Unit 9: Thermodynamics

Unit 9 in one idea

Thermodynamics is Unit 9, the opening unit of AP Physics 2. The CED weights it at 15 to 18 percent of the multiple-choice section and suggests roughly 10 to 16 class periods for it. Two other units, Electric Force, Field, and Potential (Unit 10) and Electric Circuits (Unit 11), sit in the same 15 to 18 percent band; the remaining four units of the course are each 12 to 15 percent.

One idea holds the six topics together, and the unit opener states it in its first line: in Unit 9 "students investigate what they cannot see by examining the properties of ideal gases." You learn to describe the same gas twice. Microscopically it is a swarm of atoms with a spread of speeds. Macroscopically it is three numbers you can measure, pressure, volume and temperature, plus one you infer from them, internal energy. The opener is direct about what that costs you: "Translation between models and representations is key in this unit."

The rest of the unit is bookkeeping on energy crossing a system boundary: how much crosses it (Topic 9.4), how fast it crosses (Topic 9.5), and which way it goes on its own (Topic 9.6). Get in the habit of naming your system and its boundary before you write anything down, and most of this unit turns routine.

What the CED requires across the unit

Six topics carry nine learning objectives between them. Here is the Unit at a Glance list with the suggested science-practice skill codes the CED prints against each topic.

Every one of those nine objectives starts with the verb "describe", and the CED is explicit that the verb covers graphical, mathematical and verbal work: students should be able to describe a physical concept in all three ways, not just calculate with it. The unit opener singles out four skills as the ones Unit 9 develops. Derive a symbolic expression by following a logical mathematical pathway (2.A). Predict new values or factors of change using functional dependence between variables (2.D). Apply a law, definition, relationship or model to make a claim (3.B). Justify or support that claim with evidence (3.C).

Three of the six topics carry a boundary statement, and each one cuts work out of the unit. Quoted in full, exception clauses included:

• Topic 9.1: "AP Physics 2 only expects students to perform qualitative and quantitative analysis of collisions in one and two dimensions. Students are not expected to know the functional form of the Maxwell-Boltzmann distribution but are expected to be familiar with how features of the distribution are related to the temperature of the gas."
• Topic 9.5: "AP Physics 2 will model specific heat as independent of temperature."
• Topic 9.6: "Only qualitative treatment of the second law of thermodynamics is within the scope of AP Physics 2."

Read the whole of each. The 9.1 statement excuses you from the functional form of the Maxwell-Boltzmann distribution and then, in the same sentence, requires you to be familiar with how features of the distribution are related to the temperature of the gas. That is a graph-reading job, not a memorization one, and the first half of the statement on its own would tell you the wrong thing.

How the six topics build on each other

Take them in CED order. Each one supplies something the next needs.

[9.1 Kinetic Theory of Temperature and Pressure](/ap-physics-2/unit-9-thermodynamics/9-1-kinetic-theory-of-temperature-and-pressure) defines the two headline quantities in terms of atomic motion. Pressure is the ratio of the sum of the magnitudes of the perpendicular components of the forces the gas's atoms exert on a surface to the area of that surface, and the CED adds that pressure exists throughout the gas itself, not just at the boundary with the container. Temperature is characterized by the average kinetic energy of the atoms in the system, which the sheet ties to speed:

The collisions behind gas pressure are analyzed with the same conservation of momentum principles you used in Physics 1, and the definition of pressure is the one from Physics 1 fluids.

[9.2 The Ideal Gas Law](/ap-physics-2/unit-9-thermodynamics/9-2-the-ideal-gas-law) names the assumptions behind the classical model of an ideal gas: instantaneous atomic velocities are random, the volume of the atoms is negligible compared with the volume the gas occupies, the atoms collide elastically, and the only appreciable forces are the ones during collisions. Those assumptions buy you one equation of state:

The topic also asks you to pull properties of a gas out of graphs of pressure, temperature and volume, and to extrapolate a graph of pressure against temperature back to the temperature at which an ideal gas would exert zero pressure.

[9.3 Thermal Energy Transfer and Equilibrium](/ap-physics-2/unit-9-thermodynamics/9-3-thermal-energy-transfer-and-equilibrium) carries no equation in the CED at all. It is definition work, and it is where the vocabulary the rest of the unit uses gets fixed. Two systems are in thermal contact if they may transfer energy by thermal processes; heating moves energy in and cooling moves it out; the three named thermal processes are conduction, convection and radiation. Energy moves spontaneously from the higher-temperature system to the lower-temperature one, and the CED explains why in terms of collisions: energy is most likely to pass from higher-energy atoms to lower-energy atoms, and after many collisions the most probable state is the one where both systems are at the same temperature. Thermal equilibrium is then defined as no net energy transfer between two systems in thermal contact, which is not the same as nothing happening.

[9.4 The First Law of Thermodynamics](/ap-physics-2/unit-9-thermodynamics/9-4-the-first-law-of-thermodynamics) carries the arithmetic. Internal energy is the kinetic energy of the objects making up the system plus the potential energy of their configuration, and the CED then rules the second part out for an ideal gas: its atoms do not interact through conservative forces, so an ideal gas has no internal potential energy and its internal energy is purely kinetic.

The first law itself is described as a restatement of conservation of energy that accounts for energy transferred by work, heating or cooling. The topic then adds PV diagrams, isotherms, work as the area under the curve, and the four special processes: isovolumetric, isothermal, isobaric and adiabatic.

[9.5 Specific Heat and Thermal Conductivity](/ap-physics-2/unit-9-thermodynamics/9-5-specific-heat-and-thermal-conductivity) puts numbers on Topic 9.3. How much energy a temperature change costs is $Q = m c \Delta T$; how fast conduction delivers it is $\frac{Q}{\Delta t} = \frac{k A \Delta T}{L}$. The CED calls both specific heat and thermal conductivity intrinsic properties of the material that depend on the arrangement and interactions of its atoms, which is the reason two objects of the same mass can need different amounts of energy for the same temperature rise. The conduction equation has the same shape as the resistance of a wire in Topic 11.3: more length resists, more cross-section helps.

[9.6 Entropy and the Second Law of Thermodynamics](/ap-physics-2/unit-9-thermodynamics/9-6-entropy-and-the-second-law-of-thermodynamics) closes the unit and, like 9.3, carries no equation in the CED. The second law states that the total entropy of an isolated system can never decrease and is constant only when every process the system undergoes is reversible. Entropy is described qualitatively as the tendency of energy to spread out, or as the unavailability of some of a system's energy to do work; localized energy tends to disperse; entropy is a state function; and maximum entropy occurs when a system is in thermodynamic equilibrium. The distinction to hold on to is that an isolated system's entropy never decreases, while a closed system's entropy can, because energy can be transferred into or out of it.

Which Unit 9 equations are printed on the sheet

The AP Physics 2 equation sheet carries 129 entries in seven blocks. One of them, Thermal Physics, is almost exactly Unit 9. It has eight lines, and here they are counted off the sheet with the topic each one serves:

That is the entire block. Pressure appears a second time elsewhere on the sheet, in the Mechanics and Fluids table that the Physics 2 sheet reprints in full, so a fluids question and a kinetic-theory question reach for the same printed line.

Four entries in the sheet's Constants and Conversion Factors table belong to this unit: Avogadro's number $N_0 = 6.02 \times 10^{23}\ \text{mol}^{-1}$, the universal gas constant $R = 8.31\ \text{J/(mol} \cdot \text{K)}$, Boltzmann's constant $k_B = 1.38 \times 10^{-23}\ \text{J/K}$, and $1\ \text{atm} = 1.0 \times 10^5\ \text{N/m}^2 = 1.0 \times 10^5\ \text{Pa}$.

Two things printed alongside those equations are easy to walk past and both settle arguments about signs and symbols. The Thermal Physics block has its own symbol key, and that key defines $W$ as "work done on a system" and $Q$ as "energy transferred to a system by heating". It also defines $k$ in this block as thermal conductivity, while the constants table elsewhere on the sheet uses the same letter for the Coulomb constant. Second, the sheet's opening page, alongside those constants, lists the conventions the exam uses unless a question says otherwise, and one of them is "Ideal gases are monatomic." That is why $U = \frac{3}{2} n R T$ is safe on any ideal gas the exam hands you.

What the sheet does not print matters just as much. There is no entropy equation on it, no efficiency formula, and no Celsius-to-Kelvin conversion. The absent entropy equation lines up with the Topic 9.6 boundary statement, which puts only a qualitative treatment of the second law in scope. The absent conversion does not let you off: it is on you. The whole sheet is laid out on the AP Physics 2 formula page.

The sign convention to fix before anything else

Two thermodynamic sign conventions are in circulation and they differ by a minus sign. AP Physics 2 uses one of them, and the CED states it inside Topic 9.4 rather than leaving it to your textbook.

Essential knowledge 9.4.B.1.ii: for a closed system, the change in internal energy is the sum of energy transferred to or from the system by heating, or work done on the system. Its relevant equation is

Essential knowledge 9.4.B.1.iii then defines that work. The work done on a system by a constant or average external pressure that changes the volume of that system, a piston compressing a gas being the CED's own example, is

So $W$ means the work done on the gas. Compress it and $\Delta V$ is negative, so $W$ comes out positive: you put energy in. Let it expand and $W$ comes out negative. Never write the first law as $Q - W$ on this exam. That is the other convention, in which $W$ means the work done by the gas, and a page that mixes the two produces a sign error that survives all the way to the boxed answer.

The routine for turning a PV diagram into numbers, reading work off the area when pressure is not constant, handling closed cycles, and working the isobaric, isochoric, isothermal and adiabatic cases, is set out step by step in the thermodynamics and PV diagrams guide. It uses the same sheet convention as this page, so you can move between the two without re-deriving anything.

Traps that span more than one topic

Each of these bites in more than one topic, which is why they are worth fixing at unit level rather than topic by topic.

• Absolute temperature versus a temperature interval. $PV = nRT$, $K_{\text{avg}} = \frac{3}{2} k_B T$ and $U = \frac{3}{2} n R T$ all need $T$ in kelvins, because each one would give nonsense at a negative Celsius reading. But $Q = m c \Delta T$ and $\frac{Q}{\Delta t} = \frac{k A \Delta T}{L}$ contain a difference, and a difference is the same number of kelvins as it is of Celsius degrees. Converting a $\Delta T$ is harmless; forgetting to convert a $T$ is not.
• Moles or atoms, $R$ or $k_B$. $R = 8.31\ \text{J/(mol} \cdot \text{K)}$ pairs with $n$ in moles; $k_B = 1.38 \times 10^{-23}\ \text{J/K}$ pairs with $N$, the number of atoms. The bridge is Avogadro's number, $N = n N_0$. Both halves of $PV = nRT = Nk_BT$ are printed, so pick the one that matches the quantity the question gave you instead of converting.
• Temperature is not energy, and heat is not something a system contains. Temperature is characterized by the average kinetic energy per atom; internal energy is the total. Two samples at the same temperature have the same $K_{\text{avg}}$ and can have wildly different $U$. And the sheet's own key calls $Q$ "energy transferred to a system by heating": it is a transfer, so a system never has a stock of $Q$ to spend.
• Same temperature does not mean same speed. $K_{\text{avg}} = \frac{3}{2} k_B T$ has no mass in it, so any two gases at the same temperature have the same average kinetic energy per atom. $K_{\text{avg}} = \frac{1}{2} m v_{\text{rms}}^2$ does have mass in it, so at that same temperature the heavier atoms are the slower ones.
• $k$ is two different constants on one sheet. In the Thermal Physics block $k$ is thermal conductivity, in watts per meter per kelvin. In the constants table $k$ is the Coulomb constant. Read which block the equation came from.
• Intrinsic property versus how much you have. Specific heat and thermal conductivity are properties of the material and do not change when you take a bigger piece. What changes is the energy required ($m$ in $Q = m c \Delta T$) and the rate ($A$ and $L$ in the conduction equation). The CED's own essential question for this unit, why a tile floor feels colder than a bathroom mat, is a question about $k$, not about temperature.
• Isolated versus closed, in Topic 9.6. The entropy of an isolated system never decreases. The entropy of a closed system can decrease, because energy can be transferred into or out of it. A freezer does not violate the second law; the freezer plus its surroundings is the isolated system.
• Pressure is not only a wall effect. The CED states that pressure exists throughout the gas itself, not just at the boundary between the gas and the container, so a diagram that shows pressure acting only at the walls is an incomplete representation of the gas.

How Unit 9 is assessed

The AP Physics 2 exam runs 3 hours. Section I is 42 multiple-choice questions in 85 minutes and is worth 50 percent of the score; Section II is 4 free-response questions in 95 minutes and is worth the other 50 percent. A four-function, scientific, or graphing calculator is allowed on both sections. The four free-response formats are fixed in advance: Question 1 Mathematical Routines, Question 2 Translation Between Representations, Question 3 Experimental Design and Analysis, Question 4 Qualitative/Quantitative Translation.

The 15 to 18 percent figure is stated for the multiple-choice section only, so on a 42-question Section I it works out at roughly 6 to 8 questions. The free-response section is not weighted by unit, so thermodynamics can turn up in any of the four.

The unit opener flags one of those formats in particular. It describes the Qualitative/Quantitative Translation as the fourth free-response question, where you first make a claim and support it with evidence and reasoning without reference to equations, then derive an equation or set of equations for the same scenario, then connect the claim to the mathematics. The CED warns that "students exposed primarily to numerical problem solving often struggle with the QQT because it requires them to express a conceptual understanding of course content and representations." Unit 9 gives that question plenty to work with: three of its six topics list both 3.B, apply a law or model to make a claim, and 3.C, justify a claim with evidence, among their suggested skills.

The opener also gives the flavor of Unit 9 prediction the CED has in mind: predict the temperature of an ideal gas in a sealed container from the change in volume and the heat added to or removed from the gas, then justify the prediction with physics principles. That single task pulls in the first law, the ideal gas law and kinetic theory at once.

How to work through the unit

The order in the CED is the order to study in, because 9.4 needs the internal-energy expression that 9.1 and 9.2 build, and 9.5 puts numbers on the transfer that 9.3 describes.

1. Start with 9.1 and 9.2 together. Get to the point where you can go from $P$, $V$ and $T$ to $n$, $N$, $K_{\text{avg}}$, $v_{\text{rms}}$ and $U$ without stopping to think about which constant goes where.
2. Do 9.3 in words. Write your own one-sentence answers to what thermal contact means, why energy flows the way it does, and what is and is not happening at thermal equilibrium. There is no equation to fall back on in this topic, so the wording is the answer.
3. Lock the sign convention before touching 9.4, then work PV diagrams until reading them is automatic, using the thermodynamics and PV diagrams guide for the procedure and cycle problems.
4. Treat 9.5 as two separate skills: an energy calculation with $Q = m c \Delta T$ and a rate calculation with $\frac{Q}{\Delta t} = \frac{k A \Delta T}{L}$. Practice the functional-dependence version of each, since skill 2.D is listed for this topic: what happens to the rate if the slab is twice as thick, half the area, or made of a material with three times the conductivity.
5. Finish with 9.6. Because the boundary statement keeps the second law qualitative, the work here is precision of language, not algebra.
6. Keep the AP Physics 2 formula sheet open while you practice so the eight thermal lines and their symbol key become familiar, and check the rest of the course at the AP Physics 2 hub.

The energy accounting in this unit is the same accounting as conservation of energy in Physics 1, with one new channel added for heating and cooling. If the Physics 1 version is solid, Unit 9 is mostly a matter of learning where the boundary of the system is and which sign each transfer takes.