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Internal Energy | 28 mins | 0 completed | Learn |

Introduction to Heat Capacity | 23 mins | 0 completed | Learn |

First Law of Thermodynamics | 23 mins | 0 completed | Learn |

Intro to Thermal Processes | 21 mins | 0 completed | Learn |

Work & PV Diagrams | 14 mins | 0 completed | Learn |

Concept #1: Introduction to Heat Capacity

**Transcript**

Hey guys, in this video we're going to talk about a measurement of a substance called the heat capacity which is closely related to the specific heat so we're going to draw some parallels between the two. Alright, let's get to it. Remember guys when we talked about calorimetry which was talking about how the temperature of a substance changes for a given heat input. So we talked about inputting heat that leads us to a temperature change. We discussed the specific heat of certain substances, the specific heat of water, the specific heat of ice, etc. This is the amount of heat per unit mass needed to change the temperature of a substance and it was really important that it's per unit mass because then the specific heat doesn't actually depend upon the mass of the substance, it's an inherent quality of the substance that's independent of mass. There's another important value that's closely related to the specific heat which we call the heat capacity. While the specific heat is per unit mass that heat capacity is the amount of heart per unit mole to change the temperature and because it's per unit mole once again it's independent of how much of the substance is there. The heat capacity doesn't change when you have more of a substance because it's already per unit mole just like the specific heat doesn't change when there's more of a substance because it's per unit mass. The heat capacity for substances given by the heat input or the heat removed from a substance divided by the number of moles of that substance times the subsequent temperature change or that the amount of heat needed to change the temperature of a substance is equal to the number of moles times the heat capacity times delta T. Now something that I sort of exaggerated right here but I want you guys to remember, it's very important to remember, is that heat capacity is given by an uppercase C whereas specific heat is given by a lowercase C. So I tried to make it look intentionally super large very upper case because it's very important to realize that when you see an upper case C it's heat capacity but when you see a lower his C it's specific heat. The units of heat capacity are the same joules per Kelvin as a specific heat but it's joules per mole Kelvin where specific heat is per kilogram Kelvin because it's per unit mass for specific heat and per unit mole for heat capacity alright. Let's do a quick example. 4 moles of an unknown substance Y requires 100 joules of heat to change the temperature by 5 Kelvin. How much heat will be required to increase the temperature of 1 and a half miles of Y 27 degrees Celsius to 100 Celsius? So we want to know how much heat is required to change the temperature of Y. That amount of heat is just the number of moles of Y we're told is 1.5 times the heat capacity of Y times delta T. We know the number of moles is 1 and a half, we know delta T is from 27 degrees Celsius to 100 degrees Celsius but we don't know the specific heat sorry the heat capacity of Y. However they tell us that Y requires 100 joules of heat to change 5 Kelvin if there are 4 moles. That's enough to find the heat capacity. The heat capacity of Y which is going to be given by Q over N delta T is 100 joules of heat input to increase the temperature of 4 moles by 5 Kelvin. So this ends up being 5 joules per mole Kelvin. Now that we know the heat capacity and we know the number of moles right 1 and a half right here and we know the change in temperature right here we can find how much heat is required to change that temperature. So this N C Y delta T, this is going to be 1.5 moles it's not 4 moles because those 4 moles told us that 100 joules was required to change a temperature by 5 Kelvin but now we want to know how much heat's required to change 1 and a half moles. The specific heat is 5 and the change in temperature is a 100 degrees Celsius minus 27 degrees Celsius. Remember for changes in temperature we don't need to convert to Kelvin, for differences in temperature that 273 is going to be subtracted from both sides. So we can keep it like this and this becomes 547.5 joules. That's how much heat is required to increase the temperature of 1 and a half moles of substance Y from 27 degrees Celsius to 100 degrees Celsius. As I said both specific heat and heat capacity are inherent qualities of a substance. They don't change with the amount of substance they only change with what substance you have so all copper no matter how much you have is going to have the specific heat capacity. All water is going to have a specific heat capacity and specific heat. All ice is going to have a particular heat capacity and specific heat. And this is due specifically to the fact that specific heat's per unit mass and heat capacity's per unit mole. Very important to remember. There's a quick and easy conversion between the two, between these two. It's just the heat capacity is equal to the mole or mass, this capital M, times the specific heat remember guys upper case is heat capacity. Lower case is specific heat. So heat capacity equals the specific times the mole or mass of a substance. Let's do a quick example of that. 5 moles of hydrogen gas requires 10.25 kilograms of heat to raise the temperature 50 Kelvin. What is the specific heat of molecular hydrogen? Assuming that the hydrogen gas doesn't change phase during this process. Now to find the specific heat we need per unit mass. We're not told the mass of hydrogen, we're told the number of moles of hydrogen. So we can do this one of two ways we can convert the number of moles to the amount of mass of hydrogen or we can find the specific heat, sorry the heat capacity and then convert that to specific heat. So those are the two ways of doing it either way you need to know the molar mass of hydrogen which we can find. The molar mass of hydrogen is just the atomic mass number. So hydrogen molecular hydrogen has 2 H's in it each of which has an atomic mass of 1 so that's 2 and this is grams per mole. Now let's do it with the heat capacity first because we're talking about heat capacity. We know the heat capacity is just the amount of heat divided by the number of most times delta T so it's 10.25 kilojoules divided by 5 moles and 50 Kelvin. So if we plug that in sorry I can't use that I need to use 10250. So kilo remember is just 1000 so if I multiply by 1000 I get it in joules. This equals 41 joules per mole Kelvin. If I kept it as kilojoules the units would be kilojoules per mole Kelvin but it'd be 0.041. So I'm just keeping it as joules. Now that we know that heat capacity we can say that the specific heat is just the heat capacity divided by that molar mass. So that heat capacity is 41, the molar sorry the specific heat divided by the molar mass which is 2 and what's very important is to remember the units that are used here. The heat capacity is in joules, the molar mass is in grams so this is 20.5 sorry joules per gram Kelvin. Now we want to make this in terms of kilograms because that's a SI unit for the specific heat. So if we add kilo in the denominator we can add a kilo in the numerator to balance it out so that's 20.5 kilojoules divided by kilograms Kelvin. That's the easiest way to convert them, you add a kilo in the numerator, you had a kilo in the denominator. Alright guys that wraps up this introduction to heat capacity. Thanks for watching.

Example #1: Heat Capacity and Specific Heat of Unknown Substance

**Transcript**

Hey guys, let's do an example. Some substance X has a heat capacity of 15 kilojoules per mole Kelvin. How much heat is required to change the temperature of 700 micrograms of X from 0 to 100 degrees Celsius? The molar mass of X 50 grams per mole and assume that X doesn't undergo a phase change. We know from our M CAT equation that if we know the specific heat then we can find out how much heat is required to change the temperature for a given mass. Now luckily we're given the heat capacity and the molar mass because the conversion between them is pretty simple. This is specific heat on the left and heat capacity on the right. The heat capacity is in kilojoules per mole Kelvin so I'm gonna write that as 15000 joules per mole Kelvin. I got rid of the kilo and added 1000. This is 50 grams per mole so this becomes 300 joules per gram Kelvin. Now we want this per kilogram so the quickest way to convert that is just to add a kilo in the numerator and a kilo in the denominator. Kilojoules per kilogram Kelvin. Now we can use our M CAT equation because we know the specific heat. 700 micrograms is times 10 to the -6 grams. Now if we're in grams we need to use this conversion right here because it's also in grams not this. It turned out that we didn't need this form, we needed it in grams or you could use this form right here in kilograms but you'd have to convert between micrograms and kilograms. Now this is the conversion between micrograms and grams. So this is 300 but this is grams per sorry joules per gram Kelvin. And the temperature is increased from 100 degrees Celsius sorry from 0 to 100 degrees Celsius so that change in temperature is 100 degrees Celsius which is equivalent to 100 Kelvin. Remember they're equivalent for changes. Plugging this in we get 21 joules. That's how much heat is required to change the temperature of 700 micrograms of X from 0 Celsius to 100 Celsius. Alright guys, that wraps up this problem. Thanks for watching.

Concept #2: Heat Capacity at Constant Volume for an Ideal Gas

**Transcript**

Hey guys, in this video we're going to talk about the heat capacity at constant volume for an ideal gas. Let's get to it. We want to answer the question when he enters the substance, what is that heat doing? When we talked about calorimetry when heat entered the substance it changed the substance temperature. However heat doesn't have to change the temperature it doesn't have to do this. It only does this when the volume of the gas is constant. So this heat going in to the internal energy which causes a temperature change this chain of events only happens if the volume is constant because you cannot say that this is guaranteed to happen otherwise. When the volume is constant we can say that all of this heat is used to change that temperature and it does this as I indicated up here by changing the internal energy. That means that all that heat is what causes the change in internal energy, that the heat input is the change in internal energy. Now more properly our calorimetry equation should be in terms of moles and specific heat. Specific heat sorry heat capacity. Specifically heat capacity at constant volume. That's what that sub V means. This is the heat capacity at constant volume. So if I just isolate the number of moles times the specific heat I get Q divided by delta T and then if I substitute Q using this equation I get delta U over delta T. Something interesting is what delta U over delta T actually means. If you were to plot U vs T as I did in this figure here you would see that delta U over delta T is actually equal to the rise divided by the run. That means that in C sub V, the number of mole times the heat capacity at a constant volume which is equal to delta U over delta T is actually equal to the slope of U vs T and this is a general result this is not actually specific to ideal gases but for an ideal gas we know what the internal energy looks like it's F over 2NRT. This is a straight line it's U linearly dependent upon temperature. So if you were to plot this it would look like a straight line exactly as the figure above me. What is the slope of the straight line? It's just what multiplies the temperature. The slope is F over 2NR but the slope is not equal to that heat capacity, the slope is equal to the number of moles times the heat capacity. So that heat capacity at constant volume is the slope divided by N so if I take the slope right here and I divided by N I get that heat capacity at constant volume is F over 2 times R and this is only for ideal gases because we had to use specifically the internal energy for an ideal gas. Alright let's do an example. What is it heat capacity at constant volume for a 3D solid lattice with elastic bonds? Assume that the equation for ideal gases apply. So we've already seen a problem similar to this before and we know that for a 3D lattice each molecule is going to have three independent directions were they can vibrate. We can vibrate delta X, delta Y and delta Z. Furthermore there are three independent elastic potential energies 1 half K delta Z squared, 1 half K delta X square and 1 half K delta Y squared. So those three vibrational directions and those three independent elastic potential energies means that the degrees of freedom is six and that means that the heat capacity at constant volume which is F over R I mean sorry F over 2 times R is 6 over 2 times R which is just 3R. That's the heat capacity at constant volume for this 3D solid in a lattice where the bonds are treated elastically. Alright guys that wraps up this talk about heat capacity at a constant volume for ideal gases. Thanks for watching.

Practice: Two samples of gas, each containing n moles, are in separate, sealed containers. One sample is of gaseous H^{2} (considered to be bound elastically), and the other sample is of gaseous Ne. If the hydrogen gas required 50 J of heat to raise the temperature by 1.4 K, by how much will the temperature of the neon gas change if it absorbs the same amount of heat?

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