelvins (K), our si unit of temperature.0136

If Kb is Boltzmann's constant, that is 1.38 × 10-23 J/K -- the temperature is in Kelvins, the si unit of temperature again, not Celsius, not Fahrenheit, but Kelvins.0157

Now it is important to note that even though two objects can have the same temperature and therefore the same average kinetic energy, they may have different internal energies, depending on what those particles that are moving are.0172

Let us take a look at some temperature scales.0186

We have Fahrenheit (F) on the left, Celsius (C) in the center, and Kelvins (K) on the right.0189

Now they all work in the same basic way, but they have different values at different key temperature readings.0194

The Fahrenheit scale has water freezing at 32 ° F and water boiling at 212 ° F and if you extrapolate that back, you get to what is known as absolute 0 at about -459.7 ° F, where absolute 0 is a theoretical minimum temperature.0199

It is the point on a volume vs. temperature graph on a gas where the extended curve would hypothetically reach 0 volume.0220

It is not specifically the absolute lack of motion of particles, it is a theoretical minimum, but for our purposes, really cold and you do not get any colder than that.0230

For Celsius, water boils at 100 ° C, freezes at 0 ° C, and absolute 0 would be -273.15 ° C.0240

Now Kelvins, the scale we are going to use here in physics -- and Kelvins has the same size of its main unit, a Kelvin -- 100 ° between water freezing and boiling, but the only difference is we are going to start 0 at absolute 0.0250

That means that water freezes at 273.15 K and it boils at 373.15 K.0268

Converting between temperature scales is fairly straight forward.0277

If you know Celsius, to get Kelvins, you add 273.15.0280

If you know Celsius and you want Fahrenheit, multiply the Celsius temperature by 9/5 and add 32.0286

If you know Fahrenheit and you want Celsius, take the Fahrenheit temperature, subtract 32 and multiply by 5/9.0293

Now let us talk about heat as the transfer of thermal energy from one object to another object due to their difference in temperature.0304

That is typically accomplished through some sort of particle interactions or collisions in which momentum is transferred from one object to another.0313

That is conduction.0319

Now energy is typically transferred from higher energy to lower energy particles and after many collisions, both systems of particles likely have the same average temperature.0322

That does not mean that every collision works this way, but on the average it goes in that general direction.0330

Because the particles comprising objects have a distribution of particles, velocities, and energies, on the microscopic scale, this transfer of energy is a probabilistic process.0335

So as you look at it more and more closely, you have to get more and more into statistics of distribution.0345

So methods of heat transfer -- We can transfer heat from one object to another by three different methods.0353

Conduction is the transfer of energy along an object to the particles comprising the object colliding.0359

Think of sticking an iron rod in the fire.0364

Okay, the fiery end is going to get hot real quick, but if you hold that long enough, the other end that started off cool is going to get pretty hot.0368

That is by the transfer of the energy from particle to particle to particle in that object.0374

Convection is the transfer of energy as a result of energy or heated particles moving from one place to another, like the convection ovens -- heated air molecules move from one place to another.0379

And radiation is the transfer of energy through electromagnetic waves.0389

Now as we try and quantify heat transfer in conduction, we can get a look at the rate of heat transfer, (H) in J/s or also watts (W).0397

Heat is K × A × δt/L, where δt is your temperature gradient, the difference in temperature; (A) is the cross-sectional area, (L) is the length of your object, and (K) is a thermal conductivity depending on the material, typically something you would look up, a material property.0407

Now, I have put in here a table of some thermal conductivities of selected materials and on the left we have materials such as aluminum, concrete, copper, glass, stainless steel, and water and on the right they are thermal conductivities in J/(s-m-K).0432

For example, copper has a much higher thermal conductivity than something like water.0447

Let us see how we can put this into practice.0455

What is the average kinetic energy of the molecules in a steak at a temperature of 345 K?0458

Well the average kinetic energy is given by 3/2 times Boltzmann's constant times the temperature.0464

So that will be 3/2 × 1.38 × 10-23 (Boltzmann's constant) and a temperature of 345 K is going to give us an average kinetic energy of about 7.14 × 10-21 J.0474

Let us take a look at another example this time dealing with body temperature.0498

Normal canine body temperature is 101.5 F. What is normal canine body temperature in degrees C and K?0504

Well let us convert temperature in degrees C -- is 5/9 times the temperature in degrees F minus 32.0513

So that will be 5/9 × 101.5 - 32 = 38.6 ° C.0523

Now let us convert that to K.0539

Temperature in K is the temperature in degrees C plus 273.15, so that will be 38.6 + 273.15 = 311.75 K.0543

Let us look at the temperature of space.0570

The average temperature of space is estimated as roughly -270 ° C, that is cold.0574

What is the average kinetic energy of the particles in space?0579

Well, first thing we are going to do is convert to K, so the temperature in K is the temperature in degrees C plus 273.15, so that will be -270 + 273.15 = 3.15 K.0583

Average kinetic energy then, is going to be 3/2 times Boltzmann's constant times our temperature, or 3/2 × 1.38 × 10-23 × 3.15 K (temperature) or 6.5 × 10-23 J.0608

All right, let us look at the temperature of the sun.0644

Given the average kinetic energy of the particles comprising our sun is 1.2 × 1019 J.0648

Find the temperature of the sun in K.0654

Well, if average kinetic energy is 3/2 Kbt, then that means the temperature must be 2 times the average kinetic energy divided by 3 times that Boltzmann's constant, Kb.0658

Or 2 × 1.2 × 1019 J/3 × 1.38 (Boltzmann's constant) × 10-23, or 5800 K.0675

Let us take a look at a heat transfer problem.0698

Let us find a rate of heat transfer through a 5 mm thick glass window with a cross-sectional area of 0.4 m2 if the inside temperature is 300 K and the outside temperature is 250 K.0701

Well the rate of heat transfer (h) is Kaδt/L, where if we look up (K) for glass, we can find that the thermal conductivity of glass is about 0.9.0713

So that is going to be 0.9 times the cross-sectional area (0.4) times δt, the change in temperature, is 50 K (temperature gradient) divided by our length (5 mm or 0. 0.005 m).0729

So our heat transfer rate is going to be about 36 J/s or 3600 W.0748

Let us look at heat transfer across a rod.0761

One end of a 1.5 m stainless steel rod is placed in an 850 K fire.0764

The cross-sectional area of the rod is 1 cm and the cool end of the rod is at 300 K.0770

Calculate the rate of heat transfer through the rod.0776

Well, first let us figure out that cross-sectional area.0779

Area is πr2, so that is going to be π times our radius (1 cm), so that is 0.1 m2 or 3.14 × 10-4m2.0782

Now we are also going to need the thermal conductivity of steel and there are different conductivities depending on the types of steel, but let us just assume an average thermal conductivity of steel, rough estimate of about 16.5.0801

Our rate of heat transfer (h) is Kaδt/L, where (K) for steel is 16.5.0818

Our cross-sectional area, we just determined was 3.14 × 10-4m2 and our temperature gradient from 850 K to 300 K is 550 K...0828

...divided by the length of our rod 1.5 m or 1.9 J/s or 1.9 W.0842

All right, so you know when objects are heated, they tend to expand and when they are cool, they tend to contract and at higher temperatures, objects have higher average kinetic energies so their particles vibrate more.0857

At those higher levels of vibrations those particles are not bound as tightly to each other, so the object expands -- exact opposite, as it cools down, they do not vibrate as much and they are bound a little bit more tightly, so they contract.0870

This is why if you have a stuck jar of pickles or something and you are trying to open it and you cannot quite untwist it, go try and run it under hot water because if you run it under hot water, the lid is going to start expanding.0885

It is going to expand at a faster rate than the glass, so if you run it under hot water, you give yourself a little bit more room and you loosen it up so hopefully, now you are strong enough to undo the lid.0898

Linear expansion -- the amount of material expands is characterized by the materials coefficient of expansion.0912

One-dimensional expansion, we use the linear coefficient of expansion which gets the symbol α.0918

So the change in an object's length due to linear expansion is this -- linear coefficient of expansion times its initial length times its change in temperature (δt).0925

For volumetric expansions, the amount of material that expands is again characterized by the coefficient of expansion, but if it is three-dimensional expansion, you use the volumetric coefficient of expansion, which gets the symbol β.0938

The change in volume is that coefficient of expansion, the volumetric coefficient of expansion, β times the initial volume times the change in temperature.0951

In most cases the volumetric coefficient of expansion is roughly 3 times the linear coefficient of expansion and that change in temperature can be provided in either ° C or K because the sign of the individual units are the same and we are looking at a relative change, not an absolute C or K -- it does not really matter for these problems.0963

So, some coefficients of thermal expansion again.0984

We have aluminum, concrete, diamond, glass, stainless steel, and water and we have the linear coefficient of expansion and the volumetric coefficient of expansion.0987

Now water is a little bit tricky here.0996

Although I have included it here, it actually expands when it freezes, so calculations near the freezing point of water require a little more detailed analysis than is provided here.0999

There is a window of a couple of degrees in water, that make it a little bit more complicated, so just keep that in mind, that this is not the full story for water.1008

Let us take a look at a contracting railroad tie.1020

A concrete railroad tie has a length of 2.45 m on a hot sunny 35 ° C day.1023

What is the length of the railroad tie in the winter when the temperature dips to -25 ° C?1029

Well, if it is a concrete railroad tie, let us find the linear coefficient of expansion for concrete and for concrete, that just so happens to be about 12 × 10-6.1036

So, δL is equal to α L-initial δT.1050

That is 12 × 10-6 × 2.45 m (initial length) × -60 (temperature change) or a total of -0.0018 m.1059

So what is the new length of the railroad tie?1078

Well, δL is equal to (L) - L-initial -- δ anything is the final value minus the initial.1080

Therefore the final value is going to be δL + L-initial...1087

...which will be -0.0018 m + 2.45 m, so its new length will be about 2.448 m.1095

All right, pretty straightforward.1115

Let us take a look at the expansion of an aluminum rod.1117

An aluminum rod has a length of exactly 1 m when it is at 300 K.1121

How much longer is it when placed in a 400 ° c oven?1125

Well, a couple of things I am going to need to know here.1130

First I am going to need to convert this temperature to K, because I start at K and then I am crossing over to C, that is kind of tough to tell the temperature difference between the two.1133

First, let us convert that -- our temperature in K is our temperature in ° C + 273.15, so that is going to be 400 ° C +273.15 or 673.15 K.1143

For dealing with the expansion of aluminum, I am also going to have to know that α, the linear coefficient of expansion for aluminum is about 23 × 10-6, so now I can find that shift in length.1164

Δ L is α L-initial δT or 23 × 10-6 × 1 m (initial length) × 673.15 (change in temperature) to 300 K...1182

...is going to be about 373.15 or a total change in length of about 0.0086 m.1200

How about looking at some volumetric expansion.1218

A glass of water with volume 1 liter is completely filled at 5 ° C.1221

How much water will spill out of the glass when the temperature is raised to 85 ° C?1227

Well, we have to realize here that both the glass and the water are going to expand, so let us see how much each expands and find the difference between those two.1234

If we start with the water, the change in volume is going to be β, the volumetric coefficient of expansion times its initial volume times our difference in temperature and the volumetric coefficient of expansion for water (β) is 207 × 10-6, so that is 207 × 10-6 × 1 L × 80 ° or about 0.0166 L.1244

The glass is a slightly different story.1267

Change in volume is (β)(V0δT) again, but the volumetric coefficient of expansion for the glass is 27 × 10-6 × 1 L × 80 ° or about 0.0022 L.1291

We have considerably more expansion from the water than the glass, so how much is going to spill out?1312

We are going to take the difference of these two, 0.0166 L - 0.0022 L to find out that we have 0.0144 L spilling out.1318

All right, let us take a look at an example problem where we are looking at some graphs of average kinetic energy vs. temperature.1338

Which graph best represents the relationship between the average kinetic energy of the random motion of the molecules of an ideal gas in its absolute temperature.1344

Well, first of all, let us write down that relation.1353

The average kinetic energy is 3/2 Boltzmann's constant times (t).1356

Notice that we have a direct linear relationship between the average kinetic energy and the temperature.1365

As temperature goes up, average kinetic energy goes up.1372

There it is -- our direct linear relationship.1376

Let us take a look at one more.1382

Jodie cannot remove her wedding ring.1387

If she runs the entire ring under hot water, what is going to happen to the hole in the middle>?1389

Will it expand, contract, or stay the same? Well, here is how we are going to treat this.1394

We are going to find what happens if we treat this as two rings, an outer ring and an inner ring.1400

Let us treat it as a circle, a bigger circle and a more little circle.1407

The big circle is going to expand and the inner circle is also going to expand.1410

Let us expand them both and then we are going to recombine them and when we do that what we are going to find is if the inner one has expanded and the outer one has expanded, of course this is where the finger goes inside that one and that one is expanded as well, therefore, they both expand.1418

In linear expansion, every linear dimension of an object changes by the same fraction when it is heated or cooled.1434

That is a good way to get the ring off -- run it under hot water, hopefully it expands -- maybe try a little bit of dish soap or some lubricant there as well.1441

Hopefully that gets you a good start on temperature, heat, and thermal expansion.1450

Thank you so much for your time and make it a great day!1454

We will see you soon.1457