Gas Laws

When you compress a gas into a smaller space, the pressure goes up. Pull back on the syringe plunger and the pressure drops. This inverse relationship between pressure and volume at constant temperature is Boyle's law.

Mathematically, for a fixed amount of gas at constant temperature:

If you double the pressure, the volume halves. If you triple the volume, the pressure drops to one-third. The product P × V stays constant as long as temperature and moles don't change.

Boyle's law explains everyday phenomena: your lungs expand (volume up, pressure down) to draw air in during inhalation, then contract (volume down, pressure up) to push air out during exhalation. Scuba divers rely on this relationship when managing buoyancy at different depths.

Heat a balloon and it expands. Cool it and it shrinks. Charles's law captures this direct proportionality between volume and temperature for a gas at constant pressure.

The critical detail: temperatures must be in kelvin. The kelvin scale starts at absolute zero (0 K = −273.15 °C), the theoretical temperature at which gas volume would reach zero. Using Celsius in gas law calculations produces incorrect results because the Celsius scale doesn't start at a true zero.

Charles's law means that if you double the kelvin temperature of a gas while holding pressure constant, the volume doubles. Hot air balloons work because heating the air inside the balloon increases its volume, making the balloon less dense than the surrounding cooler air.

Real-world gas problems rarely involve changing only one variable. When both pressure and temperature change simultaneously, the combined gas law handles the situation by merging Boyle's and Charles's laws into a single equation:

This equation works for any situation where the amount of gas (moles) stays constant but pressure, volume, and temperature all change. To use it:

1. Convert all temperatures to kelvin
2. Identify the known values (P₁, V₁, T₁, and two of P₂, V₂, T₂)
3. Solve algebraically for the unknown

The combined gas law is especially useful when comparing gas samples under different sets of conditions, such as predicting the volume of a gas at standard temperature and pressure (STP: 273.15 K and 1 atm) from measurements taken at lab conditions.

The ideal gas law is the master equation that unifies all the individual gas laws into one relationship:

Here, P is pressure, V is volume, n is the number of moles, T is temperature in kelvin, and R is the universal gas constant. The value of R depends on the pressure units you use:

• R = 0.08206 L·atm·mol⁻¹·K⁻¹ (when P is in atm and V in liters)
• R = 8.314 J·mol⁻¹·K⁻¹ (when working with energy units)

The ideal gas law assumes that gas molecules have no volume and exert no attractive forces on each other. Real gases approximate this behavior well at low pressures and high temperatures. At STP, one mole of any ideal gas occupies about 22.4 L — the standard molar volume.

This equation can solve for any one of the four variables (P, V, n, or T) when the other three are known, making it the most versatile tool in gas-phase calculations.

The ideal gas law can be rearranged to connect density directly to molar mass. Starting from PV = nRT and substituting n = m/M (mass divided by molar mass), we get:

where d is density (g/L), P is pressure, M is molar mass (g/mol), R is the gas constant, and T is temperature in kelvin.

This equation works in two important directions:

• Known gas → density: If you know what the gas is, plug in its molar mass to find the density at any given T and P.
• Unknown gas → molar mass: Measure the density of a gas experimentally, then solve for M to help identify the substance.

At STP, denser gases have higher molar masses. For example, CO₂ (M = 44 g/mol) is denser than N₂ (M = 28 g/mol), which is why CO₂ accumulates in low-lying areas.

In a gas mixture, each gas behaves independently and contributes its own pressure as if the other gases weren't there. Dalton's law states that the total pressure equals the sum of all the individual (partial) pressures:

The partial pressure of any component is related to its mole fraction (X), the fraction of total moles that the component represents:

A practical application arises when collecting gases over water. The collected gas is always mixed with water vapor, so the measured total pressure includes the vapor pressure of water. To find the pressure of the dry gas alone, subtract the water vapor pressure (a temperature-dependent value found in reference tables) from the total pressure.

Gas stoichiometry extends the mole-based calculations you already know to gas-phase reactions, using the ideal gas law as the bridge between moles and measurable quantities (P, V, T).

The general workflow:

1. Write and balance the chemical equation
2. Convert given gas data (P, V, T) to moles using PV = nRT
3. Use stoichiometric ratios from the balanced equation to find moles of the desired substance
4. Convert moles back to the requested quantity (volume, mass, etc.)

A useful shortcut applies when gases are compared at the same temperature and pressure: volume ratios equal mole ratios, thanks to Avogadro's law. So if the balanced equation shows a 1:3 ratio of N₂ to H₂, then 1 L of N₂ reacts with 3 L of H₂ (at the same T and P).

At STP, the molar volume of 22.4 L/mol provides another convenient conversion factor for quick calculations.

The kinetic molecular theory (KMT) provides the microscopic explanation for why the gas laws work. It is built on five postulates:

1. Gas molecules are in continuous, random, straight-line motion
2. Molecules are negligibly small compared to the distances between them
3. Pressure results from molecular collisions with container walls
4. Molecules exert no attractive or repulsive forces on each other (collisions are elastic)
5. Average kinetic energy is directly proportional to kelvin temperature

These postulates explain each gas law at the molecular level:

• Boyle's law: Compressing a gas means molecules hit the walls more often → higher pressure
• Charles's law: Hotter molecules move faster and need more volume to maintain the same collision rate → volume increases
• Dalton's law: Molecules of different gases act independently, each contributing its own wall collisions → pressures add up
• Avogadro's law: More molecules at constant T and P require proportionally more volume to keep collision frequency per unit area constant

Real gases deviate from ideal behavior when the assumptions of KMT break down — specifically at high pressures (molecules are close together and their volume matters) and low temperatures (molecules move slowly enough for intermolecular attractions to take effect).

The compressibility factor Z = PV/(nRT) quantifies this deviation. For an ideal gas, Z = 1. When Z < 1, attractive forces dominate and the gas is more compressible than expected. When Z > 1, molecular volume dominates and the gas is less compressible.

The van der Waals equation corrects the ideal gas law with two terms:

The constant a corrects for intermolecular attractions (larger for polar molecules like H₂O) and b corrects for molecular volume (larger for bigger molecules). At low pressures and high temperatures, both corrections become negligible and the van der Waals equation reduces to PV = nRT.

Effusion is the escape of gas molecules through a tiny opening into a vacuum. Diffusion is the broader spreading of gas molecules through space due to random motion. Both rates depend on molecular mass, and both are described by Graham's law.

Lighter molecules move faster and effuse more quickly. Hydrogen (M = 2 g/mol) effuses four times faster than oxygen (M = 32 g/mol) because √(32/2) = 4.

Graham's law has powerful applications:

• Identifying unknown gases: Compare effusion rates to calculate the unknown molar mass
• Isotope separation: Uranium enrichment historically used gaseous diffusion of UF₆ to separate lighter ²³⁵UF₆ from heavier ²³⁸UF₆

Since effusion rate is inversely proportional to the square root of molar mass, time of effusion is directly proportional to √M — heavier gases take longer to escape through the same opening.

Gas molecules at any temperature have a range of speeds described by the Maxwell-Boltzmann distribution. The root mean square (rms) speed is the most useful average because it connects directly to kinetic energy:

where R = 8.314 J·mol⁻¹·K⁻¹, T is temperature in kelvin, and M is molar mass in kg/mol (not g/mol — watch the unit conversion!).

Key relationships:

• Raising the temperature increases u_rms and shifts the speed distribution toward higher speeds
• Lighter gases have higher u_rms at the same temperature (He moves faster than Xe)
• Average kinetic energy per mole is KE_avg = (3/2)RT — it depends only on temperature, not on the identity of the gas

For example, N₂ at 30 °C (303 K) has u_rms ≈ 519 m/s — faster than the speed of sound. Despite these enormous speeds, gas molecules in a room don't travel in straight lines because of billions of collisions per second with other molecules.

Choosing the right gas law is the first and most important step:

1. Identify what changes and what stays constant. If amount and temperature are fixed, use Boyle’s law. If amount and pressure are fixed, use Charles’s law.
2. If all four variables (P, V, n, T) are involved, use the ideal gas law PV = nRT.
3. If a gas mixture: use Dalton’s law to find partial pressures, then apply the ideal gas law to individual components if needed.
4. For stoichiometry problems involving gases: convert between moles and gas volume using PV = nRT (or molar volume at STP as a shortcut).
5. Always convert temperature to Kelvin before substituting into any gas law equation.

Common mistakes: using Celsius instead of Kelvin (this is the single most common gas-law error), choosing the wrong value of R for the pressure units given, applying the ideal gas law at very high pressures or very low temperatures where real-gas deviations are significant, using molar volume (22.4 L/mol) at conditions other than STP, and confusing total pressure with partial pressure in mixture problems.