PV-Diagrams chart the change in pressure and volume of some gas as it undergoes some thermodynamic process. They are typically a plot with pressure on the vertical axis and volume on the horizontal. They show transitions from one state ($P_i, V_i$) to another state ($P_f, V_f$).

Here is a video to illustrate an object undergoing non-Uniform Circular Motion.

https://www.youtube.com/watch?v=qVAvmieRM1E

Pre-lecture Study Resources

Watch the pre-lecture videos and read through the OpenStax text before doing the pre-lecture homework or attending class.

BoxSand Introduction

Thermo  |  PV Diagrams and Ideal Gas Processes


PV-Diagrams are a useful tool for understanding thermodynamic processes. These diagrams are typically shown with pressure along the vertical axis and volume along the horizontal. 

This is a P.V. graph with pressure on the y axis and the volume on the x axis and a point at some initial pressure and some initial volume with the words one possible state above.

If you had an ideal gas for instance ($PV=Nk_BT$), and the number of molecules was known, then for a given pressure and volume, there would be a unique temperature. So a point on a PV diagram tells you a great deal about the state of the system. Similarly a change from one point to another point tells you a great deal about the changes in state variables and the energy transfers during that transition.

This is a P.V. graph with pressure on the y axis and the volume on the x axis and a point at some initial pressure and some initial volume and another point with a final pressure and final volume which are both lower than the initial point. It shows a curved arrow from the initial state to the final state.

In the above diagram the pressure and volume both decreased. Assuming the number of particles in your system remained constant (an assumption always made here), with P and V both decreasing, than so must T according the ideal gas law. So this diagram helps you understand that the temperature decreased during this transition. Since internal thermal energy is related to temperature, you also know that $E_{th}$ has decreased. You can also know information about the work during this process.

Work is equal to the area under the PV-diagram curve

work is positive if the volume decreases and negative if the volume increases

So the you know the sign of the work is positive in this case, meaning the environment did work on your system. Think about compressing a gas, you'd have to push the gas into a smaller volume. You would lose energy doing the compression, which means you (the environment) is losing energy (Wenvironment is negative) so the system (the gas) is gaining that energy and thus Wsystem is positive. Put all that together with the first law of thermo $\Delta E_{th}=Q+W$: you know that $\Delta E_{th}$ is positive, and so is W, so what is the sign of the heat Q and how large is it? In this case Q could be positive or negative and to find the magnitude, you'd have to calcuted the change in thermal energy and the work, then use the first law to solve for Q. If $\Delta E_{th}$ is a larger positive than W, then Q must be postive. If $\Delta E_{th}$ is a smaller positive than W, then Q must be negative.

It's important to note that there are other important thermodynamic curve diagrams, pressure vs. temperature or temperature vs. volume among others, but we have decided to focus only on the PV-diagrams here.

Ideal Gas Processes

Isobaric: Pressure is sometimes measured in a non-SI unit called Bars. Iso is a prefix meaning constant. So, isobaric refers to a constant pressure process.

This is a P.V. graph with pressure on the y axis and the volume on the x axis. There are two points at an initial volume and a final volume that is higher than the initial volume both at the same pressure. The area underneath the two points shows the work done from the system. The graph shows an expansion at constant pressure.

Work = area under PV-curve,   for isobaric:   $W=P \Delta V$

With pressure constant and volume increasing, the temperature must also increase. Since the temperature increases, so does the thermal energy. In an expansion, like that shown above, the work is negative. With the first law ($\Delta E_{th}=Q+W$), heat (Q) is equal to the the change in thermal energy minus the work. With a positive $\Delta E_{th}$ and negative work (W), heat (Q) must be positive and greater in magnitude than the work. So in an isobaric expansion, energy goes in via heat more quickly than it goes out via work and thus the thermal energy increases.

Isochoric: A process where the volume remains constant is considered an isochoric process. From the ideal gas law, if the volume remains constant but the pressure increases, like in the diagram below, then the temperature must increase.

This is a P.V. graph with pressure on the y axis and the volume on the x axis. There are two points at an initial pressure and a final pressure that is higher than the initial pressure both at the same volume. Because there is no area underneath this curve, the work done and on the system is zero. The graph shows a heating at constant volume.

Work = area under curve,   w/ $W = 0$,   $Q = \Delta E_{th}$

Since work is the area under the PV-curve, the work in an isochoric process is zero. With work equal to zero, the heat is equal to the change in thermal energy (1st law). Since the temperature increased in the above diagram, the thermal energy must also have increased. Heat entered the system while zero work was done by the system, as a result it increased its temperature.

Isothermal: A process where the temperature remains constant is considered an isothermal process. The ideal gas law shows us that if the temperature is constand the volume increases, then the pressure must decrease. Because of the inverse relationship between pressure and volume when temperature (and number) is constant, isothermal transitions are always curvy lines in a PV-diagram.

This is a P.V. graph with pressure on the y axis and the volume on the x axis. There are two points at an initial high pressure and initial low volume and a second point with low final pressure and a high final volume. They are connected by a curved graph because they are at the same temperature. The area underneath this curve shows the work done onto the system. This graph shows an expansion at a constant temperature.

Work = area under curve,  for isothermal:   $|W|=Nk_BT |\ln{\frac{V_f}{V_i}}|$

In an isothermal process the internal thermal energy remains constant, so from the 1st law, the heat must be equal to negative the work. If the system is expanding, like that shown above, the work is negative so the heat must be positive. Heat is entering into the system at the same rate work is taking that energy out of the system.

Adiabatic: A process where no heat is transfered into or out of the system is considered an adiabatic process. On a PV-diagram adiabatic curves are curvy like isotherms, but they are steeper.

This is a P.V graph with the initial point at a higher initial pressure, lower initial volume and higher initial temperature and a lower final pressure, higher final volume and a lower final temperature. There is a linear line that connects these two points which is called an adiabatic process. The area underneath this curve is the work done onto the system.

Work = area under curve,   w/ $Q = 0$,  $W = \Delta E_{th}$

In an adiabatic process there is no explicit equation for the work in terms of the state variables. In these transitions, you can use the first law and the fact that the heat is zero to find the work equal to the change in internal thermal energy. For a monoatomic gas we know that $\Delta E_{th}=\frac{3}{2}Nk_B \Delta T$, an equation found in the kinetic theory of gases.

Key Equations and Infographics

A representation with the words heat engine on the top. There is an equation that shows that the efficiency of a heat engine is equal to the absolute value work done by the system divided by the absolute value heat entering the system. This is also equal to the difference of the absolute value heat entering the system and the absolute value heat leaving the system, divided by the absolute value heat entering the system. This is also written in words below.

A representation with the words Carnot cycle on the top. There is an equation that shows that the theoretical maximum efficiency is equal to one minus the temperature of the cold reservoir divided by the temperature of the hot reservoir. This is also written in words below.

A representation with the words thermodynamic efficiency on the top. There is an equation that shows that the thermodynamic efficiency of cycle is defined as the what you get out (net work) divided by what you had to pay in (heat in). Applied to a cycle, this is also equal to the difference between the heat out and the heat in divided by the heat in. This is also written in words below.

Now, take a look at the pre-lecture reading and videos below.

BoxSand Videos

OpenStax Reading


OpenStax Section: none

Fundamental examples

 

1. (a) and (b): which type of thermodynamic process is sketched? (c) Sketch the thermodynamic process in which temperature does not change. (d) Sketch the thermodynamic process in which heat neither enters or nor leaves the system.

This is an image of two P.V. diagrams. The first P.V. diagram labeled as A shows an initial point with an initial pressure and a low initial volume and a final point with the same pressure and a higher final volume. The second P.V. diagram labeled as B shows an initial point with a high initial pressure and initial volume and a low final pressure and the same volume.

2. During a thermodynamic process, the pressure in a container goes to $\frac{2}{3}$ it's original pressure: $P_f = \frac{2}{3} P_i$. Likewise, $T_f = 2T_i$. (a) What is the final volume (in terms of $V_i$)? (b) Is $\Delta E$ for this process positive or negative? (c) Does heat enter or leave the system during this process, and if so, in which direction does heat flow?

3. Using the diagram below, find the following: (a) the temperature $T_1$ at point 1 and the pressure $P_3$ at point 3. (B) For each step of the process - from points 1 to 2 and from points 2 to 3 - was the change in thermal energy positive or negative? The work? Did heat enter or leave the system during this stage, and how much? Hint: make a table.

Solutions

Short foundation building questions, often used as clicker questions, can be found in the clicker questions repository for this subject.

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Post-Lecture Study Resources

Use the supplemental resources below to support your post-lecture study.

Practice Problems

BoxSand practice problems

Conceptual problems

BoxSand's multiple select problems

BoxSand's quantitative problems

Recommended example practice problems 

  • The Openstax section for has several practice exercises at the bottom of the page, Website Link 
  • Large set of pv diagram related problems, answer key at the bottom, PDF Link 

For additional practice problems and worked examples, visit the link below. If you've found example problems that you've used please help us out and submit them to the student contributed content section.

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Additional Boxsand Study Resources

Additional BoxSand Study Resources

Learning Objectives

Summary

Summary

Atomistic Goals

Students will be able to...

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YouTube Videos

These videos discuss first what is a PV-diagram, and then how to use a PV-Diagram to determine the work done on the sytem for the four types of processes

https://www.youtube.com/watch?v=PxYmvFf834g

This Video is a continuation of the previous and focuses on Isothermal Isometric and Adiabetic processes,

https://www.youtube.com/watch?v=fdmbB3XbSWw

Other Resources

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Simulations


Here is a link to PCCL's basic simulation demonstration the connection between pressure and volume 

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For additional simulations on this subject, visit the simulations repository.

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Demos


For additional demos involving this subject, visit the demo repository

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History


Oh no, we haven't been able to write up a history overview for this topic. If you'd like to contribute, contact the director of BoxSand, KC Walsh (walshke@oregonstate.edu).

Physics Fun

 

Other Resources


Boston University's page for the first law of thermodynamics discusses the use of pv-diagrams.

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This PDF covers PV-Diagrams and the four process types, and how to find the work done,

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Hyperphysics's reference page for PV diagram snad heat cycles,

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Problem Solving Guide

Use the Tips and Tricks below to support your post-lecture study.

Assumptions

There are a LOT of things going on in analyzing a thermodynamic cycle. Here are some things that are always true to help you:

Remember the names of processes and tie them to their meaning. Iso- is a prefix that means "same", if it is there, that means the word that follows it is held constant in that process. Isothermal means process with constant temperature, isobaric means process with constant pressure, isochoric means process with constant volume. Also remember the shapes that go with each (check out the bakc of the physics survival sheet for this section for the most useful table for these processes: Survival Sheet.

There are a number of easily identifiable 0's that show up when analyzing thermodynamic cycles, here are the most frequent ones:

1) Delta Eth (change in thermal energy) for a cycle that returns to where it starts is always 0!

2) Delta Eth for an Isothermal process is always 0! This always means for an isothermal process W = -Q!

3) Work (W) for a isochoric process is always 0 (vertical line with no area underneath on a PV diagram means W = 0)!

4) Heat in an adiabatic process is always 0.

When drawing a PV diagram, an Isothermal curve must ALWAYS curve AWAY from the P and V axes (or get closer to them as P or V increases if you prefer to think of it that way). It is a frequent mistake since isothermal curve are parabola-like and bow towards or away from the origin, but they must ALWAYS bow towards the origin (open away from it).

The 1st law is always true, both for the total cycle and each step along the way, use it everywhere to save a lot of time and energy once you know 2 out of 3 of work, heat, or change in thermal energy in any section of the cycle.

The area underneath the curve in a PV diagram is work! Use geometry to find it unless the curve is isothermal, then use the equation for it!

There are two ways to tell a heat engine from a heat pump/refrigerator 1) If it is a cycle (returns to the place it starts on a PV diagram) if it progresses clockwise then it is a heat engine, counter-clockwise is a heat pump/refrigerator. 2) Heat engines do net work, heat pump/refrigerators take in net work. Therefore, you can tell by the sign of the work, negative is a heat engine, positive is a heat pump/refrigerator. Without calculating anything, you can tell this by finding the sign of work for the process on the top of the diagram (has most area underneath it, so it determines the sign of work). For expanding processes where V increases, work is negative. For contracting processes where V decreases, work is positive.

 

Checklist

For any thermodynamic cycle:

1) Breakdown how many different processes there are and what type each is (isobaric, isothermal, isochoric, etc.)

2) Draw the PV diagram for the cycle

3) Make a table where the rows are each process it undergoes and the columns are for work, heat, and thermal energy.

4) Start where you know the most information in the cycle (usually where you know 2 out of 3 of P, V, and T), and begin using your equations and geometry to fill out the work calculations, then find heat and Eth afterwards depending on which are easier. Remember, you know Delta Eth for the entire cycle is zero, and for every process you can apply the 1st law (Delta Eth = Q + W), so you should only ever need to know 2 out of 3 of Delta Eth, W, and Q for any process and can use simple addition and subtraction with the 1st law to find the last one!

Misconceptions & Mistakes

  • Not making a table

  • Forgetting which equations you have at your disposal: first law, second law, equipartition theorem, and ideal gas law. (I think this is an exhaustive list..? I'll watch lecture vids tomorrow and see if more come up)

Pro Tips

  • Make a table! You will be so lost without one. A table helps you identify which information you have and what you lack. It will also make the relationships between different steps more obvious - recall when doing energy analysis, we could often fill in "potential energy = 0" to multiple entries. The same thing happens with PV Diagram tables. This helps you determine which variables you have are actually unkown, which helps you find the correct equations to apply.

  • PV Diagrams are tedious, don't expect them to be anything else when digging into one. You will not have a linear path to solving the problem (you probably can't just go left-to-right and top-to-bottom filling out your table) and will probably have to jump between different points in the cycle. This is okay, this is how they're supposed to be done.

  • Lots of systems change their thermal energy by units of $\frac{3}{2}Nk_{B}T$, and the thermal energy of a system is often $\frac{3}{2}Nk_{B}T$. For lots of book problems/mastering physics problems/homework problems, the entry in one of the boxes is just $\frac{3}{2}Nk_{B}T$, where T is the temperature at the point (probably labeled with a subscript so  $T_A$, $T_B$, or something like that). Get used to seeing and writing $\frac{3}{2}Nk_{B}T$.

  • The only equations you need are the ideal gas law, the first law, the second law, and the equipartition theorem. A few of these combine to make the virial equation, which is also sometimes useful. 

  • $Nk = nR = pV$. That means $3/2 nKT = 3/2 pV$. This is sometimes useful <needs formatting>

Multiple Representations

Multiple Representations is the concept that a physical phenomena can be expressed in different ways.

Physical

Physical Representations describes the physical phenomena of the situation in a visual way.

 The graphical nature of the PV diagram describes the relationship between pressure and volume with respect to the state temperature.

This is an image of a P.V. diagram with two points on the graph. The initial point starts at a high initial pressure, low initial volume, and high initial temperature. The final point is at a low final pressure, high final volume, and a low final temperature. There is a linear line that connects both initial and final state.

Mathematical

Mathematical Representation uses equation(s) to describe and analyze the situation.

$PV=nRT = N K_b T$

Pressure ($P$), Volume ($V$), Numer of mols ($n$), Rydberg constant ($R$), Temperature ($T$), Numer of particles ($N$),

Boltzman constant ($K_{b}$)

A representation with the words thermal process: isobaric on the top. There is an equation that shows that the in an isobaric reaction (constant pressure) the work is equal to the negative pressure multiplied by the change in volume. This is also written in words below with a note that says isobaric refers to a thermodynamic process in which there is constant pressure.

A representation with the words thermal process: isochoric on the top. There is an equation that shows that the in an isochoric process (constant volume) the heat is equal to the change in thermal energy because the work is zero. This is also written in words below with a note that says isochoric refers to a thermodynamic process in which there is constant volume.

A representation with the words thermal process: isothermal on the top. There is an equation that shows that the absolute value of the work is equal to the product of the number of particles, Boltzmann’s constant, and the temperature multiplied by the absolute value of the natural log of the final volume divided by the initial volume. This is also written in words below with a note that says isothermal refers to a thermodynamic process in which there is constant temperature and the Boltzmann’s constant is equal to one point three eight times ten to the negative twenty third power in units of meters squared times kilograms divided by seconds squared and Kelvin.

A representation with the words thermal process: adiabatic on the top. There is an equation that shows that in an adiabatic process, (zero heat) work is equal to the change in thermal energy. This is also written in words below.

Graphical

Graphical Representation describes the situation through use of plots and graphs.

 The graphical nature of the PV diagram describes the relationship between pressure and volume with respect to the state temperature.

This is an image of a P.V. diagram with two points on the graph. The initial point starts at a high initial pressure, low initial volume, and high initial temperature. The final point is at a low final pressure, high final volume, and a low final temperature. There is a linear line that connects both initial and final state.

Descriptive

Descriptive Representation describes the physical phenomena with words and annotations.

 PV diagrams are directly related to the ideal gas law which describes the relationship between pressure (P), volume (V), and temperature (T). Namely,  pressure and volume are inversely proportional to each other when the temperature is held constant. In addition, there is a direct relationship between pressure and temperature when the volume is held constant, or conversely there is a direct relationship between volume and temperature when the pressure is held constant.

Experimental

Experimental Representation examines a physical phenomena through observations and data measurement.

 A quick experiment could be to measure the pressure of a piston relative to the volume inside. As we can see from the graph, when the temperature is held constant at say $T_{i}$, then we can see the explicit inverse relationship between pressure (P) and volume (V).