### Energy, Enthalpy, and the First Law of Thermodynamics

The Relationship Between Free Energy and the Equilibrium Constant . is more likely to be spontaneous; if Delta H is negative, this makes it. This allows us to derive relations between variables without regard for the So " delta h" means the change of "h" from state 1 to state 2 during a process. Then. Time-saving video by Brightstorm on Understanding the Difference between Delta H and Delta S.

So a certain amount of heat gets added to your gas. How much does the temperature increase? That's what the heat capacity tells you. So capital C is heat capacity and it's defined to be the amount of heat that you've added to the gas, divided by the amount of change in the temperature of that gas. And actually, something you'll hear about often is the molar heat capacity, which is actually divided by an extra n here. Pretty simple but think about it.

If we had a piston in here, are we going to allow that piston to move while we add the heat, or are we not going to allow the piston to move? There's different ways that this can happen, and because of that there's different heat capacities. If we don't allow this piston to move, if we weld this thing shut so it can't move we've got heat capacity at constant volume, and if we do allow this piston to move freely while we add the heat so that the pressure inside of here remains constant, we'd have the heat capacity at constant pressure.

And these are similar but different, and they're related, and we can figure them out. So let's clear this away, let's get a nice, here we go, two pistons inside of cylinders. We'll put a piston in here, but I'm going to weld this one shut. This one can't move.

## Can anybody explain the difference between Delta H, and Delta T?

We'll have another one over here, it can move freely. So over on this side, we'll have the definition of heat capacity, regular heat capacity, is the amount of heat you add divided by the change in temperature that you get. So on this side we're adding heat, let's say heat goes in, but the piston does not move and so the gas in here is stuck, it can't move, no work can be done.

Since this piston can't move, external forces can't do work on the gas, and the gas can't do work and allow energy to leave. Q is the only thing adding energy into this system, or in other words, we've got heat capacity at constant volume is going to equal, well, remember the first law of thermodynamics said that Delta U, the only way to add internal energy, or take it away is that you can add or subtract heat, and you can do work on the gas.

So there's no work done at all so the heat capacity at constant volume is going to be Delta U over Delta T, what's Delta U? Let's just assume this is a monatomic ideal gas, if it's monatomic we've got a formula for this.

That's not the only way I can write it. Remember I can also write it as three halves NK Delta T over Delta T, and something magical happens, check it out the Delta T's go away and you get that this is a constant. That the heat capacity for any monatomic ideal gas is just going to be three halves, Capital NK, Boltzmann's constant, N is the total number of molecules.

Or you could have rewrote this as little n R Delta T. The T's would still have cancelled and you would have got three halves, little n, the number of moles, times R, the gas constant.

So the heat capacity at constant volume for any monatomic ideal gas is just three halves nR, and if you wanted the molar heat capacity remember that's just divide by an extra mole here so everything gets divided by moles everywhere divided by moles, that just cancels this out, and the molar heat capacity at constant volume is just three halves R. So that's heat capacity at constant volume, what about heat capacity at constant pressure? Now we're going to look at this side. Again, we're going to allow this gas to have heat enter the cylinder, but we're going to allow this piston to move up while it does that so that the pressure inside of here remains constant and this is going to be the heat capacity at constant pressure.

What's W going to be? Remember W is P times Delta V. So this is a way we can find the work done by the gas, P times Delta V, so this is going to equal Delta U, we know what that is. If this is again, a monatomic ideal gas, this is going to equal three halves nR Delta T plus this is P times Delta V, but we have to be careful, in this formula this work is referring to work done on the gas, but in this case, work is being done by the gas, so I need another negative.

- Kinetic molecular theory of gases
- What Is Free Energy?
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Technically the work done on the gas would be a negative amount of this since energy is leaving the system. So what do we get? Look what I'm left with. I'm left with C.

### Heat capacity at constant volume and pressure (video) | Khan Academy

Heat capacity at constant pressure is going to be equal to three halves nR plus nR, that's just five halves nR, and if I wanted the molar heat capacity again I could divide everything, everything around here by little n, and that would just give me the molar heat capacity constant pressure would be five halves R.

And notice they're almost the same.

The heat capacity at constant volume is three halves nR, and the heat capacity at constant pressure is five halves nR. They just differ by nR.

### Enthalpy - Chemistry LibreTexts

So the difference between the heat capacity at constant volume which is three halves nR, and the heat capacity at constant pressure which is five halves nR, is just Cp minus Cv which is nR, just nR, and if you wanted to take the difference between the molar heat capacities at constant volume and pressure, it would just be R.

Or it can be as imaginary as the set of points that divide the air just above the surface of a metal from the rest of the atmosphere as in the figure below.

Internal Energy One of the thermodynamic properties of a system is its internal energy, E, which is the sum of the kinetic and potential energies of the particles that form the system. The internal energy of a system can be understood by examining the simplest possible system: Because the particles in an ideal gas do not interact, this system has no potential energy. The internal energy of an ideal gas is therefore the sum of the kinetic energies of the particles in the gas.

## Heat capacity at constant volume and pressure

The kinetic molecular theory assumes that the temperature of a gas is directly proportional to the average kinetic energy of its particles, as shown in the figure below.

The internal energy of an ideal gas is therefore directly proportional to the temperature of the gas. The internal energy of systems that are more complex than an ideal gas can't be measured directly. But the internal energy of the system is still proportional to its temperature.

We can therefore monitor changes in the internal energy of a system by watching what happens to the temperature of the system. Whenever the temperature of the system increases we can conclude that the internal energy of the system has also increased. Assume, for the moment, that a thermometer immersed in a beaker of water on a hot plate reads This measurement can only describe the state of the system at that moment in time. It can't tell us whether the water was heated directly from room temperature to Temperature is therefore a state function.

It depends only on the state of the system at any moment in time, not the path used to get the system to that state. Because the internal energy of the system is proportional to its temperature, internal energy is also a state function.

Any change in the internal energy of the system is equal to the difference between its initial and final values. Energy can be transferred from the system to its surroundings, or vice versa, but it can't be created or destroyed. First Law of Thermodynamics: It says that the change in the internal energy of a system is equal to the sum of the heat gained or lost by the system and the work done by or on the system.

When the hot plate is turned on, the system gains heat from its surroundings. As a result, both the temperature and the internal energy of the system increase, and E is positive. When the hot plate is turned off, the water loses heat to its surroundings as it cools to room temperature, and E is negative. The relationship between internal energy and work can be understood by considering another concrete example: When work is done on this system by driving an electric current through the tungsten wire, the system becomes hotter and E is therefore positive.

Eventually, the wire becomes hot enough to glow. Conversely, E is negative when the system does work on its surroundings.