Sign In | Subscribe
Start learning today, and be successful in your academic & professional career. Start Today!
Loading video...
This is a quick preview of the lesson. For full access, please Log In or Sign up.
For more information, please see full course syllabus of High School Physics
  • Discussion

  • Study Guides

  • Download Lecture Slides

  • Table of Contents

  • Transcription

  • Related Books & Services

Bookmark and Share
Lecture Comments (2)

1 answer

Last reply by: Professor Selhorst-Jones
Thu Mar 14, 2013 10:30 AM

Post by Adil Garad on March 13, 2013

There is a question on my homework sheet that I cannot figure out:
Q: "What mass of copper at 90'C, when added to 200g of water at 15'C contained in a 100g aluminum calorimeter, will give a final temperature of 20'C?"
I want to know how to include the calorimeter into the question. Can you show me the steps for solving this question?
Thank You.

Intro to Temperature & Heat

  • All atoms and molecules have some vibrational motion in them, even solid objects. We call the average of this motion in a substance temperature (T).
  • Since temperature is based on motion, the lower limit of this is when the objects stop moving: this is called absolute zero.
  • We measure temperature in kelvin (K). Absolute zero is 0K. One kelvin is the same "size" as a Celsius degree, but they have very different starting points. We can convert between them with
    TC = TK − 273.15.
  • Heat (Q) is the transfer of thermal energy. Heat is positive when the environment puts thermal energy into the system, and negative when the environment takes energy out.
  • One calorie (cal) is the amount of heat required to raise one gram of room temperature water by one kelvin/one degree Celsius.
  • Since thermal energy is a form of energy, we can covert calories to joules:
    1  cal = 4.1868 J.
  • Different materials take different amounts of heat to have the same change in temperature. The specific heat (c) of an object tells us the proportion of heat to temperature change. Different substances have different specific heats.
  • The more mass an object has, the more thermal energy it takes to increase its temperature. In general, the amount of heat required for a change in temperature is
    Q = cm(∆T).

Intro to Temperature & Heat

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
  • Absolute Zero 1:50
    • Absolute Zero
  • Kelvin 2:25
    • Kelvin
  • Heat vs. Temperature 4:21
    • Heat vs. Temperature
  • Heating Water 5:32
    • Heating Water
  • Specific Heat 7:44
    • Specific Heat: Q = cm(∆T)
  • Heat Transfer 9:20
    • Conduction
    • Convection
    • Radiation
  • Example 1: Converting Temperature 13:21
  • Example 2: Calories 14:54
  • Example 3: Thermal Energy 19:00
  • Example 4: Temperature When Mixture Comes to Equilibrium Part 1 20:45
  • Example 4: Temperature When Mixture Comes to Equilibrium Part 2 24:55

Transcription: Intro to Temperature & Heat

Hi, welcome back to Today we’re going to be talking about temperature and heat.0000

All atoms and molecules have some vibrational motion in them. They’re shaking around just a slight amount. Even solid objects still have some of this motion.0005

Well we can’t see this motion without eyes, it is happening on an atomic level. This vibration has a huge impact on how substances interact with one another and how they behave on their own.0015

What do we call this slight shaking motion that’s so integral in the very nature that chemistry behaves?0024

We call it temperature. This is a strange thing at first but it’s that motion, that slight shaking motion that is heat.0032

That is heat actually being something that’s slightly different that we’re about to talk about.0040

That is what is warm, that is why something feels hot or something feels cold is how much of this shaking that is.0044

We call the average of the motion is in a substance, the average of this microscopic atomic level motion in a substance, temperature.0051

You denote it with the t. Note, this is the average of many different microscopic, super microscopic, not the sort of microscopic that you can see with a microscope, but something on the super micro level.0058

This is the average of many different tiny, tiny motions. On a macro level it seems like the substance is one unified temperature.0070

On a micro level one molecule might be moving slightly faster or slower than the next.0079

So on a micro level each one is moving slightly different than all of its neighbors, but when we look at the giant scale, we’re seeing so many things happening at once.0083

We just take the average because that’s what it seems like to us because they’re so many tiny particles in there each doing slightly, slightly different behavior.0091

From our point of view we can’t notice the tiny particles. It’d be talking about a beach but trying to talk about every single grain of sand.0099

We just sort of notice it as sand under our feet supporting us on the whole.0106

With this idea of temperature being based on super microscopic motion, we can see that there has to be a lower bound to that temperature.0111

When molecules completely stop moving we can’t get below the point where they aren’t moving.0118

There’s no stiller thing than just being motionless. So once their motionless we’ve gone down as far as temperature can go.0124

We call this lower limit absolute zero and it’s going to form the base of our temperature scale.0131

The base of our temperature scale will be the lowest that we can get to, the lowest amount of motion we can have as nothing as so zero is nothing.0136

What scale are we going to use? For length we use meters. For mass we use kilograms. What do we use for temperature?0145

So far we’ve talked a lot in terms of centigrade. We’ve talked about change, centigrade is actually Celsius, we’ve talked about a lot of things in terms of Celsius and what is that?0153

Well kelvin is actually what we’re going to switch over to. We’re going to have a starting point for temperature and now we can introduce temperature measurement and we’re going to use kelvin.0163

So kelvin is the exact same size, the distance on a kelvin from 0 to 1 and 1 to 2. It’s the same thing.0172

One change in a kelvin degree is the same thing as one change in a Celsius degree.0181

There are still 100 kelvin degrees to get from the frozen point of water to the boiling point of water; I mean the end of the frozen point to the beginning of the boiling point.0185

It’s not going to start at the same place so Celsius has its zero set at frozen water, a reasonable thing when you’re living in normal daily life.0195

When you want to do laboratory experiments, you’re going to want to have it set down so your kelvin is set at zero, is set at the very base of where temperature can go.0204

It’s going to matter a lot when we talk about certain other things later on.0213

The Kelvin scale is 0k is absolute zero. 0 degrees Celsius on the other hand is a freezing point of water at 1 atmosphere of pressure.0216

To convert between the two we have the temperature in Celsius is equal to the temperature in kelvin minus 273.15.0225

If you want to go from kelvin to Celsius, you just add 273.15 and you’ll get what you’re number…sorry, if you want to go from kelvin to Celsius, you subtract.0232

If you want to go from Celsius to kelvin then you’re to add 273.15 because you need to get from the fact that your 0 of frozen water is actually 273.15 above where the stopping of motion is or the end to temperature is.0240

I think it’s kind of the end. Heat versus temperature. In everyday life we often talk about heat and temperature as totally interchangeable ideas.0260

I accidentally slipped up and did it at the very beginning of this lesson in fact.0269

In physics we make a distinction, heat we denote it with a q. Heat is the transfer of thermal energy.0273

Heat is positive when we put thermal energy into a system and it’s negative when the environment takes it out.0280

So environment puts it in, you’ve got positive heat, positive q, positive thermal energy.0287

Its negative when the environment takes it out. So if the system gets cooler, it’s going to be negative heat, it’s going to be a heat flow out of it. It’s going to be negative thermal energy, negative q.0293

Notice the similarity to work in energy. When we were dealing with energy, energy was the fixed quantity that moved up and down.0304

Here temperature is the fix, once again these things aren’t really fixed, they clearly move around a lot.0311

But temperature was the thing that moved up and down based on how much work we put in.0315

Positive work when we put energy into the system, negative work meant we took energy out of the system.0318

It’s the same thing here, positive heat, positive q means that we put thermal energy into the system. Negative q means we take energy out of the system.0324

It’s the exact same thing. If we want to heat stuff, how much heat do we have to put into it?0331

If we want to heat water, if we want to say raise the temperature on a pot of water, how much heat do we have to put in to it?0337

Clearly from experience the more water in the pot the more heat we need to raise. If you have a small pot of water it’s going to boil way faster than if you have a giant, giant pot of water.0343

We’re used to this at this point. Clearly the amount of the object, the mass of the object is going to have some effect on it.0352

There’s also some other stuff coming into it. For water, one way to measure that heat is the calorie, which is shortened to Cal when we’re putting it in as a unit.0359

1 calorie is the amount of heat required to raise 1 gram, and notice that’s gram, not kilogram.0369

1 gram of room temperature water by 1 kelvin or 1 degree Celsius.0377

If you’re at room temperature and you want to increase the temperature by 1 degree Celsius or kelvin, not Fahrenheit.0382

If you want to increase it by 1 degree, you put in 1 calorie. If you have 1 gram, 1 degree gets 1 calorie.0390

Thermal energy is just energy though, so we can also use the joule, the conversion between calories and joules is 4.1868 joules to the calorie.0397

That’s a defined thing because the calorie, the amount of energy, the specific amount of energy that you’re going to have to put in to get a temperature raise of 1 degree actually varies a little bit as the temperature of the water goes up.0408

For our purposes, we’ll be perfect fine to call it one calorie, but notice there is a very slight change as we move around. As we get farther and farther away from room temperature.0418

One calorie won’t be quite enough to make the exact same change. If we want to be really precise scientist is able to use joules instead because calorie is defined to make 4.1868.0427

Having it mean 1 calorie having…for 1 calorie to heat 1 gram, 1 degree isn’t going to always be the case.0441

It’s actually more correct to base it just on ideas of energy and then we’ll have to do lots more specific measurements if we want to be really careful about this.0450

For our propose we can treat it as always being 1 gram, 1 degree. Specific heat, the amount of heat needed to raise one objects temperature is going to be different than the amount of heat needed to raise another objects temperature.0459

Water is different than steel is different than wood is different than granite is different than rubber.0473

Each one of these is going to need different amount of heat. They’re going to have a different heat and that’s based on the chemical composition and the really deep molecular atomic structure of these things.0478

That’s something we defiantly won’t be getting into. Just for our proposes, it’s enough to know that there’s going to each one’s going to need a different coefficient.0487

We define this coefficient as specific heat, c. That will give us some proportion; this proportion is going to vary based on the substance.0495

We’ll have to get it for each problem or look it up in a table if we want to find out what it is.0503

We are able to look these things up and then figure out how much heat energy we’d have to put in to an object.0508

The amount of heat needed for a given mass, m, to have a temperature change, delta t, is given by the equation q, the heat we put in, is equal c, that’s specific heat, times the mass of the object times the change in temperature.0512

This makes a lot of sense. Each object’s going to have a different type, c, different amount proportion for their heat that they need.0526

Each thing is going to care how much mass it is. A small pot of water takes a different amount of time than a large pot of water.0534

There’s also going to be a big difference if you want to raise the temperature in that water just a little bit or if you want to get it all the way from frozen ice to boiling water.0540

Totally different numbers are going to be needed in each one of these and so we take account with that with c for the proportion, m for the mass, the amount of the thing and change in t, the delta t for just how much we want to make a difference in the temperature.0548

Heat can be transferred through one of three of methods.0561

The first one is conduction. Direct contact, motion in the atoms is directly passed to adjacent atoms. If I heat one end of this pencil, pen, I’m not really sure what to call it since it writes on a tablet, but if I were to heat one end of this, over time that would heat would slowly make it way through the object, all the way through.0565

Some things are going to conduct heat at different rates. You’ve probably see this before if you use a wooden spoon to stir a pot, the heat gets transferred to the end of that spoon way slower than if you use a metal spoon.0583

They’ve got different rates of conduction. One again that’s going to be based on the chemistry involved. A really good example would be if we were to put an empty pot to heat on a stove, so if we just put an empty pot and heat it on the stove.0595

This isn’t a good idea to do at home because it probably won’t hurt you but it is going to possibly ruin a nice pot. If you were to heat an empty pot on the stove though the temperature, either the hot coils or the hot gas flame would heat the bottom of the pot and that heat would spread through the metal.0606

It would be spread directly through conduction. The next one is convection. This is fluid motion doing the work. A combination of fluid motion happening and conduction.0624

If a fluid manages to have a hot pocket and then that hot pocket gets spread through, it’ll manage to conduct way faster than if it’s trying to conduct layer through layer through layer.0636

It’s a combination of direct conduction and the fluids mixing due to pressure differentials from temperature variation.0645

Hotter water has a slightly different pressure than colder water so that hot water is going to rise which means that its’ going to spread out through the colder layers that are now above it and it’s going to spread it and it we’re going to get convention currents.0651

If fire at the bottom of a chimney, it’s going to heat the air directly above it. That hot air will rise and will now easily touch the air at the top of the chimney making it hotter.0664

Its not direct conduction, it’s not having to make it through each layer of air atoms. It’s that it’s heating the hot air and then that hot air has a different amount of pressure in it.0674

So that gust of hot air, that packet of hot air will move up the chimney and it will manage to mix with the colder air at the top and so it will heat that hot air more easily.0683

Finally, radiation, electromagnetic waves. Hot objects emit EM waves that can be received by other objects increasing their internal thermal energy.0695

Great example of this would be to go out on a sunny day. That sun, the reason that you can feel any heat from that sun is because of electromagnetic waves.0704

That sun doesn’t have anything to conduct through. It doesn’t have convention. It’s got the dead vacuum of space for a long distance between us and it.0711

The only way that heat manages to make it to us is because it’s able to directly shot it at us.0719

It uses light to shot it at us. Infrared is one of the main carries of heat energy, as one of the first thing that a hot object starts to emit.0723

Its also going to emit it through a variety of spectrums. If you’ve ever heard of something being white hot, that’s because it’s literally been heated to the point where it manages to emit white light.0731

When something is red hot it’s at less energy because it’s only emitting a lower wave length. It’s emitting red which is less than the entire spectrum.0740

White light being the entire spectrum of light seen a once. When we manage to make something really, really hot temperature, if you’ve ever seen a glowing piece of steel either in a movie or in real life that’s because it’s so hot it actually managing to broadcast light to us.0748

That’s what happening to the flame too. That thing is so hot it’s managing to broadcast light to us.0766

It’s not able to take anyone of these on its own. That fire at the bottom of chimney, it’s able to heat some of those bricks at the top of the chimney by directly shooting electromagnetic waves at them.0773

Which is then also going to be able to conduct to the air next to it. Anyone of these, they’re going to come together.0785

You can’t just say there’s just one at a time because they’re all working together. They’re all working in concert.0790

Each one works slightly differently and it’s an interesting to have an understanding of how heat is moving around.0795

We’re ready for some examples. If we had a block of steel that had a temperature of 540 kelvin what would that be in Celsius?0802

Remember, the temperature in kelvin minus 273.15 is equal to the temperature in Celsius.0808

Interesting point about kelvin is when we talk about kelvin we don’t say degrees kelvin. We just say number kelvin.0819

So 540 kelvin. When we talk about Celsius though we say degrees Celsius.0824

It’s just a thing that we do, it’s the way we describe it.0830

If a block of steel is the temperature of 540 kelvin, how do we convert that?0834

Well we’re going to need to move that down because kelvin has a higher number for this same temperature.0838

We move that down 540 subtract by 273.15, we get 266.85 degrees Celsius.0844

If on the other hand we’re going from -15 degrees Celsius, if we had a cloud of air at -15 Celsius and we needed to know what that was in kelvin.0854

We would add that same amount 273.15. So we add 273.15 and we get 258.15, no degree mark, just kelvin directly.0862

If we’re got something that’s in kelvin and we need to convert it to Celsius the number we get lower because kelvin has a lower starting base.0875

If we have something that’s in Celsius and we convert it to kelvin the number will get higher because Celsius has a higher starting base.0885

The number that we convert with is 273.15.0891

A calorie in food, notice the capital C. This is an interesting point, a calorie in food, if you look at the back of a box.0896

If you’re in the United States, I’m not quite sure about some of Europe. But if you’re in the United States and you see a calorie of food you’re going to see that it’s not actually the same thing as the calorie we were talking about.0904

We see calories on the back. Other countries though will actually stop kilocalories because what a calorie is with a capital C is it’s actually a kilocalorie.0917

It’s 1,000 of those calories that heat water that we were talking about before.0927

Some countries in fact, they don’t need calories because they could also just talk about in straight energy.0931

Other countries will use joules on the back. We’ll talk about the kilojoules that the food has that you’d be looking at.0934

It’s an interesting point of view. It’s an interesting point; we couldn’t look into this as anything as long as we’re looking in terms of the energy that one of these things can impart.0940

If we’ve got calories, as kilocalories, and a person burns 2,000 kilocalories in a day. What’s their average power output going to be?0948

Remember we had 1 calorie is equal to 4.1868 joules. SO what would one kilocalorie be?0957

It would be 1,000 times that. If we want to see the energy of the kilocalorie is then we’ve got 2,000, sorry energy of 2,000 kilocalories.0968

That’s what we want to look at, the 2,000 kilocalories from here. So energy of 2,000 kilocalories is going to be 2,000, the number of kilocalories times 10^3 because we’ve got a kilo, we’ve got a 1,000 of them, calories.0982

The conversion between calories, so at this point we’ve got 2,000 times 10^3 plain calories.0994

If we want to convert it to joules we multiply by another 4.1868 and this will tell us what the number of joules is in 2,000 kilocalories.0998

We get, multiply it out, and we get 8.374 x 10^6 joules.1007

The person goes through 8.374 x 10^6 joules in a day. If we want to know what the power is, we need to figure out how many joules you go through per second.1018

So what would be the average output over the day? Well it’s going to be power is equal to the change of energy, the amount of energy we use, this number right here.1026

Divided by the amount of time. Well the change in energy, we’ve already figured that out, we use up 8.374 x 10^6, at least for this average person using 2,000 kilocalories a day.1036

Not necessarily average the amount that you need, depends on your personal lifestyle, your personal metabolism, and how much work you had to do.1049

On a cold day, a linebacker, a heavy athletic person. On a cold day practicing can use something like 6,000 calories in a day.1058

If you’re living...if you’re a very small person, living in a warm climate, you might only need 1,500.1066

It depends on your personal life. 8.374 x 10^6 joules, we want to divide that by, well how many hours are in a day?1070

24 hours in a day. How many minutes in an hour? 60 minutes an hour. How many seconds in a minute? 60 seconds.1078

We put that all the together, we divide it out and we get 96.92 watts.1084

That’s what the person average power output is. Keep that in mind, think about that.1092

A watt, a bulb where used to using a 100 watt bulb to light our house. If you have incandescent bulb at 100 watts, that things actually pretty warm to the touch.1097

It can burn you. That’s how much heat your body is putting out pretty much. Your body is going to be putting out almost certainly at least 96.92 watts at any time.1106

That’s why a crowd of people, if you’ve ever noticed that it’s a cold day but you’re standing with a crowd of people its noticeable warmer.1116

Or if you’re in a closed classroom for a long time. It starts to get noticeably hot. Part of that is just because of the raw amount of heat that a bunch of bodies sitting around make.1123

A bunch of live people put out a lot of heat. That’s what you’re seeing, you’re actually seeing waste heat of a bunch of people just standing around.1133

Granite. Granite has a specific heat of .79 kilojoules; notice this is kilojoules, not joules.1142

If you have a 2.5 kilogram block of granite currently at temperature 280 kelvin and you want to raise the temperature by 20 kelvin, how much thermal energy needs to be put in?1149

First off, I’d like to point out; do we need to know what the starting and ending was if we know what the proportion is?1158

If we know what the specific heat is for the level we’re starting at, we don’t actually need to know the temperature.1164

We know we’ve got 280 kelvin going to 20 kelvin, so we end at 300 kelvin, but the important thing is the change in t is equal to 20.1169

That’s all that we really care about. If change in t equals 20, c is equal to .79 kilojoules per kilogram times kelvin and 2.5 kilograms is the amount of our mass, then we have q is equal to the specific heat times the mass times the change in the temperature.1177

We plug everything in, ah .79 kilojoules, so if we want to do this in joules, what we’re used to doing.1193

We’ll want to change it to kilojoules with .79 x 10^3. So it’s joules per kilogram per kelvin.1199

We multiply by the mass, 2.5 kilograms times the change we want to effect. So we want to change 20, we put it all together and we get 39,500 joules.1208

If we were curious how many kilojoules that would be, we just divide by 10^3, move that decimal over by three, 39.5 kilojoules.1221

Either way you want to look at it, same thing.1230

You use the specific heat that you’ve got, the mass that you’re working with and the temperature change. Put them all together and that tells you how much heat needs to be put in to get the change.1234

Finally, we have a perfectly insulated, this means the environment won’t effect it so we don’t have to worry about it radiating heat or heat being put in from the environment.1246

We know that we only have to care about what’s happening inside. We’ve got a perfectly insulated container that is 4 liters of water at 10 degrees centigrade.1255

You place a 3 kilogram brick of iron that is currently at 400 degrees centigrade, whoops typo right here. This should be 400 degrees centigrade, into the water.1262

Of the specific heat of iron is .47 kilojoules per kilograms times kelvin. What temperature will the water/iron brick mixture be when it comes to equilibrium?1272

First thing, do we have what the mass of water is? We don’t have it yet actually but one more thing that we haven’t mentioned before.1283

What you might have learned in pervious science classes, is that 1 milliliter of water, so 1 milliliter of water is the same thing at most pressures and most temperatures.1291

1 milliliter of water is the same thing as 1 cubic centimeter of water, which is the same thing as 1 gram of water.1301

Thus, 1 liter has got to be 1 kilogram since there is 1,000 milliliters in a liter and there is 1,000 grams in a kilogram. Then we know that 1 liter of water at normal temperature and pressure is what we’re dealing with in this problem.1315

We know that we’ve would have 4 kilograms of water for this problem. So the mass of our water is equal to 4 kilograms.1328

We know what the mass of the iron is. Do we know what the coefficient for the iron is? We know what the coefficient for the iron is.1339

Do we know what the coefficient for the water is? Well once again, we know that the coefficient for water is in general 1 calorie per gram per kelvin.1344

If we want to switch from calories to joules, we have 4.1868 joules per gram per kelvin and notice we couldn’t do this if we were in calories because we’ve got a different thing over here.1357

We’ve got a different thing for our iron, so we’d have to convert into calories there.1371

We have to make sure that we’re dealing with this same thing for both of them.1375

If we want to convert from joules per gram per kelvin into kilojoules per kilogram, well 1,000 but also divided by 1,000, so it’s going to wind up being the exact same thing since its kilo both on the top and the bottom.1378

Kilojoules per kilogram times per kelvin. At this point, we’ve got what the coefficient for water is.1391

We know what its specific heat is. We know what the specific heat of iron is.1400

One more idea, if we put iron into water it’s going to conduct its heat both through, we’re going to have conduct, we’re going to have convection, and we’re going to have electromagnetic waves bouncing around inside of that container.1405

What will be the final heat? Sorry, what will be the final temperature, not the final heat?1421

What will be the final temperature for that container? Are they going to have to agree on a temperature at the end? Yeah, when they hit equilibrium, there’s going to have to be at the same temperature otherwise they‘re going to continue transmit heat.1426

Only when something is at the same thing are they not transmitting heat back and forth.1439

What is…what’s that going to be? We know that the amount of energy we put into the water, the amount of thermal energy that gets put into the water because it’s going to have its heat, it’s going to have to take positive heat in because it’s having its temperature raised.1445

It’s going to be the amount of energy that the iron loses. If the iron, it’s going to have a negative amount because it’s putting energy out of itself. Energy is coming out of it so it’s going to have a negative heat.1459

That means that we need to change that to a positive for it to be the same thing as the water.1472

The amount…the total absolute amount of energy that the iron loses is the amount of energy that the water gains because we’ve got this perfect insulator around it.1478

Negative heat becomes the positive heat of the water. This relationship right here is how we’ll be able to solve this problem.1488

At this point we’ve finally ready to work on it. Here’s all of the things we need.1496

The mass of water is equal to 4 kilograms. The mass of the iron is equal to 3 kilograms. The temperature of the water is 10 degrees centigrade. The temperature of the iron is 400 degrees centigrade.1500

The coefficient for the water is 4.1868 kilojoules per kilogram per kelvin. The specific heat for the iron is .47 kilojoules per kilogram per kelvin.1510

What is our ending temperature? We need to know that the amount of energy that goes in for a specific heat for the amount of heat that has to go in for a temperature change is going to be the specific heat times the mass times the change in that temperature.1520

One quick question, do we have to convert from Celsius into kelvin before we can do this problem?1539

In this case we don’t actually have to do it because we’re only looking at the change, the difference between those two numbers are going to be the same difference whether we’re looking at this in kelvin or if we’re looking at this in Celsius.1545

The difference between a Celsius, 1 Celsius and 1 kelvin is the same distance temperature wise.1555

If we’re looking at the change in temperature, it’s the same thing if we do it as 400 Celsius or 673.15 kelvin.1562

Ultimately when we look at the difference it’s going to be the same difference whichever way we measure this to begin with.1571

We can actually continue to work in Celsius. We know that the change in the heat for the water is equal to the opposite of the heat for the iron because the iron heat is negative.1579

We set this two equal to one another and we’ve got that c of the water times the mass of the water times the temperature difference.1590

We want to actually use the temperature difference because we want end to show up minus the temperature of the water is equal to negative specific heat of the iron times the mass of the iron times the ending temperature.1600

Its going to be the same ending temperature on both sides minus the starting temperature for the iron.1613

Let’s start moving things around, make this a little bit easier. We’ve got –cimi x tn so we can multiply that in.1618

We distribute the right side and add it over here. We’ve got cwmw t end plus because it’s negative on the right side, plus cimi t end.1624

Then we’ll move everything that’s not t end based because we’re solving for t end to the right side.1643

We’ve got cwmw x –tw, so we add that over here, we have cwmw tw and then over here that –cimi hits the negative t end, so we get positive.1648

So we’ve got positives on both sides, cimi ti. Let’s actually take a quick pause and look at this for a moment.1662

Notice what we’ve got here. We’ve got cwmw t end plus cimi t end, so this the total inertia to speak.1669

The thermal inertia for the system as a whole, the iron and water together at the end is going to be the temperature times its specific heat times the amount of mass.1679

On the right side, we’re going to have…it’s going to become a…we’ve got what the total thermal inertia is here plus the thermal inertia here.1689

We know that the total amount of heat energy, sorry the total amount of thermal energy for the iron plus the water is going to have to be the same no matter what because we aren’t losing any thermal energy because we’ve got that perfect insulation.1697

The total amount of thermal energy at the beginning is going to be equal to the total amount of thermal energy at the end.1712

We start solving for t end. Pull out the t end, we have cwmw plus cimi equals all that same stuff over.1718

We divide by cwmw plus cimi, whoops I accidently got confused again, a lot of letters here.1734

Cwmw tw plus cimi ti all over cm, ah keep doing that. Cwmw plus cimi.1747

At this point I’d also like to point out what we’ve basically got, is we’ve got a weighted average. We’ve got a way of weighting how much the water temperatures matters because we’ve got cw times mw.1766

What sort of that thermal block is like, how much mass it has along with its specific heat, to see how hard it is to push that mass around temperature wise.1777

Along with its starting temperature plus cimi ti, what its thermal inertia is, times that starting place.1785

Then we divide it, we average it, we have to take that out because we want to just get temperature at the end.1793

If you remember back to finding the center of mass, there is a lot of similarities to center of mass here.1797

What we’re looking for in the end when we solve it out like this is we’re really looking for the weighted average between these things is.1802

We have to work this out with algebra because it’d be too easy to just go 400 plus 10 divided by 2 won’t be the correct answer.1808

Because we’ve got different amounts of water then we have iron. We’ve got difference specific heats for water and iron. We’ve got to work this thing out with math, but it is going to be a weighted average.1814

It’s how important the heat, how important the temperature in that water is versus how important the temperature in the iron is and what it becomes when we put them together.1823

We finally substitute in a huge mess of numbers, we have cw is .47, now once again, what are we working with?1833

We’re working with kilojoules. So we probably want to switch to joules because we know joules are the friendly SI unit. Just in case we’ll switch it over.1844

We’ll see in the end that wasn’t going to be too important but we’ll switch it over for now.1854

Specific heat of water, 4.1868 x 10^3 to deal with that kilojoules time the mass of the water, 4 kilograms times the starting temperature of the water, 10 degrees Celsius plus ci.1857

So .47, but once again we’ve got kilojoules so we multiply that by 10^3 times the mass of the iron, 3 kilograms times the starting temperature of the iron, 400 degrees Celsius.1873

Then we divide that by 4.1868 x 10^3 x 4 + .47 x 10^3 x 3. Now at this point notice we’ve got 10^3 showing up everywhere.1887

10^3 here, here, and here. SO every single one of our additive elements there had a 10^3 factor so we can cancel it out.1906

Ultimately in this case because we had this specific heat using the same unit on the top and the bottom and we had specific heat showing up everywhere we’re able to completely cancel out that unit.1915

So that all that matter was the coefficient in front of that unit. That’s a dangerous thing to do just like toss around so you don’t want to just use unit without understanding what you’re using.1924

It’s generally the safest thing to do is to use the SI unit, whatever it is.1935

Once again, that’s why we had to talk about why we were using Celsius instead of converting the kelvin first.1940

If we think about it, it might be the case that we can get away with using non perfect SI unit, but sometimes you’re going to completely ruin your problem by thinking you can.1944

And you won’t be able because it’s inherent in the way that the formulas we have, we have expectations.1952

We finally punch all this into a calculator and the number that we get out is 40.3 degrees Celsius.1958

40.3 degrees Celsius, notice how low that is. That iron block was almost the same mass of our water.1968

Almost the same mass as our water and it was 400 degrees Celsius when we put it into that water and yet we managed to raise the temperature of the water by 30.3 degrees.1976

Hardly anything. That’s less than a 10th of the temperature difference to the block of iron.1987

Why is that? Because the specific heat of iron is so much lower than the specific heat of the water.1993

More accurately, it’s because the specific heat of water is so freaking high. The specific heat of water is really, really, really high compared to most other things.1998

Most other things are at least below 2, if not below 1. The specific heat of water is really high number.2009

It’s actually one of the reasons that life can exist is because you can go out on a hot day and be basking in the sun and your body temperature won’t jack up because you’ve got all that water to give you this nice thermal inertia that will keep you from moving suddenly from one temperature to another.2014

It’s one of the great things that make the possibility for life here on Earth. The fact that we’ve got this wonderful stable water temperature because we’ve got this wonderful high specific heat for water.2028

Really interesting. Hope you enjoy this, hope you learned something great and we’ll meet again at later. Bye.2041