**Concept:** Introduction to Internal Energy

Hey guys, in this video we're going to talk about internal energy which is another very important energy measurement in thermodynamics. Let's get to it. Now energy can be divided into two broad categories we can talk about internal energies and we can talk about external energies. What internal and external mean are internal and external to a particular system. Just a collection of particles. Individual kinetic energies and interaction energies of these particles within our system are internal energies. The kinetic energy of the whole substance, of our whole system and any interactions with outside particles particles external to our system are external energies and we typically call that external energy just the traditional mechanical energy of that system. What we just said was the kinetic energy of the system plus the potential energy due to any outside forces. Now the internal energy which for some reason is also given by a U, this is not the same thing as potential energy so watch out for that. The internal energy of a system is just the sum of all of these kinetic and potential energies that are internal to the system, these energies the energies within our system alright. Internal energy changes with the internal state of our system for an ideal gas the state is defined by the pressure, the volume and the temperature of the gas and since we typically deal with ideal gases in thermodynamics, that's what we're going to be looking out for. Mechanical energy however changes with the external state so mechanical energy depends on the external state of the system, internal energy depends on the internal state of the system. Internal and mechanical energy are independent of one another. There is no shared energy contribution between them, there is no overlap between the internal and external energy or the internal and the mechanical energy as we call it. For instance, if we consider a system moving with some speed as a whole, then we wouldn't consider all the individual particle energies sorry all the individual particle speeds internal to that system because they would basically average out. There is no contribution to the overall kinetic energy of the substance by those individual particles within the substance. The only thing that contributes to that external kinetic energies, that mechanical energy is the overall motion V of that system. Now if there is a heat transfer into a substance, we know that that's going to change the kinetic energy right if heat goes in, the kinetic energy will probably increase. If heat goes out, the kinetic energy will probably decrease. This change in kinetic energy always leads to a change in temperature because the temperature is a measurement of that average kinetic energy of each particles and the internal energy depends on the temperature of the substance because it depends on these internal kinetic energies. So what we figure out from this is that changes in temperature are always going to be linked to changes in internal energy. That dealt T and delta U are always going to be paired up with one another alright and that's a very important thing that we're going to see over and over through our discussion of temperature and internal energy. Internal energy is known as what we would call a state function and what's really important to know about a state function is that all state functions, their change is path independent. So I'm starting at some initial state P1V1 and I'm moving to some final state P2M2, it doesn't actually matter how I go from the initial state to the final state, the change in that state function delta U, delta U, delta U, they're all going to be the same regardless of which path you actually take. Furthermore if I were to take a cyclic path as the orange path indicates starting at the initial state, going in a circle and coming back to the initial state, there is no change in that state function because I began and ended at the same initial state. Potential energy is a common example of a state function because we know so much about potential energy from talking about mechanical energy before. The change in the potential energy we know is path independent. It only depends upon the initial position and the final position of the object. As I just mentioned cyclic processes, processes that start and end at the same state produce no change in internal energy. That change internal energy for a process that begins and ends at the same state for a cyclic process is always zero. State functions as the name implies are functions that depend only they only depend upon the state of the system. Now as I said for ideal gases, the state is determined by pressure, volume and temperature but really it's determined by the temperature because the pressure and the volume were all related to the temperature by that ideal gas law. So the temperature is the most important thing when assessing the state of an ideal gas and the internal energy of an ideal gas depends only on that temperature. So as I said changes in internal energy and changes in temperature are going to be linked but for ideal gases which don't have any interaction energy remember that the definition of an ideal gas is that there's, or one of the properties of an ideal gas, is that there's no interaction between the particles so the only thing that ideal gases have are particle kinetic energy and so the entirety of the internal energy is going to depend upon the temperature because those individual particle energies depend upon the temperature. Let's do an example on the following graph of P versus V, lines of constant internal energy have been drawn. What is the change in internal energy for the path indicated in the figure delta U? So where do you begin? Well you begin at 10 joules right that's what this line is, 10 joules of internal energy and then you jump to 15 joules and then you jump to 25 joules and then you jump to 20 joules and then you end down at 15 joules again. So does that change in the internal, does delta U depend upon the path taken? No all that it depends upon is the change in internal energy which would be identical if we just jump straight from the initial state and went to that final state. The point is delta U, regardless of that path, is the final internal energy 15 joules minus that initial internal energy 10 joules which is just 5 joules and that's completely path independent. It doesn't matter who followed the green line or the red line that I drew there. Alright guys, that wraps up this introduction into internal energy. Thanks for watching.

**Concept:** The Equipartition Theorem and the Internal Energy of Ideal Gasses

Hey guys, in this video we want to talk about the specific form of the function for the internal energy of an ideal gas and we're going to talk about that by utilizing something called the equipartition theorem. Let's get to it. Now remember for an ideal gas, the internal energy depends only on the temperature. It's very important to remember the line of reasoning behind this that the internal energy is made up for any system of the individual particle energies and the particle interactions. One of the fundamental properties of an ideal gas is that there are no particle interactions in an ideal gas so the internal energy is only made up of the particle energy, particle kinetic energy sorry and the particle kinetic energy as we know depends upon the temperature of the gas. So that's that line of reasoning. Now the equipartition theorem tells us how exactly to construct the internal energy for an ideal gas. The internal energy can be given as either F over 2 little N RT or equivalently F over 2 capital N KT. This is how to describe it in terms of either the number of moles little N or the number of particles big N because remember you can describe it as either where capital R is the ideal gas constant and case of B it's the Boltzmann constant. Now F is something new. F is what we call the degrees of freedom for the system and the equipartition theorem tells us specifically how to assign these degrees of freedom to different systems. A system is assigned the degree of freedom one degree of freedom for each of the following. The independent directions of a translational rotational and vibrational motion so each direction of translational, motion, rotational motion or vibrational motion that a particle in an ideal gas can undergo it's one degree of freedom. Additionally we get one degree of freedom for the number of elastic potential energies. For each elastic potential energy that you find there's one contribution to the degrees of freedom so for instance if a particle exists in a two dimensional lattice where the bonds are described as being elastic then there are two elastic potential energies. There's the one vertically and the one to the right. You wouldn't consider the one below it or the one to the left because those are the same potential energies as the one above and the one to the right they're in the same direction. So let's look at a few examples. For monoatomic ideal gases there are only three degrees of freedom. There are three directions of translational motion for a monoatomic ideal gas, ideal gas produced by a single atom. Monoatomic gases cannot have any rotational motion because they are considered point masses and a point by definition can't actually rotate around itself so there's no rotational motion. The internal energy per particle then of a monoatomic ideal gas is just three halves RT. I got rid of the number of moles N, this is just that the amount per particle. Now for a diatomic gas which are composed of two atoms that are bound together if we consider the molecular bond to be rigid, we consider not to be elastic but to be rigid there are actually five degrees of freedom. The reason that there are five is because there are the three translational degrees of freedom because those two atoms in ideal gas can translate in any direction they want to plus they have two additional rotational motions, it can rotate around its own axis sorry the bond, it can rotate about the bond or it can rotate perpendicular to the bond. Those two additional directions of rotational motion give two additional degrees of freedom so there are five total and the internal energy per unit particle is five halves RT, just F over 2. Lastly if we want to consider diatomic molecules sorry diatomic ideal gases again but this time with an elastic molecular bond. This time instead of considering that bond to be fixed now these particles can move slightly away from one another or slightly closer to one another but there is an elastic connection so it requires energy to move apart or requires energy to move closer in this case they have the same three translational degrees of freedom, they have the same two rotational degrees of freedom but now they have an additional degree of freedom due to vibrational motion. They can vibrate along the axis connecting them so that plus one for vibrational motion and plus one for the single elastic potential energy. So that seven total degrees of freedom is an additional two. As I said in addition a translation and rotation there's also a single vibrational motion just along the line connecting the two atoms. There is also a single elastic connection so that gives us one additional degree of freedom for a total of seven. So the internal energy per unit sorry per particle as we know is just F over 2RT which is seven halves RT. Pretty straightforward how to count the degrees of freedom. Alright let's do an example, what is the internal energy per particle for a 3D solid lattice with elastic connections as shown below? Assume the equations for ideal gases apply. So each of the atoms in the solid is bound it to its nearest neighbor by an elastic connection. So that molecular bond between them is elastic which means that there can be some vibrating about them. So how many elastic true independent elastic connections does each atom have? It has three. One in the X direction, one in the Y direction and one in the Z direction. So there are three contributions to the degrees of freedom due to vibrational motion. So three from vibrational motion. Further there are three elastic connections so there are three elastic potential energies. There's one half K delta X squared. We can say one half K delta Y squared and we can say one half K delta Z squared. Those three independent directions. So there are three additional from elastic potential energies? So what does that mean? That means that the number of degrees of freedom for this solid lattice is 6 and so the potential energy per particle is 6 over 2RT which is just 3RT. So the degrees of freedom divided by 2 which determines the function of internal energy for the gases and this solid because I said that the equations for ideal gases apply that equation is determined by the number of degrees of freedom divided by 2 and one half the number degrees of freedom for the solid lattice is 3. So we have this results. Alright guys that wraps up this specific discussion of what the function for internal energy looks like for ideal gases due to the equipartition theorem. Thanks for watching.

**Problem:** An ideal monoatomic oxygen gas containing 5 x 10^{25} particles is stored in a closed jar. If the temperature in the jar were initially 27°C, how much heat would you have to add to the gas to raise the temperature to 75°C?

**Example:** Heating a Gas of Diatomic Nitrogen with Rigid Connections

Hey guys, let's do an example 3 moles of an ideal gas N2 of nitrogen is stored at 0.003 cubic meters sorry in a 0.003 cubic meter container open to the air if the initial temperature was 27 degrees Celsius and the gas was heated up by 50 degrees Celsius What is the new internal energy of the gas treat the molecular bonds as rigid it's a diatomic gas right there are 2 atoms and we're told to treat them as rigid we know the sorry degrees of freedom for the internal energy should be 5 if we were told to treat the molecular bonds as elastic the degrees of freedom for the internal energy would be 7. Now what does open to the air mean like what's going on in this problem well we have an initial temperature that's raised by 50 degrees Celsius so we're changing the temperature of the gas and the internal energy of the temperature sorry of the gas is going to change with the change in temperature well what else is changing because we know the temperature is changing open to the air means that the pressure is always going to equal atmospheric pressure so the pressure doesn't change but it also means gas molecules can come in and leave the jar. So it means the number isn't constant that's a very important things to understand now the volume is a constant because the container doesn't change volume so what we have here let's just keep track of this P is constant, volume is constant. Right what's changing the number of moles changes and the temperature changes. So when we construct our ideal gas law we know that the whole left side is a constant and the right side changes except for the ideal gas constant so n T equals P V over R is a constant this is very important to understand and it's very important to use the ideal gas law this way because now we know that in the initial state of the gas N1 T1 has to equal the final state of the gas N2 T2 because the number of moles times the temperature is a constant for this gas. This means because we know the initial moles we know right it says 3 moles of gas is stored in a container we know the initial temperature its 27 degrees we know the final temperature it's 50 degrees higher we can find the second number of moles after the gas has been increased in temperature so this is T1 over T2 N1 now the temperature we need to convert to Kelvin initially it was 27 plus 273 right that'll be in Kelvin. After it was increased by 50 degrees Celsius it's 77 degrees Celsius right plus 273 to convert it to Kelvin and the initial number of moles is 3 so this tells us that after the temperature is increased there are 2.57 moles of gas and this number actually makes sense as the temperature increases the pressure is going to want to increase for the gas but the pressure is constant it's always atmospheric pressure so instead of the pressure increasing the volume of the gas increases and it expands out of the jar. So you lose that some number of moles contained by the jar as the gas expands and some of those moles leave and that's what's going to happen with a rise in temperature. What we want to know is what's the final internal energy for this gas well we know that the internal energy for any gas is going to be F over 2 NR T. We know that this is a diatomic gas that's bound rigidly to one another so that's 5 for the degrees of freedom and the final gas is going to depend upon the final number of moles which we calculated right right here and the final temperature which we know was 77 degrees Celsius don't forget that we do have to convert that into Kelvin but ideal gas constance 8.314 this was 77 degrees Celsius right plus 273 to make it Kelvin and so that final internal energy is 18696 joules or 18.7 kilojoules that's that final internal energy of this gas. Alright guys that wraps up this problem. Thanks for watching.

A constant amount of ideal gas undergoes an isochoric process, and as a result, its change in internal energy is –100J. What could be the heat added to the system (Q) and the work done by the system (W) in this process?

A. Q = -50 J, W = 50 J

B. Q = -100 J, W = 0

C. Q = -100 J, W = 200 J

D. Q = 100 J, W = 0

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*n* moles of an ideal gas initially has pressure *p*_{1 }= 5.0 x 10^{3} Pa and volume *V*_{1 }= 1.0 x 10^{-3} m^{3}. The gas undergoes a process for which the final pressure is *p*_{2 }= 2.0 x 10^{3} Pa and the final volume is 4.0 x 10^{-3} m^{3}. In this process, the internal energy of the gas

(a) increases

(b) decreases

(c) stays the same

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You add equal amounts of heat to two identical cylinders containing equal amounts of the same ideal gas. Cylinder A is allowed to expand while cylinder B is not. How do the temperature changes of the two cylinders compare?

(a) Cylinder A will experience a greater temperature change.

(b) Cylinder B will experience a greater temperature change.

(c) The two cylinders will experience the same temperature change.

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9.00 moles of a monatomic ideal gas expands isobarically at 3.00 x 10 ^{5} Pa from an initial volume of 0.300 m^{3} to a final volume of 0.400 m^{3}. What is the change in the internal energy of the gas? In this process, does the internal energy of the gas increase or decrease?

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(a) Four moles of a monoatomic ideal gas undergoes a process in which the temperature of the gas increases from 300 K to 500 K while the pressure is kept constant at 2.00 x 10^{5} Pa. What is the change in internal energy of the gas? Indicate whether the internal energy increases or decreases.

(b) Four moles of the gas undergoes a process for which the volume is 3.00 m ^{3} and is kept constant while the pressure decreases from 4500 Pa to 3600 Pa. What is the heat flow Q for this process? Does heat flow into the gas or out of the gas?

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Four moles of a monatomic ideal gas undergoes a process during which 700 J of heat flows into the gas. Calculate the work *W* done by the gas and the change Δ*U* in internal energy of the gas if the process is isobaric (constant *p*). Note: the heat capacity of the gas (C_{p}) equals 5/2R.

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The temperature of 6.00 moles of a monatomic ideal gas is increased from 20.0°C to 80.0°C in a constant pressure process. During this process the change in internal energy of the gas is

(a) zero

(b) 4490 J

(c) 1497 J

(d) 8980 J

(e) 7483 J

(f) none of the above answers

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