Video: Specific Latent Heat

In this lesson, we will learn how to use the formula 𝐸 = 𝑚𝐿 to calculate the amount of energy absorbed or released by a change in the state of a material.

16:17

Video Transcript

In this video, we’re talking about specific latent heat. As we’ll see, this term refers to an amount of energy that is exchanged, either absorbed or released, when an object changes state without changing its temperature. We see an example of this on screen, where we have solid water, ice, at zero degrees Celsius, melting into liquid water, which is also at zero degrees Celsius. In this case, heat is being absorbed by the ice to cause it to melt without raising its temperature. To see how this works, let’s consider a block of ice that we put inside a dish, and we put that dish on top of a hot plate, a device whose surface we can heat up to some temperature we set.

Then, say that we turn on the hot plate and we set its temperature to be 200 degrees Celsius. The hot plate begins to heat up to that temperature, and it transmits this heat to the dish and to the block of ice. Now, as time goes on, say that we keep track of two quantities; first, we track the temperature of the water, which is currently the block of ice. And we also keep track of how much energy is absorbed by the water over time. If we set up a graph like this so that data points would appear on it in real time, we might see a graph looking like this. As points began to appear, we would see that more and more energy is going into the water and its temperature is going up.

After some time, though, we would notice two things changing. First, we would see that our curve starts to level out, indicating that energy is being added to the water but its temperature is not going up. And we would also notice that our block of ice is starting to melt. And this would continue on until eventually the entire block was completely melted. The water is now all liquid. Once that took place, as we continue to watch our curve, we would see that once again it starts to slope upward. This means we’re continuing to add energy to the water, and now its temperature is going up in response. If we continued on and kept heating the water closer and closer to its boiling point, eventually, when the water got hot enough, it would start to evaporate. And as it did, we would once more see this curve start to level of.

Now, if we were to stop our experiment right here and look at the graph so far, we would see that there are portions of the graph where we’re adding energy to the water and its temperature is increasing. But there are also portions where we are adding energy, but its temperature is not changing at all. On those portions, where the temperature of the water is going up, we can understand where the added energy is going. It’s going into increasing the temperature of the water. But what about these regions? These are places where the energy being added isn’t going into increasing the temperature. It must be doing something else. And in fact, we’ve seen what it was doing. It was going into changing the state of matter of the water, first from ice to water and then from water to steam.

And if we think about the total energy that went into changing the water from a solid to a liquid and then from a liquid to a gas, we can give names to these amounts of energy. We could say that the total energy input to melt the water without raising its temperature is 𝐸 sub melt, and that the total energy that went into boiling the water, again without raising its temperature, we can call 𝐸 sub boil. If we wanted to show these quantities on our graph, then they would correspond to the flat portions of our curve. The energy difference from here, where the water started to melt, to here, where it was completely melted, is 𝐸 sub melt.

And then, if we consider 𝐸 sub boil, we see that we have truncated our experiment here. But if we had kept going, this line would’ve stayed flat for some distance farther. Let’s say we had run the experiment up till here, at which point all of the water was evaporated out of the dish. It’d been boiled away. In this case, we can figure out how much energy 𝐸 sub boil is by dropping vertical lines from this flat portion of our curve. Just like before, the distance between these two lines on the horizontal axis indicates how much energy went into the water simply to cause it to boil. That is, to change the state of all the water in the dish from liquid to gas without raising its temperature.

Now, there are a couple of things worth noticing about these two energy amounts, 𝐸 sub melt and 𝐸 sub boil. First, we can notice that they’re not equal to one another. They don’t represent the same amount of energy. In fact, we can see that for water, 𝐸 sub boil is greater than 𝐸 sub melt. And we know that because this distance here, representing 𝐸 sub boil, is greater than this distance here. Physically, this tells us that it takes more energy to boil, say, a gram of water than it does to melt a gram of water. Or put it more precisely, it takes more energy to take one gram of liquid water at 100 degrees Celsius and convert it to a gas than it does to take one gram of frozen water at zero degrees Celsius and convert it to a liquid. It’s important to note, though, that this is true for water, but not necessarily for all materials.

The second thing we’d like to say about these results is that each one of these amounts of energy, 𝐸 sub melt and 𝐸 sub boil, corresponds to what is called a latent heat. The energy it took to melt our ice into water without raising its temperature is called the latent heat of fusion. The reason for this name, latent heat, is because this word latent indicates that the heat is in some sense hidden. And indeed, since it doesn’t raise the temperature of the water, it is. In general, the energy that it takes to convert a material from its solid phase to its liquid phase is called its latent heat of fusion. The particular value of the latent heat of fusion depends on the material. Water has one value, for example.

Now, at this point, let’s recall that we were adding energy into our system to lead to this state change from a solid to a liquid. In this case then, energy was being absorbed by the water as it was changing state. We can imagine, though, that it’s possible to move in the opposite direction. That is, it’s possible to start out with liquid water and then cool that sample down until it freezes. In that case, the same amount of energy is involved in changing the state of the water. But, when we go from a liquid to a solid, this energy is being released rather than absorbed.

So, just to be clear, for any given material, whenever it transitions from a liquid to a solid or a solid to a liquid, the amount of energy involved in the exchange is called the latent heat of fusion. That amount of energy doesn’t change, but whether the energy is being absorbed or released does change. In the case of cooling a liquid down until it freezes, in that case, energy is being released by the liquid. But going the opposite direction and heating a solid up until it melts, energy is being absorbed by the material. So, this amount of energy for a given material can be either released or absorbed but the total magnitude of that energy will be the same either way. And it’s called the latent heat of fusion.

Knowing all this about 𝐸 sub melt gives us good background to consider similar ideas about 𝐸 sub boil. In general, the amount of energy it takes to transition a material from its liquid phase to its gas phase is called the latent heat of vaporization of that material. And just like with the latent heat of fusion, it can go either way. The material can either be transitioning from a liquid to a gas or from a gas to a liquid. That will affect whether this amount of energy is being absorbed or released by the material. But it won’t change the total magnitude of the energy involved. It won’t change the latent heat of vaporization for that material.

Now, in our experiment so far, we’ve been talking about some amount of water, but we haven’t been specific about how much water is involved. We just know that however much water we were melting and then boiling, some amount of energy we called 𝐸 sub melt was required to melt it, and then some greater amount of energy we called 𝐸 sub boil was required to make it steam. But let’s say that instead of dealing with an unknown mass of water, we were working with a known mass of exactly one kilogram. In that case, the energy required to melt our one-kilogram block of ice is no longer just called the latent heat of fusion of water, but it’s now called the specific latent heat of fusion. The word specific tells us that this is an amount of energy corresponding to the phase transition of one kilogram of this material, in our case, water.

And the same thing, by the way, happens with the energy required to boil this one kilogram of water. We’re no longer talking about the latent heat of vaporization, but rather the specific latent heat of vaporization. We’re now ready to give a definition for this term, specific latent heat. Specific latent heat is the energy that’s released or absorbed by one kilogram of a material during a phase transition. In the case of a transition from solid to liquid or liquid to solid, we call this the specific latent heat of fusion. And in the case of going from a liquid to a gas or a gas to a liquid, we call it the specific latent heat of vaporization.

And as we saw, the reason for these two different names is that these amounts of energy for a given material are typically not the same. For example, when it comes to water, we saw that the specific latent heat of vaporization, what we called 𝐸 sub boil, is greater than the specific latent heat of fusion, what we called 𝐸 sub melt. Knowing this definition of specific latent heat, let’s clear a bit of space on screen and see how to work with this value as a variable.

Typically, we represent specific latent heat using the symbol capital 𝐿. And we can recall from the definition of 𝐿 that this is an amount of energy that’s either released or absorbed by a one-kilogram mass. That is, 𝐿 represents an amount of energy per mass. So, if we wrote out the units of the specific latent heat, they would be units of energy, we’ve used joules here, over units of mass, here we’ve used kilograms.

Now, if we look up specific latent heats for various materials in a table, we’ll often see them quoted in units of kilojoules per kilogram. Know that this is still an amount of energy per an amount of mass. Knowing that specific latent heat is represented by these units, we can see that if we multiplied the specific latent heat of some material by some amount of mass of that material, then from a units’ perspective, when we multiply kilojoules per kilogram by some amount of kilograms, we see that that unit of kilograms cancels out. We’re left with a unit of energy often in kilojoules.

This shows us that if we multiply the specific latent heat of some material, either the heat of fusion or the heat of vaporization, by some amount of mass of that material, then what we’ll calculate is the energy required to change the state of that mass of material. This could be energy that’s released or energy that’s absorbed by the material. But in general, we can think of it as the energy transferred over this phase transition. Knowing all this about specific latent heat, let’s get a bit of practice with these ideas through an example.

The table lists the specific latent heat of fusion for various metals. A student has 200 grams of an unknown metal. The student heats the metal to its melting point and then measures how much energy is absorbed by the metal for all of it to melt, and gets a value of 79.6 kilojoules. Which of the four metals listed in the table does the student have?

Okay, taking a look at this table, we see that it has two rows. The first row lists four different kinds of metal, aluminum, cobalt, iron, and nickel. In the second row, the specific latent heat of fusion of those metals is given in units of kilojoules per kilogram. When we talk about the specific latent heat of fusion, that refers to the amount of energy that a substance needs to go from a liquid to a solid or a solid to a liquid. In other words, it’s an amount of energy needed to solidify or liquify one kilogram of some material.

Now, in our scenario, we’re told that a student has a 200-gram mass of some unknown metal. The metal begins as a solid but then is heated up to its melting point. And the student then measures how much energy is absorbed by the metal in order for all of it to melt. So, our 200-gram sample has now gone from completely solid to completely liquid. The student measures the energy required for that phase transition from solid to liquid to be 79.6 kilojoules. Based on this information, we want to figure out whether the student is working with aluminum, cobalt, iron, or nickel. To figure that out, let’s recall a mathematical relationship between specific latent heat, mass, and energy.

In general, the energy, 𝐸, required to affect a phase transition is equal to the mass of the substance going through that transition multiplied by the specific latent heat of that substance. Looking again at our table, we’re given the specific latent heat, in particular of fusion, for these four different metals. In other words, for these metals, we know capital 𝐿. But we don’t yet know the specific latent heat of fusion of the unknown metal that our student is working with. It’s that that we want to identify to know which of the four metals we’re using. Now, here’s what we do know about our unknown metal and the energy involved. First, we know the mass of our sample. That’s given as 200 grams. And we also know how much energy it took to completely melt this sample of metal. We can call that energy 𝐸, and we’re told that it’s equal to 79.6 kilojoules, 79.6 thousand joules.

Now, taking a look back over at our expression for the energy 𝐸 in terms of the mass and the specific latent heat. We can see that if we divide both sides by the mass involved, then that term cancels from the right and we arrive at a mathematically equivalent statement. That the energy involved in this transition divided by the mass of the substance is equal to the specific latent heat of that substance. And it’s that value, capital 𝐿, that we want to solve for for our as-yet-unknown metal.

So, to do it, we’ll divide the energy we needed to melt the metal by the mass of the sample. In other words, we’ll divide 79.6 kilojoules by 200 grams. But before we do that, notice the units in which our specific latent heats of fusion are given. They’re kilojoules per kilogram. And we have kilojoules per gram. So, before we do this division, we’ll want to convert our mass into units of kilograms. We can recall that one kilogram of mass is equal to 1000 grams, which means that to convert 200 grams into kilograms, we’ll shift the decimal place three spots to the left, at which point we can see that 200 grams is equal to 0.200 kilograms.

Looking at the units in our fraction now, we see that they’re kilojoules per kilogram. They’re a match for the units in terms of which the specific latent heats of fusion of these metals are given. So, we’re ready to divide. When we do, we find a result of 398 kilojoules per kilogram. And looking through our table, we see that this is a match for the specific latent heat of fusion of aluminum. These tells us that the metal the student is working with is aluminum.

Having seen this, let’s summarize what we’ve learned in this lesson about specific latent heat. Starting off, we saw that when a substance changes phase, that is, goes from a gas to a liquid or a liquid to gas or a solid to liquid or a liquid to solid, then that substance absorbs or releases energy. But it doesn’t change in temperature. This energy is known as latent heat. It’s the energy that a substance either absorbs or releases specifically to change state. When the latent heat of a substance is indicated per unit mass, its name changes to become specific latent heat. Specific latent heat is symbolized using a capital 𝐿. And its units are units of energy per unit of mass.

For this reason, if we multiply the specific latent heat of some substance by a given mass of that substance, then that tells us the amount of energy that’s either absorbed or released during a phase transition of that amount of that substance. And lastly, we saw that the specific latent heat of a material depends on the particular phase transition the material goes through. For a material that’s either freezing or liquefying, that is, going from a solid to a liquid or a liquid to a solid, we refer to the specific latent heat of fusion. But when a material is going from a liquid to a gas or a gas to a liquid, we refer to the specific latent heat of vaporization. For a given material, in general, these two values are not the same.

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