Pressure, Thermal Equilibrium and Ideal Gas Law — AP Physics 2
1. Pressure and the Zeroth Law of Thermal Equilibrium ★★☆☆☆ ⏱ 4 min
The SI unit of pressure is the pascal, where $1\ \text{Pa} = 1\ \text{N}/\text{m}^2$. A critical distinction for problem-solving is between **absolute pressure** and **gauge pressure**: gauge pressure measures pressure relative to atmospheric pressure.
P_{\text{abs}} = P_{\text{gauge}} + P_{\text{atm}}
The zeroth law of thermodynamics formalizes the concept of temperature: If system A is in equilibrium with system B, and system A is in equilibrium with system C, then system B is in equilibrium with system C. This establishes that temperature is the property that determines thermal equilibrium. For all gas law calculations, you must use absolute (Kelvin) temperature, with conversion: $T(\text{K}) = T(^\circ\text{C}) + 273.15$, rounded to 273 for most AP problems.
Exam tip: AP exam questions almost always give temperatures in Celsius for context, but require absolute temperature for all gas law calculations. Convert to Kelvin first, before plugging values into any formula, no exceptions.
2. Combined Gas Law for Closed Systems ★★☆☆☆ ⏱ 4 min
Four separate empirical gas laws describe gas behavior for different fixed conditions. All of these can be combined into a single general relationship, called the combined gas law, that applies to any change of state for a gas sample.
- Boyle's Law (constant temperature): $P \propto 1/V$
- Charles's Law (constant pressure): $V \propto T$
- Gay-Lussac's Law (constant volume): $P \propto T$
- Avogadro's Law (constant P, T): $V \propto n$
\frac{P_1V_1}{n_1T_1} = \frac{P_2V_2}{n_2T_2}
For a closed system (no gas added or removed, so $n_1 = n_2$), this simplifies to the commonly used form:
\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}
This formula is ideal for problems where you know all but one variable for an initial and final state of a gas sample. Intuition: reducing volume increases collision frequency and pressure; increasing temperature increases molecular speed, so volume expands at constant pressure.
Exam tip: When using the combined gas law, pressure and volume units only need to be consistent across initial and final states. Temperature must always be in Kelvin, no exceptions, even if units cancel out.
3. The Ideal Gas Law (Two Forms) ★★★☆☆ ⏱ 4 min
The combined gas law tells us that $\frac{PV}{nT}$ is a universal constant for all ideal gases, called the universal gas constant $R$. This gives the most common molar form of the ideal gas law, the equation of state for an ideal gas.
PV = nRT
Where $n$ is the number of moles of gas, and $R$ has two common values depending on units: $R = 8.314\ \text{J}/(\text{mol·K})$ for SI units (pressure in Pa, volume in m³), and $R = 0.0821\ \text{L·atm}/(\text{mol·K})$ for pressure in atm and volume in liters. A second form, used for counting individual molecules in kinetic theory problems, replaces moles with number of molecules $N$.
PV = Nk_B T
An ideal gas is defined as a gas where molecular volume is negligible, there are no intermolecular forces, and all collisions are elastic. Real gases follow this law closely at low pressure and high temperature, the standard assumption for all AP Physics 2 gas problems.
Exam tip: Always match the value of R to your pressure and volume units. Using 8.314 with liters and atm will give a pressure off by three orders of magnitude, a common mistake that costs free-response points.
4. Concept Check ★★☆☆☆ ⏱ 2 min
Common Pitfalls
Why: Most problems give temperatures in Celsius for context, and students forget gas laws depend on absolute temperature.
Why: Most practical gauges measure gauge pressure, so students forget to add atmospheric pressure to get the absolute pressure required by the law.
Why: Students memorize R but forget its value depends on the units of P and V.
Why: Students use the simplified version by default even when gas is added or removed from the system.
Why: Students rush to plug into the formula without checking proportionality.
Why: Students assume the ideal gas law applies to all gases in all problems.