Essential Gas Law Equations

Why This Matters

Gas laws form the foundation of thermodynamics in AP Physics 2, connecting microscopic molecular behavior to macroscopic properties you can measure—pressure, volume, and temperature. You're being tested on your ability to predict how gases respond to changing conditions, connect kinetic theory to observable phenomena, and apply the right equation to the right scenario. These concepts appear throughout Unit 9 and connect directly to energy conservation principles that show up across the entire course.

The key insight is that all gas laws derive from the same underlying physics: molecules bouncing around and transferring momentum to container walls. Whether you're analyzing a piston compressing air, a balloon expanding in heat, or molecules diffusing across a membrane, you need to recognize which variables are held constant and which relationship applies. Don't just memorize equations—know what physical principle each one represents and when to deploy it.

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The Master Equation: Ideal Gas Law

The ideal gas law is your starting point for nearly every gas problem. It assumes molecules have negligible volume and no intermolecular forces—a simplification that works remarkably well for most real gases under normal conditions.

Ideal Gas Law

• $PV = nRT$—the fundamental relationship connecting pressure, volume, moles, and temperature through the universal gas constant $R = 8.314 \text{ J/(mol·K)}$
• Temperature must be in Kelvin—this is the most common error on exams; always convert from Celsius by adding 273
• Derived from kinetic theory—this equation emerges from statistical mechanics, linking macroscopic measurements to molecular motion

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Constant-Quantity Laws: Isolating Variables

When one variable stays fixed, the ideal gas law simplifies to specific relationships. These "named" laws are really just special cases of $PV = nRT$ with constraints applied.

Boyle's Law

• $P_1V_1 = P_2V_2$—pressure and volume are inversely proportional when temperature and moles remain constant
• Isothermal process—"iso" means same, "thermal" means temperature; this describes compression or expansion at constant $T$
• Classic application: syringes and pistons—push a plunger in, volume decreases, pressure increases proportionally

Charles's Law

• $\frac{V_1}{T_1} = \frac{V_2}{T_2}$—volume and temperature are directly proportional at constant pressure
• Isobaric process—constant pressure means the gas can expand freely against a movable boundary
• Explains hot air balloons—heating air increases volume, decreasing density, creating buoyancy

Gay-Lussac's Law

• $\frac{P_1}{T_1} = \frac{P_2}{T_2}$—pressure and temperature are directly proportional at constant volume
• Isochoric (isovolumetric) process—rigid containers where gas cannot expand
• Safety implication—why pressurized containers warn against heating; pressure rises with temperature in fixed volumes

Combined Gas Law

• $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$—handles situations where pressure, volume, and temperature all change simultaneously
• Subsumes Boyle's, Charles's, and Gay-Lussac's—set any variable equal on both sides and you recover the individual laws
• Go-to equation for multi-variable problems—when a problem changes two or more state variables, start here

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Quantity and Mixture Laws

These equations address what happens when the amount of gas changes or when multiple gases share a container.

Avogadro's Law

• $\frac{V_1}{n_1} = \frac{V_2}{n_2}$—volume is directly proportional to moles at constant temperature and pressure
• Equal volumes contain equal moles—at STP, one mole of any ideal gas occupies approximately 22.4 L
• Foundation for stoichiometry—connects gas behavior to chemical reaction calculations

Dalton's Law of Partial Pressures

• $P_{\text{total}} = P_1 + P_2 + P_3 + \ldots$—each gas in a mixture contributes independently to total pressure
• Partial pressure—the pressure each gas would exert if it alone occupied the container
• Critical for gas collection over water—subtract water vapor pressure from total pressure to find the pressure of your collected gas

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Kinetic Theory: The Microscopic Connection

These equations bridge the gap between what molecules do and what we measure. Kinetic theory treats gas molecules as tiny elastic spheres in constant random motion.

Kinetic Theory Pressure Equation

• $PV = \frac{1}{3}Nm\bar{v^2}$—relates pressure and volume to the number of molecules ($N$), molecular mass ($m$), and mean square speed
• Pressure comes from collisions—molecules striking container walls transfer momentum, creating the force per unit area we call pressure
• Links to temperature—since $\bar{KE} = \frac{3}{2}k_BT$, temperature is fundamentally a measure of average molecular kinetic energy

Root Mean Square Speed

• $v_{\text{rms}} = \sqrt{\frac{3RT}{M}}$—the characteristic speed of gas molecules, where $M$ is molar mass in kg/mol
• Lighter molecules move faster—at the same temperature, hydrogen molecules travel much faster than oxygen molecules
• Explains diffusion and effusion rates—Graham's Law follows directly from this relationship

Molar Mass from Density

• $M = \frac{dRT}{P}$—rearrangement of the ideal gas law using density ($d = \frac{m}{V}$) instead of moles
• Identifies unknown gases—measure density, temperature, and pressure to calculate molar mass
• Density form of ideal gas law—can also be written as $P = \frac{dRT}{M}$, useful when mass rather than moles is given

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Quick Reference Table

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Self-Check Questions

1. A rigid sealed container of gas is heated. Which specific gas law applies, and why can't you use Charles's Law here?

2. Two gases at the same temperature have different $v_{\text{rms}}$ values. What single property must differ between them, and which gas moves faster?

3. Compare and contrast Boyle's Law and Charles's Law: what variable is held constant in each, and what type of proportionality (direct or inverse) does each describe?

4. If an FRQ gives you the density of an unknown gas along with temperature and pressure, which equation would you use to find its molar mass? Write it out.

5. Using the kinetic theory equation and the ideal gas law, explain why temperature can be interpreted as a measure of average molecular kinetic energy. What would you set equal to begin this derivation?