# Pressure-Volume Diagrams

by Jared Rovny

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00:01 The next topic is pressure-volume diagrams.

00:04 And this topic is one of the most important, not only for thermodynamics, but also for this entire physics course in the context of the MCAT.

00:12 These kinds of questions are very, very common, but they're also somewhat confusing if you're not aware of exactly how these pressure-volume diagrams work.

00:20 So we're really going to dive into these in some serious detail because again, these are very important especially for an exam context.

00:27 There are two ways, as we've discussed already, for a system to change the amount of internal energy it has.

00:34 First of all, as we've just discussed it can have heat either added to it or that heat can be lost from your system.

00:40 A second way for energy to leave or go into a system is for that system to do work by again applying a force over a distance as we've discussed back in the Newton's last chapters, or that work could be done on the system, giving that system more internal energy.

00:56 We represent the pressure and volume of the system as these changes happen using a pressure-volume diagram, which you can see here.

01:04 In a pressure volume diagram we always put the pressure on the vertical axis and the volume of the system on the horizontal axis.

01:11 And these two can be changing in some way, which we'll discuss shortly.

01:15 As the system evolves through some sort of a process whether work is being done on the system or whether work is being done by the system.

01:23 And to measure the amount change of energy of a system as it either gains or loses heat or does or receives work, we have that the change in the internal energy, which again is this capital letter U, will be equal to the amount of heat added, minus the amount of work done.

01:40 So you can see that by this convention we're using the work W to mean the work done by the system.

01:46 So in other words, if Q is a postive number, that means heat is being added to your system.

01:51 And so the energy change will go up, you will gain some energy.

01:54 On the other hand if the system does work, W will be a positive number as well.

01:59 Meaning that the system loses energy because the system used that energy that it had to do work, to change the environment around it.

02:07 And so be careful with the sign convention here.

02:09 Sometimes people will write this as Q plus W and use W to mean the work done on the system.

02:16 You can always make sure you have your sign right relative to a given problem by ensuring that it physically makes sense, that if a system is doing work, that's causing something to change around it, it must be giving up energy in order to accomplish that work.

02:29 The amount of work done -- and this is the key point especially in an exam setting -- will be equal to the pressure of the system times the change in the volume of the system.

02:41 And that's why we've written W in the shaded area below the blue line.

02:44 So in this system as we've graphed it here, the pressure is staying constant while the volume is increasing in your system.

02:52 So you could think of this maybe as a balloon, the pressure may be is staying constant but the balloon blows up and inflates, increasing the volume for a given amount of air.

03:00 So this would really only happen of course if the temperature of that the system were changing.

03:04 But if this is happening if the pressure is staying constant while the volume is increasing, we can think back to our Newton's laws, that the work done by a system is equal to the force that the system is applying.

03:16 In this case, it would be the force of the gas molecules and the wall of whatever container it's in, times a distance.

03:22 So the thing that they're pushing on has to go some distance.

03:25 So thinking about the container for a gas, if those gas particles are hitting the wall or applying force on the wall, and then the wall moves, changing the volume of your system, you know that that gas did work because it applied a force over a distance.

03:40 All we would have to do to find the work is a trivial sort of change in the variables of force times distance and then changing the units slightly.

03:49 We could see that it would be pressure times volume.

03:52 And we won't go over that change, that derivation because the important thing for you to memorize is just that the work done by a system will be equal to the pressure of the system times the change in that system's volume.

04:04 Geometrically, once again before we move forward, notice that in this graph this would be the horizontal axis, the change in the volume of your system times the pressure of that system which would be the vertical height of that blue line.

04:18 And so this is simply the area of a rectangle.

04:20 The base, the change in volume times the height, which is the pressure of the system.

04:24 And so the area under the blue curve where the blue curve describes what your system is doing as it's changing its volume by the given pressure, that area under the curve, the area under that line will always represent the work that's been done by your system.

04:38 It turns out that the direction of this change will always tell you what kind of work is being done, whether it's positive work or negative work.

04:47 In other words, work being done by your system or work being done on your system.

04:52 And again, you can always use your physical intuition to make sure you have the sign of this correct.

04:57 So for example, we said that this graph describes a system that is at a constant pressure, and then moves towards more volume, it's increasing its volume.

05:05 And you can think intuitively that, of course, the gas must be doing work pushing on the walls of its container, making those walls go out to increase the volume of the system, meaning that this must be positive work done by your system.

05:18 The work done by the system that's a positive number is going to also be the energy lost by your system, which we can see in the first equation that we've already introduced here, that the change in the energy of your system will depend on the negative of the work done by the system, again, just because the energy is being put into doing that amount of work.

05:38 If the pressure-volume curve is blue line, it's more complicated.

05:43 It might seem harder to find the work, but it turns out that the area under the curve argument that we just gave will still apply no matter how complicated the pressure-volume line is.

05:53 It could be all sorts of a squiggly line or a curve like this one, and the work done by the system will still be the entire area under the curve from the line down to the axis.

06:04 And this is how you would always find the work done by a system.

06:08 We can choose to keep different quantities constant as we change the volume and the pressure of our system.

06:14 So for example in this system right here, as the volume is changing, the pressure is staying constant.

06:22 When we have a constant pressure system, we call it an isobaric system.

06:26 So you can remember this in terms of barometer.

06:29 If you heard of a barometer, which is a way of measuring the pressure of a system -- this is how weather people also measure the pressure of systems outside using a barometer.

06:38 We would say that the barometer, the pressure of the system is not changing, so it's isobaric, meaning that the pressure is not changing as the volume or other variables in your system can change.

06:49 Another variable we could keep constant is instead the volume.

06:53 So if the volume stayed constant in this graph while the pressure increased, you could see we would just have a vertical line.

06:58 We call this an isochoric change. And this kind of change is much less common unless we're talking about a component of a cycle, and we'll get to cycles very shortly.

07:08 We could also keep the temperature of our system the same.

07:11 So what happens if we have the temperature of the systems staying constant while the pressure in volume are changing? Going back to our chapter on gases we already talked about Boyle's Law, Where Boyle's Law was something that came from experiments where the temperature was held constant for a given system.

07:27 while the pressure and the volume of the system were allowed to change.

07:30 So we already know exactly the equation, the expression for this.

07:33 And we can see that on our pressure-volume diagram, it follows the exact same shape that we saw back when we were discussing gases and isothermal changes.

07:42 Finally, we could have what's called an adiabatic change.

07:46 In an adiabatic change, what were saying is that we don't add any heat to the system.

07:52 So if we have no heat transfer either into the system or out of the system, then the change in the energy of the system will simple be equal to, minus the work done by the equation that we already introduced for the change of energy of a system.

08:05 So this line that you can see here.

08:07 The pressure volume curve is going to be slightly different than it was for those isothermal curves that were given by Boyle's Law.

08:15 So it's good to know that for these isothermal changes, these red lines where we're keeping the temperature of our system constant the adiabatic lines sometime called an adiabat, will lie between the two isotherms -- one defined by the beginning point and one defined by the end point.

08:31 Because again, for an adiabatic change we're not allowing any energy, any heat energy to be added to or subtracted from the system, which means it is not able to maintain the same amount of internal energy because we're not allowing it to change the heat while its work is changing.

### About the Lecture

The lecture Pressure-Volume Diagrams by Jared Rovny is from the course Thermodynamics and Thermochemistry.

### Included Quiz Questions

1. Amount of work done = pressure x change in volume
2. Amount of work done = pressure x change in energy
3. Amount of work done = pressure x change in temperature
4. Amount of work done = pressure x change in specific heat
5. Amount of work done = pressure x change in internal energy
1. Work done by a system: W > 0, is the energy gained by the system
2. Work is pressure times change in volume
3. Direction of change determines the sign
4. Work is calculated by the area under the curve on a pressure volume graph
5. Work can be represented as a curve with pressure on Y axis and volume of X axis
1. When the volume is constant
2. When the temperature is constant
3. When pressure is constant
4. When there is no heat transfer
5. When work is constant
1. Between two isotherms
2. Above the two isotherms
3. Below the two isotherms
4. Between two isobaric curves
5. Between two isochoric curves

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