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 $V_2<V_1$, so at constant $T$, $P_2>P_1$.

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.