Topic 9.2: The Ideal Gas Law

What Topic 9.2 requires

Topic 9.2 has one learning objective, 9.2.A: describe the properties of an ideal gas. Four essential knowledge statements sit under it, and none of them has sub-statements.

• 9.2.A.1 The classical model of an ideal gas assumes that the instantaneous velocities of atoms are random, the volumes of the atoms are negligible compared to the total volume occupied by the gas, the atoms collide elastically, and the only appreciable forces on the atoms are those that occur during collisions.
• 9.2.A.2 An ideal gas is one in which the relationships between pressure, volume, the number of moles or number of atoms, and temperature of a gas can be modeled using the equation $PV = nRT = Nk_BT$.
• 9.2.A.3 Graphs modeling the pressure, temperature, and volume of gases can be used to describe or determine properties of that gas.
• 9.2.A.4 A temperature at which an ideal gas has zero pressure can be extrapolated from a graph of pressure as a function of temperature.

Topic 9.2 prints no boundary statement. Three of the other five topics in Unit 9 do print one, 9.1, 9.5 and 9.6, so the absence here is worth noticing: nothing in the CED narrows what you can be asked about the ideal gas law itself.

There is one genuine oddity, and it is in the CED rather than in this page. The suggested skills for Topic 9.2 are listed twice in the same document, version 1, and the two lists do not match.

Both are printed in the Effective Fall 2024 course and exam description. The other five topics in Unit 9 agree with themselves across the two lists; only 9.2 disagrees. Treat the union as the safe reading, since every skill named is a skill the course assesses: create diagrams and schematics (1.A) and quantitative graphs with appropriate scales and units (1.B), compare quantities across scenarios (2.C), predict new values using functional dependence between variables (2.D), create experimental procedures (3.A), apply a law or model to make a claim (3.B), and justify a claim with evidence (3.C). Read together, that list describes this topic accurately: it is the graphing, predicting and experimental-design topic of the unit.

Unit 9 is weighted at 15 to 18 percent of the multiple-choice section, with a suggested 10 to 16 class periods.

The four assumptions that make a gas ideal (9.2.A.1)

Essential knowledge 9.2.A.1 is one sentence with four assumptions inside it. Splitting them out is worth doing, because each one names a different way the model can fail.

1. The instantaneous velocities of the atoms are random. No net drift, no preferred direction. This is why pressure is the same on every wall of a container, and it is what makes an average over atoms meaningful in the first place.
2. The volumes of the atoms are negligible compared to the total volume occupied by the gas. The atoms are treated as points. The $V$ in $PV = nRT$ is therefore the volume of the container, with nothing subtracted for the space the atoms themselves occupy.
3. The atoms collide elastically. No kinetic energy is lost to internal structure in a collision, so a sealed sample left alone does not cool itself down.
4. The only appreciable forces on the atoms are those that occur during collisions. Between collisions an atom travels in a straight line at constant velocity. There is no attraction pulling atoms together and no long-range repulsion pushing them apart.

Where the model breaks is the useful part.

• Squeeze a gas hard enough and assumption 2 fails: the atoms' own volume stops being negligible next to the container's, and the real gas resists compression more than $PV = nRT$ predicts.
• Cool a gas far enough, or use one with strong intermolecular attraction, and assumption 4 fails: the attractions start to matter at the low speeds involved, which is why real gases condense into liquids and ideal gases never do.
• Both failures are why the temperature in essential knowledge 9.2.A.4 has to be reached by extrapolation. No real gas is still a gas when its extrapolated pressure would reach zero.

Two vocabulary points. First, the CED says atoms, not molecules, throughout Unit 9, and the AP Physics 2 equation sheet's own symbol list defines $N$ as the number of atoms. That is deliberate: the Table of Information lists, among the conventions used on the exam unless otherwise stated, that ideal gases are monatomic. Second, the microscopic picture behind all four assumptions is Topic 9.1, which is where the collisions become a pressure and the speeds become a temperature.

Two ways to count the gas: n and R, or N and k_B (9.2.A.2)

This is one equation written twice, and the only difference is how you count the gas.

The bridge between them is Avogadro's number, also printed on the sheet: $N = nN_0$ with $N_0 = 6.02 \times 10^{23}$ per mole.

The two constants are not independent, and checking that is a good five-second use of a calculator in the exam room. Multiply the sheet's own numbers:

which is the sheet's $R$ to three significant figures. So $nRT$ and $Nk_BT$ are the same quantity of energy, arrived at by counting moles or by counting atoms. If a calculation gives you different answers by the two routes, the error is yours.

Which one to use is decided by the question, not by preference. A problem quoting grams or moles wants $n$ and $R$. A problem quoting a number of atoms, or asking for one, wants $N$ and $k_B$. A problem doing neither, such as a ratio question where the amount of gas is fixed, needs neither constant at all.

One more reason $Nk_BT$ earns its place on the sheet: it is the form that connects straight to Topic 9.1. Since $K_{\text{avg}} = \frac{3}{2}k_BT$ for one atom, $Nk_BT$ is two-thirds of the total translational kinetic energy in the sample. The macroscopic state equation and the microscopic energy statement are the same physics written at two scales.

Units: the three that decide whether the answer is right

$R = 8.31$ J/(mol K) is quoted in joules, so every other quantity has to be in the matching SI unit or the joules do not cancel. Three conversions cover almost every problem.

• Pressure in pascals. $1 \text{ Pa} = 1 \text{ N/m}^2$. The equation sheet prints 1 atm $= 1.0 \times 10^5$ N/m$^2$ $= 1.0 \times 10^5$ Pa, so use that conversion rather than a more precise value you may have learned in chemistry. Kilopascals need a factor of 1000.
• Volume in cubic meters. $1 \text{ L} = 1 \times 10^{-3}$ m$^3$ and $1 \text{ cm}^3 = 1 \times 10^{-6}$ m$^3$. Leaving a volume in litres puts the answer out by a factor of a thousand, and it still looks plausible, which is what makes it dangerous.
• Temperature in kelvin. Every time, without exception, and before any arithmetic.

The kelvin rule is not a convention you could choose differently. $PV = nRT$ says the pressure of a fixed sample at fixed volume is proportional to $T$, so $T = 0$ must be the temperature at which the pressure vanishes. Celsius puts its zero at the freezing point of water instead, which is nowhere near that point, so a Celsius reading substituted into the equation is not proportional to anything. The CED never prints the offset between the two scales and neither does the Table of Information; the unit-symbol table lists "degree Celsius" and "kelvin" and no relation between them. Carry it yourself.

A dimensional audit is a fast way to confirm a rearrangement. On the left, $[\text{Pa}][\text{m}^3] = (\text{N}/\text{m}^2)(\text{m}^3) = \text{N} \cdot \text{m} = \text{J}$. On the right, $[\text{mol}][\text{J}/(\text{mol}\cdot\text{K})][\text{K}] = \text{J}$, and equally $[\text{J}/\text{K}][\text{K}] = \text{J}$ for the $Nk_BT$ form. Both sides of the ideal gas law are an energy. If your rearranged expression does not come out in joules on both sides, stop and find the slip.

When a gas changes state: the ratio form

When a problem gives you a gas in one state and asks about another, you never need $n$ at all. Write the law twice, once for each state, and divide:

That equation is not printed on the AP Physics 2 equation sheet. Only $PV = nRT = Nk_BT$ is; check the full sheet rather than trusting a memory of one. Deriving the ratio form takes one line, and it is worth doing that line on paper rather than recalling the result, because the derivation is where the condition lives: $n$ and $R$ cancel only if the amount of gas does not change. A leaking container, a valve opened between two vessels, or gas pumped in makes this equation false and sends you back to $PV = nRT$ applied separately to each state.

Three special cases fall straight out, and they are the ones named in Topic 9.4 as thermodynamic processes.

The CED's own wording for the third of these is worth copying: essential knowledge 9.4.B.3 calls it constant volume (isovolumetric). Many textbooks say isochoric, and both appear in the CED's sample instructional activities, so recognise either.

Skill 2.D, predict new values or factors of change of physical quantities using functional dependence between variables, is precisely this kind of question, and it is on the Unit at a Glance list for Topic 9.2. The efficient way to answer one is by factors rather than by absolute numbers: if the volume is cut to two-fifths and the absolute temperature rises by a factor of 1.2, the pressure is multiplied by $2.5 \times 1.2 = 3.0$, and you never need to know $n$.

Reading PV, PT and VT graphs (9.2.A.3)

Essential knowledge 9.2.A.3 says graphs modeling the pressure, temperature, and volume of gases can be used to describe or determine properties of that gas. Three graph types cover almost everything you will be shown, and each is just $PV = nRT$ with one variable frozen.

• Pressure against volume, at fixed $T$ and $n$. Rearranged, $P = nRT/V$, so the curve is a hyperbola: $P$ falls as $V$ rises, and the product $PV$ is constant everywhere along it. The CED calls a line of constant temperature on a PV diagram an isotherm (essential knowledge 9.4.B.2.i). A higher temperature gives a curve further from the origin, and isotherms for different temperatures never cross.
• Pressure against temperature, at fixed $V$ and $n$. Now $P = (nR/V)T$, a straight line through the origin with slope $nR/V$. A smaller container gives a steeper line.
• Volume against temperature, at fixed $P$ and $n$. $V = (nR/P)T$, again a straight line through the origin, slope $nR/P$.

The "through the origin" part is true only if the horizontal axis is in kelvin. Plot the identical data against degrees Celsius and you get a straight line with a positive vertical intercept, which is a different-looking graph carrying the same information. Confusing the two is what the next section is about.

A caution for graph questions: the slope of a $P$-against-$T$ line is $nR/V$, so reading a slope off such a graph tells you $n/V$ and nothing else on its own. To get $n$ you need $V$ from somewhere else in the problem. Treating that slope as a single variable is a quiet way to get the wrong number.

Skill 1.B, create quantitative graphs with appropriate scales and units including plotting data, is a Unit at a Glance skill for this topic, and the CED gives a matching lab: groups take a 10 mL syringe with a Luer-Lok tip and cap, set it to 10 mL, cap the end, stand it vertical, stack books on top, calculate the absolute pressure of the books plus the atmosphere, and plot pressure against volume. Note the word absolute in that instruction. Gauge pressure will not produce a hyperbola.

The other thing a PV diagram is used for, area under the curve as the work done on the gas, belongs to Topic 9.4. The thermodynamics and PV diagrams guide works through that routine and its sign conventions in full.

Extrapolating to zero pressure (9.2.A.4)

Essential knowledge 9.2.A.4 reads: a temperature at which an ideal gas has zero pressure can be extrapolated from a graph of pressure as a function of temperature. It is one sentence, and it is doing more work than it looks.

Here is the experiment behind it. Seal a fixed amount of gas in a rigid container so $V$ and $n$ cannot change. Put it in a series of baths at known temperatures and record the pressure at each. Plot pressure against temperature in degrees Celsius. The points lie on a straight line. Extend that line backwards, past the coldest bath you could actually make, until it crosses the horizontal axis. That crossing point is the temperature at which the model says the pressure would be zero.

Three things are worth pulling out of that.

• The intercept does not depend on your choices. Change the gas, change how much of it you used, change the volume of the container, and the line's slope changes but the intercept does not. That is not obvious in advance, and it is the evidence that the temperature axis has a natural zero rather than an arbitrary one.
• It is an extrapolation, and the CED says so. Every real gas liquefies well before that temperature, so the final stretch of the line describes no measurement that anyone can make. The verb in 9.2.A.4 is "extrapolated" for exactly this reason.
• That intercept is the zero of the kelvin scale. It sits near $-273$ degrees Celsius, which is why $T$ in kelvin makes $P$ proportional to $T$ while $T$ in Celsius does not.

This is also the CED's own framing. One of the four essential questions printed on the Unit 9 opener is "How cold can something get?" Essential knowledge 9.2.A.4 is the course's answer, and it arrives through a graph rather than through a definition. The third worked example below runs the extrapolation with real numbers.

What the equation sheet gives you, and which topic owns it

The Thermal Physics block of the AP Physics 2 Table of Information holds exactly eight equations. Counting them is worth doing once, because it is a short enough list to know completely.

Seven of those eight are new at AP Physics 2, because the AP Physics 1 sheet has no thermal physics section. The exception is $P = F_{\perp}/A$, which AP Physics 1 also prints among its fluids equations. The AP Physics 2 sheet reprints the whole AP Physics 1 mechanics and fluids table alongside its own thermal, electricity, magnetism, waves and modern physics blocks, so that one equation ends up printed twice on the same page.

The constants for this topic are printed too: $R = 8.31$ J/(mol K), $k_B = 1.38 \times 10^{-23}$ J/K, $N_0 = 6.02 \times 10^{23}$ per mole, and 1 atm $= 1.0 \times 10^{5}$ Pa. See /formulas/ap-physics-2 for the rest.

Notice the row above and below $PV = nRT$. The internal energy of a monatomic ideal gas, $U = \frac{3}{2}nRT$, contains the same $nRT$ product, so a single number computed for one equation is immediately available to the other: $U = \frac{3}{2}PV$. That identity is not printed and is not required, but it explains why the first law and the ideal gas law get used in the same breath. What the first law actually asks you to do with it, including the $\Delta U = Q + W$ sign convention where $W$ is work done on the gas, is Topic 9.4.

How Topic 9.2 is tested, and where it leads

Taking both of the CED's skill lists together, four kinds of question follow.

• Plug and solve (2.C, and 2.B elsewhere in the unit). Find $n$, $N$, $P$, $V$ or $T$ from the other three. These are won or lost on units, not on algebra. Convert litres and Celsius before you touch the calculator.
• Factor-of-change questions (2.D). "The volume is halved and the absolute temperature is tripled; what happens to the pressure?" Answer by ratios, never by finding $n$ first.
• Graph questions (1.B, 1.A). Identify which variable is held constant, decide whether the relationship should be linear or hyperbolic, and check whether the axis is Celsius or kelvin before reading an intercept.
• Experimental design (3.A) with a claim to justify (3.B, 3.C). The syringe activity and the constant-volume thermometer are both in the CED's own materials, and both end in a graph you have to interpret rather than a number you have to produce.

The unit opener also flags where these land on the exam. It names the fourth free-response question, the Qualitative/Quantitative Translation, and describes it as asking students to make a claim with evidence and reasoning without reference to equations, then derive an equation or set of equations for the same scenario, then connect the two. It goes on to say that students exposed primarily to numerical problem solving often struggle with that question. Topic 9.2 is a natural setting for it, because the same gas can be described in words, as a graph, and as $PV = nRT$. The exam is 3 hours long: 42 multiple-choice questions worth 50 percent in 85 minutes, then 4 free-response questions worth 50 percent in 95 minutes, calculator allowed throughout.

Where it goes next:

• Topic 9.1 is the microscopic half of the same story: why pressure exists at all, and what temperature is measuring.
• Topic 9.3 takes two gases at different temperatures and puts them in contact.
• Topic 9.4 is where PV diagrams stop being a way to display a state and start being a way to compute work. Start with the thermodynamics and PV diagrams guide for that procedure.
• Topic 9.6 closes the unit, and its boundary statement limits AP Physics 2 to a qualitative treatment of the second law.