# Heat Capacity: Example

by Jared Rovny

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00:01 So let's look at an example of how we could apply this idea of the heat capacity to a particular example.

00:07 We could ask how much energy does a one kilogram block of metal take from your hand in going from one temperature to another, specifically in this case, from freezing cold, all the way up to 10 degree Celsius.

00:19 We can then ask how much energy with the same volume of liquid water and it's important that word volume there which we'll see.

00:26 How much energy would that same volume of water take in a same process? And then we're given some constants about the problem including the specific gravity of the metal as well as the heat capacity of the metal and of the water.

00:38 One important thing to notice in the statement of this problem is that we say the heat capacity, while if you look at the number that's given, we have joules per gram Kelvin which we saw as the specific heat capacity referring to also the mass because we have those units of grams in the denominator rather than just Kelvins in the denominator.

00:58 So the reason for this is just so that you're aware that you should always look at the units of the number that you are given to understand this exact physical significance because it is the case that sometimes the word heat capacity is use in different ways.

01:12 Sometimes correctly and sometimes incorrectly.

01:15 So you're always best off looking at the unit of whatever numbers you are given to be sure that you're properly accounting for whatever variables you have in your problem.

01:23 So use our definition of heat capacity as we've just discussed them, and see if you can find the amount of energy that given amount of metal or given amount of water would have to take from you in order to raise its temperature.

01:35 If you gone through this hopefully it looks something like this.

01:38 We have, first of all, let's take our metal, we have a particular mass of metal.

01:43 Let's catalog some of the things that we're given, we have a kilogram and mass.

01:46 We know that our temperature change, our delta T is the final temperature which we said was 10 degrees, minus the initial temperature which is freezing, zero degrees.

01:57 So this difference is a simple difference of 10 degree Celsius.

02:01 Now, here's where we get to do something tricky and say that the change in temperature is also 10 Kelvin.

02:08 And the reason for this is as we had said, even though the system for temperature in degree C and for temperature in Kelvin is different the size of a degree Kelvin and a degree, I'm sorry, degree Celsius and degree Kelvin is actually the same.

02:24 And so the difference in temperature between one point and another will still be the same, so a difference in 10 Celsius is the same temperature difference as a difference in 10 Kelvin.

02:32 So we have our temperature difference as well.

02:35 We have our mass, we should also write that heat capacity, the specific heat capacity, let's just call it M, the heat capacity of the metal, which was given as a half, a joule per gram Kelvin.

02:50 And we also know from our equation that the specific heat capacity is the heat added divided by the temperature difference that we get for that heat times the mass of the material.

03:01 What we're looking for in this problem is how much heat is taken, how much heat we need so we can rearrange and say that the heat is equal to the specific heat times the change in temperature, times the mass of the material.

03:16 And all we have to do is simply plug in our numbers here, so we have 0.5 and we can now also write our units just to be very cautious here, per gram Kelvin.

03:27 We have our temperature change which was 10 Kelvins and then we have our mass, you have to be careful not to plug in one kilogram for your mass here, because notice that our units of the heat capacity of grams in the denominator and not kilograms.

03:41 So we should write this in terms of grams and we know that one kilogram is 1000 grams.

03:46 So let's put in that number instead.

03:48 We have 1000 grams here.

03:50 So now, we can be extra sure to make sure all of our units cancel properly and they do.

03:55 Living us with units of joules which is great.

03:58 So putting in all these numbers we have 1, 2, 3, 4 zeros being multiplied by .5 which means that we'll have five and then three zeros joules or we could also write this as 5 kilojoules of energy.

04:12 So this is how much energy the metal took from our hand.

04:16 So let's actually make sure that we know exactly which one this is, because we're going to do water next.

04:21 So now we're going to do the exact same thing with the water except we have a few different variables here.

04:27 So now they see the heat capacity, specific heat capacity in this case of the water is 4 joules per gram Kelvin.

04:35 The question is how much mass, what's the mass of the water that we're using? What we know that we're talking about the same volume of water and since the specific gravity of the metal was given to us as being 8 times, I'm so sorry, the specific gravity is 8 which means that the mass, the density of the metal is 8 times the density of water.

04:58 Then for the same volume of water we have one eighth the amount of material, so instead of one kilogram, we have one eighth of a kilogram as our mass.

05:08 So we have to be very careful here and what variables we're using.

05:12 We do have a different specific heat capacity but we also have a different mass because we're talking about the same volume which for water will be a smaller mass.

05:20 So plugging in this number instead, we now have 4 joules per gram Kelvin.

05:26 We have the same temperature difference of 10 Kelvins.

05:31 And we have the mass which again we have to write in grams, which is a thousand grams divided by 8, because of that, again, different mass that we're talking about here.

05:42 And now what we have to do is simplify and make sure those units were account.

05:45 So the grams do work out, the Kelvin do work out, so now let's do a little bit of simplifying.

05:50 We can say that this 4 cancel with the 8, giving us 2.

05:53 We can cancel the 2 with the 10 maybe and get five, living us with 5 times 1000 or 5000 joules one more time, which is 5 kilojoules.

06:08 So interestingly, even though we've change the parameters, we get the exact same answer for the amount of heat taken by the metal, and taken by the water.

06:15 And the reason for this, it's that even though we have a different heat capacity, or a specific heat capacity for the water as for the metal.

06:23 We also have a different amount of mass for the water and for the metal, for the water we had much less mass but we also had a greater specific heat capacity.

### About the Lecture

The lecture Heat Capacity: Example by Jared Rovny is from the course Gas Phase.

### Included Quiz Questions

1. 300 J
2. 200 J
3. 250 J
4. 100 J
5. 400 J
1. 10,450 J
2. 104.5 J
3. 10.45 J
4. 1,045J
5. 2,050J
1. 22° C
2. 15° C
3. 20° C
4. 3° C
5. 5° C
1. 0.18 cal / (gK)
2. 0.36 cal / (gK)
3. 0.25 cal / (gK)
4. 0.80 cal / (gK)
5. 0.66 cal / (gK)

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