Specific heat capacity | Khan Academy

Pop Quiz! We have two pots of water at the same temperature, say room temperature of about 30° C, as we want to increase this temperature to, say, 40° C. The question is, which of the two will take more heat energy? What do you think?

Well, from our daily experience, we can probably guess that the one where there is a lot of water would require more heat energy, right? But what's the physics behind it?

Well, we know that the temperature is a measure of the average kinetic energy of the molecules. I mean, different molecules are moving at different speeds, so they have different kinetic energy. But the average, if you take the average value, that represents the temperature. So initially, the average kinetic energy must be the same because they have the same temperature.

Now, if you want to increase the temperature, we need to increase the average kinetic energy, and that's what happens when you heat. When you supply heat energy, that heat gets distributed to all the molecules, and as a result, the average energy increases. But over here, you have so many more molecules, and if you want to increase the average energy here as well by the same amount, you've got to supply more energy, mainly because there are so many more molecules compared to over here. That makes sense, right?

So here, because there is more stuff, it took more energy, more heat energy, to raise the temperature. Right? But now, let's make things interesting. What if I took the same amount of stuff, but instead of water, I took oil? I take the same mass of water and oil at the same initial temperature, and I want to increase the temperature to, say, 40° C again. Now, which one's going to require more heat energy?

Okay, they have the same mass. All right, so do you think it's going to be the same amount of heat energy, or will water require more or will oil require more? What's your guess? Now, all right, well, my intuition says that now that we have the same amount of stuff and again we're increasing the temperature by the same amount, 30 to 40° C, let's say we'll probably require the same amount of heat, right?

Well, it turns out not to be so. Water still requires a lot more heat energy compared to oil, even if I take the same mass of them. And that is true in general, even if I had some other substance, same mass, and had to increase the temperature from 30 to 40° C, the same amount, it would require a different amount of heat energy. Different materials would require different amounts of heat energy to raise their temperatures.

But why is that the case? Well, we'll try to answer that question in this video. But before we do that, let's quantify by this. Okay? Let's take the example of water and oil and let's ask ourselves how much heat energy do you need to raise the temperature of water and raise the temperature of oil? We can actually measure this, right? We can actually calculate this.

We've done that; we've done a lot of measurements. And it turns out when it comes to water, if you take 1 kg of water, if this was 1 kilogram, then you would need to supply 4,180 J of energy to raise the temperature by just 1° C. If you take a kilogram of water and you want to raise its temperature by just 1° C, you need this much amount of energy.

So, I can actually write this as 4,180 J is needed per kilogram of water per °C rise. Okay? That unit will help us understand what this number is saying. What about oil?

Well, for oil, it depends on which oil we're talking about, but it will be much lower. Actually, it will be somewhere around, say, 1,900 to 2,000 to 2,200. So we can put about 2,000. So, 2,000 J of energy would be needed for a kilogram of oil per kilogram to raise its temperature by 1° C.

And these numbers are important because these numbers tell us how much energy is needed to raise the temperature, right? And therefore, we give a name to these numbers. We call them specific heat capacity, or just specific heat, and the letter we use to represent that is small c. The value is different for water, it's different for oil, it'll be different for, say, copper, and just from the units, hopefully, we understand what the number says.

The number tells you how much energy you need to supply to increase the temperature of 1 kilogram of that substance by 1° C. Now, you can also write this in slightly different units. For example, you know a kilogram is 1,000 g, so if I were to convert this into g, then I would get 4,180 J by 1,000 g per °C, and that would turn out to be 4.18 J/g per °C.

So you can represent it in different units. What would this number tell you? This number says if you take 1 g of water to raise its temperature by 1° C, you would require 4.18 J of energy. And this is an important number for us because we can use this number to predict a lot of stuff.

So, let me just take that example. If you had to calculate how much heat energy is needed to raise the temperature of, say, 5 kg of water, okay, from 30° C to 40° C. Why don't you pause the video? No equations as of now, just try to intuitively use this number and try to answer this question.

Okay, here's how I'm thinking. I know that I need to supply a heat energy of 4,180 J if I had 1 kg of water to raise the temperature by 1° C. But I have 5 kg of water. Since I have 5 kg of water, well, the amount of heat energy will be five times as much, so I'll multiply this number by five.

Okay, but again, this would be to raise the temperature by 1° C. But how much are we raising it? We're going from 30 to 40, so we are raising it by 10° C. So this would be the number for 1° C. So if I have to raise it by 10° C, well, I have to multiply it by 10. So, I'll multiply by 10. Actually, let me write that 10.0 because we have three significant figures over here.

Anyways, if I simplify this, the Celsius cancels out, the kilogram cancels out, and I can just multiply this and that will give me 29,000 J. So, you see, I can do pretty cool calculations by knowing this number.

Okay, and so we can now even generalize. We can write an equation just based on this intuition that we have now. All right, so we can say in general, the amount of heat energy that is supplied, Q, is what we use for heat. You can use small q; some people use capital Q. But anyways, this is the heat energy that we supplied.

What is that equal to in general? Well, that equals c. This represents the heat energy per kilogram per °C. But if I have M kg, we'll have to multiply it by m like I did over here, and if I'm not increasing by 1° C but if I'm increasing it by ΔT °C, that's the change in temperature, well then, I multiply it by ΔT. Does that make sense?

This is an important equation to do a lot of cool calculations, so hopefully it makes intuitive sense. C represents the heat energy per kilogram per degree Celsius, so if you have m in kg, you multiply it by m, and if you're raising the temperature by ΔT °C, well, you multiply it by ΔT.

With that in mind, let's solve a couple of numericals. Here's the first one. Calculate the amount of heat energy required to raise the temperature of 20 g of aluminium from 25° C to 75° C. The specific heat capacity of the aluminium is given to be this much.

So let's look at what's given and what's asked of us. We are asked to calculate the amount of heat energy or Q that is necessary. We are given the mass m, we know the initial and the final temperature, so we can figure out what ΔT is, and we are given the specific heat capacity.

So we're given all these three, and we need to figure out what the Q is, so we can just substitute and find out the amount of heat energy required. So, great idea to pause the video and see if you can try this yourself first before we do it together.

All right, so let's plug in. I know the value of C, I know the value of M. How do I calculate ΔT? Well, ΔT is a change in temperature. It's the final temperature minus the initial temperature. So if I plug in, this is what it looks like. Here's the specific heat capacity, here's the mass, and here is ΔT, final minus the initial temperature.

And if I simplify this, 75 - 25 is 50. So, if I simplify, I get this. And if you look at the units, the gram cancels out over here, the degrees Celsius cancels out over here, and I'm left over with joules. That's exactly what we want. Q should be our answer in terms of joules.

And now I can plug these numbers into the calculator. Actually, I don't have to because 20 * 50 is 1,000 and 1,000 * 897 will be 897. So because we had simple numbers, our answer is going to be 897 joules. And we have to make sure this is in three significant figures because everything else is in three significant figures, and yes, it is.

So this is how much energy is needed to raise the temperature of our aluminium from 25 to 75° C. Okay, let's try one more. Pause the video and see if you can try this yourself first.

All right, so we are given that there is a copper of mass 0.25 kg. So, we know the mass, it's initially at a temperature of 25° C, so we know the initial temperature. It's heated with 5,210 J of energy, so we know Q. And if the specific heat capacity of copper is 385 J per kg °C, which means C is given to us, we need to find the final temperature of the copper.

So look at what's given. We are given Q, we're given C, we're given M, and we are also given the initial temperature. I just need to figure out the final temperature of the copper. So how do we do this?

Well, before substituting, I'm going to rearrange this to get ΔT to isolate ΔT because that's what I'm interested in. I'm interested in figuring out the final temperature. So if I rearrange this, I just divide by CM on both sides. I will now get ΔT to be Q / (CM).

And now I can plug in everything, right? ΔT is T final minus T initial, and I know the T initial. So T initial is 25, and then I've plugged in all the other stuff.

And again, I'm just going to cancel out all the units just to make sure that I'm on the right track. So the joules cancel out, the kilograms cancel out. If one of them was in grams, then we would have to convert to make sure the units are the same. So it's always a good idea to make sure that you write down your units when you're simplifying.

And you have one °C that will go in the numerator, so you'll get °C. This will have a unit of degree Celsius, which makes sense because our temperature should be in °C.

Anyways, now we'll plug this into our calculator. 5,210 divided by 385 divided by 0.25, and that gives me this. So it's 5.54. Let's say 5.54, so that is my right-hand side. And I usually like to round off after having the final answer, okay?

So the final step would be to calculate T final. I need to add 25 on both sides, so T final will be this + 25, which will be 79.1, and there you have it. That's my final answer. That will be the final temperature of the copper.

But that now finally brings up the question: Why do different materials have different specific heat capacities in the first place? In other words, if you had two objects of exactly the same mass and let's say you supply the exact same amount of heat energy, then the temperatures will not increase by the same amount. Why?

Well, to answer that question, we need to look at stuff at the atomic level or the molecular level. Each atom or molecule will have different kinetic energy in that material, but the average kinetic energy is what represents the temperature of the material.

But here's the important point. There are other forms of energy, which I'm going to show in green over here. Okay? There are other forms of energy that do not contribute to the temperature. For example, there's potential energy that is due to the electric forces, or you can think of it as due to the intermolecular bonds, or in certain gases, molecules can even rotate in certain ways that again do not contribute to temperature.

So this means over here, when I supply 100 units of energy, it's possible that only 30 units are used up in increasing the kinetic energy, and the rest 70 is used up in, you know, increasing the other forms of energy like potential energy. But in this case, it's possible that, say, 60 of that 100 is used up with increasing the kinetic energy, and the rest 40 is used up in increasing other forms.

Now, if you're wondering what decides, you know, how much portion of this heat energy is used up in increasing the kinetic energy, well, that completely depends upon the material. It depends upon the molecules, the structure, and all that stuff.

But anyways, you can now immediately see because of this difference, the increase in the average kinetic energy here and here would be very different. Another reason for the difference is that 1 kilogram of this material will have a different number of molecules compared to 1 kilogram of this material.

But whatever it is, you can clearly see if you supply the same amount of heat energy for the same amount of stuff, the average kinetic energy increase would be different, and that's why the temperature change would be different. And that's why the specific heat capacities of different materials would be different.