Key Gas Law Equations to Know for AP Physics 2 (2025) (AP)

What You Need to Know

Gas-law questions in AP Physics 2 are mostly about connecting macroscopic variables—pressure P, volume V, temperature T, amount of gas n (moles) or N (molecules)—and using those relationships to predict what changes when the gas is heated, compressed, mixed, or moved through a thermodynamic process.

The “big idea” you lean on repeatedly:

• Ideal Gas Model (works well for low-density gases):
  PV = nRT = Nk_B T

If you know when you can treat n as constant (sealed container) vs changing (adding/removing gas), and you keep units consistent (especially **Kelvin** for T), you’ll be able to handle most AP-style gas-law setups.

Critical reminder: Temperature in gas laws is always absolute temperature: T_{K} = T_{\circ C} + 273.15.

Step-by-Step Breakdown

A. Choosing the right gas-law equation (fast decision tree)

1. List what’s given and what changes: identify P_1, V_1, T_1, n_1 and P_2, V_2, T_2, n_2.
2. Decide if the amount of gas is constant:
   • Sealed container (no leaks, no gas added): n constant.
   • Open system, adding gas, chemical reaction producing gas: n changes.
3. Pick the simplest law:
   • If T constant (isothermal): use Boyle’s: P_1 V_1 = P_2 V_2.
   • If P constant (isobaric): use Charles’s: \frac{V_1}{T_1} = \frac{V_2}{T_2}.
   • If V constant (isochoric): use Gay-Lussac’s: \frac{P_1}{T_1} = \frac{P_2}{T_2}.
   • If n constant but multiple variables change: use combined gas law:
     \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}
   • If n not constant (or you want a one-step universal setup): use ideal gas law:
     PV = nRT.
4. Convert units early:
   • T to Kelvin.
   • P to Pa if you’re using R = 8.314\,\text{J/(mol·K)}.
   • V to \text{m}^3 for SI.
5. Solve algebraically before plugging numbers (reduces mistakes).

B. Partial pressures (mixtures) procedure

1. Determine which law applies:
   • For mixtures of ideal gases: Dalton’s law: P_{\text{tot}} = \sum_i P_i.
2. If you know moles, use mole fraction:
   • x_i = \frac{n_i}{n_{\text{tot}}} and P_i = x_i P_{\text{tot}}.
3. If gas is collected over water (common lab context):
   • P_{\text{gas}} = P_{\text{tot}} - P_{\text{H}_2\text{O}}.

Quick worked mini-example (method in action)

A sealed syringe: V_1 = 30\,\text{mL}, P_1 = 1.0\,\text{atm}, compressed to V_2 = 10\,\text{mL} at constant T. Find P_2.

• Constant T and constant n → Boyle’s law: P_1V_1=P_2V_2.
• Solve: P_2 = P_1\frac{V_1}{V_2} = (1.0\,\text{atm})\frac{30}{10} = 3.0\,\text{atm}.

Key Formulas, Rules & Facts

Core gas laws (macroscopic relationships)

Constants + unit anchors (know these cold)

Gas mixtures (Dalton’s law)

Kinetic theory connections (often tested conceptually + quantitatively)

Internal energy for ideal gases (ties gas laws to thermodynamics)

Common “process” equations that pair with ideal gas law

These show up when a problem describes a thermodynamic path (even if it calls it “gas law” reasoning).

If you’re not explicitly told the process (isothermal, etc.), don’t assume it—use what’s stated about what is held constant.

Examples & Applications

Example 1: Combined gas law (sealed sample)

A sealed can of air: P_1=2.0\times 10^5\,\text{Pa}, T_1=300\,\text{K}. It’s heated to T_2=450\,\text{K} with constant volume. Find P_2.

• Constant V and n → \frac{P}{T}=\text{const}.
• P_2 = P_1\frac{T_2}{T_1} = (2.0\times 10^5)\frac{450}{300} = 3.0\times 10^5\,\text{Pa}.

AP-style insight: at fixed V, pressure scales linearly with absolute temperature.

Example 2: Ideal gas law to find moles (units trap)

A scuba tank has V=12\,\text{L}, P=2.0\times 10^7\,\text{Pa}, T=300\,\text{K}. How many moles of air? (Treat as ideal.)

• Convert volume: V = 12\times 10^{-3}\,\text{m}^3.
• Use PV=nRT:
  n=\frac{PV}{RT}=\frac{(2.0\times 10^7)(12\times 10^{-3})}{(8.314)(300)}\approx 96\,\text{mol}.

Exam angle: The hard part is usually not algebra—it’s consistent SI units.

Example 3: Dalton’s law + mole fraction

A container at T and V holds n_{\text{He}}=1.0\,\text{mol} and n_{\text{Ne}}=3.0\,\text{mol}. Total pressure is P_{\text{tot}}=400\,\text{kPa}. Find P_{\text{He}}.

• Mole fraction: x_{\text{He}}=\frac{1.0}{1.0+3.0}=0.25.
• Partial pressure: P_{\text{He}}=x_{\text{He}}P_{\text{tot}}=(0.25)(400\,\text{kPa})=100\,\text{kPa}.

AP-style insight: partial pressures depend on mole fractions, not molar masses.

Example 4: RMS speed comparison (temperature + molar mass)

At the same T, compare v_{\text{rms}} for helium (molar mass M_{\text{He}}=0.004\,\text{kg/mol}) and nitrogen (M_{\text{N}_2}=0.028\,\text{kg/mol}).

• v_{\text{rms}}\propto \frac{1}{\sqrt{M}} at fixed T.
• Ratio:
  \frac{v_{\text{rms,He}}}{v_{\text{rms,N}_2}}=\sqrt{\frac{M_{\text{N}_2}}{M_{\text{He}}}}=\sqrt{\frac{0.028}{0.004}}=\sqrt{7}\approx 2.65.

Common exam prompt: “Which gas has higher typical molecular speed at the same temperature?” (Lighter molar mass → faster.)

Common Mistakes & Traps

1. Forgetting Kelvin (using T_{\circ C} directly)
   What goes wrong: You’ll predict the wrong proportional changes (sometimes even negative temperatures).
   Fix: Always convert with T_K = T_{\circ C}+273.15.

2. Mixing unit systems for R
   What goes wrong: Using R=8.314 with P in atm and V in L gives nonsense.
   Fix: Either go full SI (Pa, \text{m}^3, K, R=8.314) or full atm·L (atm, L, K, R=0.08206).

3. Using the combined gas law when n changes
   What goes wrong: \frac{PV}{T} is only constant if n is constant.
   Fix: If gas can enter/leave or moles change, use PV=nRT with explicit n.

4. Assuming “constant pressure” just because the container is open
   What goes wrong: In an open container, the gas can exchange with the environment, but the details matter; pressure may be atmospheric, but the amount of gas may change.
   Fix: Write what you know: if truly open to atmosphere, you can often take P\approx P_{\text{atm}}, but then n is not fixed.

5. Confusing partial pressure with “fraction of volume” in non-identical conditions
   What goes wrong: P_i=x_iP_{\text{tot}} assumes a well-mixed ideal gas at common T and V.
   Fix: Only use mole fraction relations when the mixture shares the same container (same T, V).

6. Using molar mass in \text{g/mol} inside v_{\text{rms}}
   What goes wrong: A factor of \sqrt{1000} error in speed.
   Fix: In v_{\text{rms}}=\sqrt{\frac{3RT}{M}}, M must be in \text{kg/mol}.

7. Thinking “higher pressure means higher temperature” (without constraints)
   What goes wrong: Pressure can increase due to decreased volume at constant temperature (Boyle).
   Fix: Always state which variables are held constant before inferring relationships.

8. Sign errors and conceptual slips with “compression” and “expansion”
   What goes wrong: You might invert ratios like \frac{V_1}{V_2}.
   Fix: Do a quick sanity check: compressing means V_2

Memory Aids & Quick Tricks

Quick Review Checklist

• You can write and use PV=nRT and know what each symbol means.
• You automatically convert to Kelvin: T_K=T_{\circ C}+273.15.
• You keep units consistent with your chosen R (SI vs atm·L).
• You know when \frac{PV}{T} is constant: only when n is constant.
• You can recognize special cases:
  • Isothermal: P_1V_1=P_2V_2
  • Isochoric: \frac{P_1}{T_1}=\frac{P_2}{T_2}
  • Isobaric: \frac{V_1}{T_1}=\frac{V_2}{T_2}
• You can do mixture problems with Dalton’s law: P_{\text{tot}}=\sum P_i and P_i=x_iP_{\text{tot}}.
• You can connect microscopic and macroscopic ideas:
  • \langle K \rangle = \frac{3}{2}k_BT
  • v_{\text{rms}}=\sqrt{\frac{3RT}{M}} (with M in \text{kg/mol})
• You do a quick sanity check: compress → P up (if T constant), heat at constant V → P up.

You’ve got this—gas laws are mostly pattern recognition plus clean units.