Unit 2

Thermodynamics

Everything in this unit is one idea wearing different hats: energy is conserved, and it spreads out. The first law is the conservation half. The second law is the spreading-out half. The formulas are bookkeeping for those two sentences.

Study order

Topic 1

Thermal expansion

Why solids grow when heated, linear and volume expansion, bimetallic strips, and why water is strange.

Topic 2

Heat, specific heat & calorimetry

Q = mcΔT, latent heat, the heating curve, and mixing-problem strategy.

Topic 3

First law & work on/by a gas

Internal energy, the sign convention that trips everyone up, and work in each process type.

Topic 4

PV diagrams

The graph gallery: isobaric, isochoric, isothermal, adiabatic, cycles, and reading work off the area.

Topic 5

Second law & entropy

Heat engines, efficiency, the Carnot limit, and what entropy actually counts.

Lab

Boyle’s law & work from a PV curve

Measure P vs V with a syringe, linearize it, and get work by integrating your own data.

Foundations: the ideal gas

Almost every problem in this unit is secretly about a gas in a container, so get these three relationships solid before anything else.

The equation of state

PV = nRT = Nk_BT n = moles, N = number of molecules, R = 8.31 J/(mol·K), k_B = 1.38×10⁻²³ J/K. T must be in kelvin.

The useful form in practice is the ratio form, because you almost never know n. For a fixed amount of gas going from state 1 to state 2:

P₁V₁T₁ = P₂V₂T₂ Cross out whatever is held constant and you recover Boyle’s law (PV = const), Charles’s law (V/T = const), or Gay-Lussac’s law (P/T = const).

What temperature actually is

Temperature is average translational kinetic energy per molecule, nothing more. That is the bridge between the microscopic and macroscopic pictures:

K_avg = 32k_BT v_rms = √3k_BT / m = √3RT / M m = mass of one molecule (kg); M = molar mass (kg/mol). Note v_rms depends on √T, so doubling speed takes four times the kelvin temperature.

The consequence students forget

At the same temperature, every gas has the same average kinetic energy per molecule. Heavier molecules therefore move slower. Helium and oxygen in the same room: same K_avg, very different v_rms.

Internal energy of a monatomic ideal gas

U = N · 32k_BT = 32nRT = 32PV The last form is the handy one on a PV diagram: internal energy is proportional to the product PV, so any two states on the same hyperbola have the same U.

Because U depends only on T, the single most useful sentence in this unit is: ΔT = 0 ⇒ ΔU = 0.

Master formula sheet

Everything the unit uses, with the condition each one needs to be legal.

The conduction equation is not on every version of the AP equation sheet but shows up in plenty of courses — check yours.
Idea	Equation	Applies when
Ideal gas law	`PV` = `nRT` = `Nk`_B`T`	Any ideal gas. `T` in kelvin, `P` absolute (not gauge).
Kinetic theory	`K`_avg = ³⁄₂`k`_B`T`	Average translational KE per molecule, any ideal gas.
RMS speed	`v`_rms = √(3`k`_B`T`/`m`)	`m` is the mass of one molecule.
Internal energy	`U` = ³⁄₂`nRT` = ³⁄₂`PV`	Monatomic ideal gas only (He, Ne, Ar).
Linear expansion	Δ`L` = `αL`₀Δ`T`	Solids, modest Δ`T`. `α` in K⁻¹.
Volume expansion	Δ`V` = `βV`₀Δ`T`, `β` ≈ 3`α`	Solids and liquids (not gases — use the gas law).
Sensible heat	`Q` = `mc`Δ`T`	Temperature is changing, no phase change.
Latent heat	`Q` = `mL`	Phase change, temperature constant.
Calorimetry	Σ`Q` = 0	Insulated system, no work done.
Conduction	`P` = `kA`Δ`TL`	Steady state through a slab of area `A`, thickness `L`.
First law	Δ`U` = `Q` + `W`	Always. `W` = work done on the gas.
Work (constant `P`)	`W` = −`P`Δ`V`	Isobaric only. Otherwise use the area under the PV curve.
Work (general)	`W` = −∫ `P` d`V`	Any quasi-static process — the signed area under the PV curve.
Engine efficiency	`e` = \|`W`_net\| / \|`Q`_H\| = 1 − \|`Q`_C\|/\|`Q`_H\|	Any cyclic engine.
Carnot efficiency	`e`_c = 1 − `T`_C/`T`_H	The maximum possible for reservoirs at `T`_C, `T`_H. Kelvin only.
Entropy change	Δ`S` = `Q`/`T`	Reversible heat transfer at (essentially) constant `T`.
Second law	Δ`S`_universe ≥ 0	Always. Equality only for a reversible process.

Constants worth memorizing

Constant	Value
Gas constant `R`	8.31 J/(mol·K)
Boltzmann constant `k`_B	1.38×10⁻²³ J/K
Avogadro’s number `N`_A	6.02×10²³ mol⁻¹
Standard atmosphere	1 atm = 1.013×10⁵ Pa
Kelvin conversion	`T`_K = `T`_°C + 273
Specific heat of water	4186 J/(kg·K)

Unit-wide traps

Celsius in a gas law. PV = nRT needs kelvin. Every time. A ΔT is the same in K and °C, but a bare T is not.
Gauge vs absolute pressure. A tire gauge reading 200 kPa means 301 kPa absolute. Gas laws want absolute.
Heat is not temperature. Heat is energy in transit because of a temperature difference. An object contains internal energy, never “heat.”
Sign of W. This unit uses W = work done on the gas, so compression is positive. See the sign section.
³⁄₂nRT is monatomic-only. Diatomic gases store energy in rotation too. AP problems say “monatomic” when they want that formula — if they do not, avoid it.