Gases

The Ideal Gas Law

The ideal gas law—PV = nRT—is the single most important equation in gas behavior on the MCAT. P is pressure, almost always expressed in atmospheres (atm) or mmHg (torr); 1 atm = 760 mmHg = 760 torr = 101.325 kPa. V is volume in liters (L). n is the number of moles of gas. R is the universal gas constant, and you must know its common values: R = 0.0821 L·atm/mol·K for calculations involving atm and liters, and R = 8.314 J/mol·K when energy in joules is relevant (thermochemistry, gas expansion work). T is absolute temperature in Kelvin (K = °C + 273). The equation applies to gases that obey the kinetic molecular theory assumptions—point particles with zero molecular volume, no intermolecular forces, and perfectly elastic collisions. Under standard temperature and pressure (STP: 0°C = 273 K, 1 atm), one mole of any ideal gas occupies 22.4 L. The MCAT frequently tests proportional reasoning under changing conditions: if V is constant, P ∝ T (pressure increases with temperature in a rigid container); if T is constant, P ∝ 1/V (Boyle's law); if P is constant, V ∝ T (Charles's law). The combined gas law—P₁V₁/T₁ = P₂V₂/T₂ for a fixed amount of gas—lets you calculate how any two variables change when another is held constant. Always convert to Kelvin before plugging into any gas-law calculation; a °C value in the denominator is a classic MCAT trap. Finally, gas density (ρ = m/V) can be derived from the ideal gas law: ρ = PM/RT, where M is molar mass—this form appears in passages about buoyancy, hot-air balloons, and atmospheric density profiles.

Partial Pressures and Mole Fractions

In a mixture of gases, each component behaves as if it alone occupies the total volume, exerting a partial pressure proportional to its mole fraction. The mole fraction of gas A, denoted X_A, is the number of moles of A divided by the total moles in the mixture: X_A = n_A / n_total. The partial pressure of gas A is then P_A = X_A × P_total. This relationship is critical because the MCAT uses partial pressures to model real respiratory scenarios. At sea level, atmospheric pressure is 760 mmHg; oxygen constitutes 21% of dry air by volume, so its partial pressure PO₂ = 0.21 × 760 mmHg ≈ 160 mmHg. However, in the trachea, air is humidified—water vapor exerts a partial pressure of 47 mmHg at body temperature—so the PO₂ in inspired air reaching the alveoli is actually (760 − 47) × 0.21 ≈ 150 mmHg. In the alveoli, PO₂ drops further to ~100 mmHg because O₂ continuously diffuses into pulmonary capillary blood while CO₂ (PCO₂ ≈ 40 mmHg) diffuses out. Understanding these cascading partial pressures is essential for MCAT passages on altitude sickness, diving physiology (decompression sickness), and oxygen-hemoglobin dissociation curves. The MCAT may also ask: if a gas mixture is collected over water, the water vapor pressure must be subtracted from the total pressure to obtain the pressure of the dry gas alone—a frequent laboratory-scenario trap.

Dalton's Law of Partial Pressures

Dalton's law states that the total pressure exerted by a mixture of non-reacting gases is the sum of the partial pressures of each individual component: P_total = P_A + P_B + P_C + ... . This is a direct consequence of the ideal gas law: each gas in the mixture has the same V and T, so P_total = (n_total)RT/V, and each P_i = n_i RT/V, summing to the same total. The law assumes the gases do not chemically react with one another. An MCAT classic is the collection of a gas over water: when zinc metal reacts with HCl to produce H₂ gas, and the H₂ is collected by water displacement, the measured total pressure in the collection vessel is the sum of P_H₂ + P_H₂O. To find the amount of H₂ produced, you must subtract the vapor pressure of water at the measured temperature (looked up from a table) from the total pressure. Dalton's law also underlies the calculation of alveolar PO₂ using the alveolar gas equation: PAO₂ = FiO₂ × (P_atm − P_H₂O) − (PaCO₂ / R), where R is the respiratory exchange ratio (~0.8). This equation integrates Dalton's law, partial pressure concepts, and metabolic gas exchange—a favorite passage topic.

Kinetic Molecular Theory

Kinetic molecular theory (KMT) provides the microscopic explanation for the macroscopic ideal gas law. Its five core postulates are tested routinely: (1) gases consist of particles (atoms or molecules) in continuous random motion; (2) the volume of the gas particles themselves is negligible compared to the volume of the container; (3) collisions between particles and with container walls are perfectly elastic (no loss of kinetic energy); (4) there are no attractive or repulsive forces between gas particles; (5) the average kinetic energy of gas particles is directly proportional to the absolute temperature and independent of molecular identity: KE_avg = (3/2) k_B T for a monatomic gas, where k_B is Boltzmann's constant (1.38 × 10⁻²³ J/K). From these postulates, the root-mean-square speed of gas particles is v_rms = √(3RT/M), where M is molar mass in kg/mol. This equation yields critical proportional relationships: at a given temperature, lighter molecules move faster than heavier ones (v_rms ∝ 1/√M), and higher temperature increases molecular speed (v_rms ∝ √T). The Maxwell-Boltzmann distribution plots the distribution of molecular speeds: it is asymmetric, skewed to the right (higher speeds have a long tail), and the most probable speed, average speed, and v_rms are three distinct values (v_rms > v_avg > v_mp). At higher temperatures, the distribution flattens and shifts right; at lower temperatures, it sharpens and shifts left. The MCAT may ask you to compare distributions for two gases at the same temperature (heavier gas has a narrower, left-shifted peak) or the same gas at two temperatures.

Graham's Law of Effusion and Diffusion

Graham's law quantifies how the rate of effusion (gas escaping through a tiny pinhole) or diffusion (gas spreading through a space) depends on molar mass. For two gases A and B at the same temperature and pressure: rate_A / rate_B = √(M_B / M_A). Lighter gases effuse faster—hydrogen (M = 2 g/mol) effuses √(32/2) = 4 times faster than oxygen (M = 32 g/mol). The MCAT often reframes this as time for effusion: if rate ∝ 1/time, then t_A / t_B = √(M_A / M_B). A heavier gas takes longer to effuse the same amount. Graham's law follows directly from KMT: at a given temperature, all gases have the same average kinetic energy, so ½ m_A v_A² = ½ m_B v_B², meaning v ∝ 1/√m, and rate ∝ v ∝ 1/√M. The MCAT typically tests Graham's law in two contexts: (1) uranium enrichment by gaseous diffusion—UF₆ containing ²³⁵U (lighter) effuses slightly faster than UF₆ containing ²³⁸U, allowing isotopic separation after many cycles; (2) biological diffusion gradients—O₂ diffuses slightly faster than CO₂ across the alveolar-capillary membrane because M_O₂ (32 g/mol) < M_CO₂ (44 g/mol), giving a rate ratio of √(44/32) ≈ 1.17. In practice, membrane solubility (Henry's law) and concentration gradients dominate gas exchange physiology, but Graham's law explains the underlying physical diffusion component.

Real Gases and the van der Waals Equation

Real gases deviate from ideal behavior under two conditions: high pressure (particles are forced close together, so molecular volume becomes significant relative to container volume) and low temperature (particles slow down, allowing intermolecular attractive forces to pull them together, reducing the effective pressure exerted on walls). The van der Waals equation corrects the ideal gas law with two gas-specific constants: [P + a(n/V)²] × [V − nb] = nRT. The term a(n/V)² adds back the pressure reduction caused by intermolecular attractions (a is larger for polarizable molecules with strong dispersion or dipole forces), and the term nb subtracts the volume occupied by the gas particles themselves (b is larger for bigger molecules). For a given gas, the a and b constants are experimentally determined. To identify which gas deviates most from ideality, ask: does it have strong intermolecular forces (large a)? Is it a large molecule (large b)? Water vapor (H₂O) deviates significantly due to hydrogen bonding (large a), and large hydrocarbons like butane deviate due to both large a (dispersion forces) and large b (molecular volume). Helium approximates ideality best because it is small, nonpolar, and has the weakest intermolecular forces. The MCAT may present a table of van der Waals constants and ask you to rank gases by deviation from ideality, or to predict whether a gas at a given P and T has Z = PV/nRT > 1 (repulsive/deviation dominated by molecular volume at very high P) or Z < 1 (attractive/deviation dominated by intermolecular forces at moderate P).

Biological Gas Applications

Gas behavior is not just a chemistry abstraction—it directly governs respiratory physiology, underwater diving, and clinical gas measurement. Henry's law states that the concentration of a dissolved gas in a liquid is proportional to its partial pressure above the liquid: C = k_H × P_gas, where k_H is the Henry's law constant (unique to each gas-solvent pair and temperature-dependent). At sea level, dissolved O₂ in blood plasma is only ~0.3 mL O₂ per 100 mL blood—far too little to sustain aerobic metabolism—which is why hemoglobin is essential: each gram of Hb binds ~1.34 mL O₂, boosting O₂-carrying capacity to ~20 mL O₂/100 mL blood. CO₂ is ~20× more soluble in blood than O₂ (higher k_H), which is why dissolved CO₂ contributes meaningfully to CO₂ transport (~10% as dissolved gas, ~70% as bicarbonate after carbonic anhydrase conversion, ~20% bound to hemoglobin as carbaminohemoglobin). Decompression sickness (the bends) is a direct application of Henry's law: at depth, high ambient pressure forces more N₂ to dissolve in blood and tissues; if a diver ascends too quickly, N₂ comes out of solution as bubbles, causing joint pain, neurological symptoms, and potentially fatal gas emboli. Hyperbaric oxygen therapy treats CO poisoning and decompression sickness by increasing PO₂ to force O₂ onto hemoglobin and dissolve additional O₂ directly in plasma. The MCAT also tests gas expansion work (w = −P_ext ΔV) in thermochemistry and how lung compliance and airway resistance rely on pressure-volume relationships described by the ideal gas law.

How it works

Every gas problem on the MCAT follows a predictable decision tree. First, identify whether the gas is ideal (point particles, no forces, perfect collisions) or real (high P, low T, strong IMFs). If ideal, write PV = nRT and identify which three variables are known—the fourth is your solve-for. If conditions change and n is fixed, use the combined gas law: P₁V₁/T₁ = P₂V₂/T₂. If the gas is part of a mixture, apply Dalton's law: P_total = ΣP_i, and each P_i = X_i × P_total. If the question involves rates of effusion or diffusion, reach for Graham's law: rate ∝ 1/√M. If the question asks about molecular speed, v_rms = √(3RT/M). If the gas is real, check whether P is high enough or T is low enough to cause deviation and apply van der Waals conceptually—the MCAT rarely requires solving the full cubic equation, but it expects you to know the direction of the a and b corrections. Finally, if the passage involves respiration, diving, or gas solubility, Henry's law bridges partial pressure to dissolved concentration. The exam rewards fluent algebraic manipulation and unit awareness far more than raw memorization. Always check your units: R = 0.0821 L·atm/mol·K forces you to use atm, liters, and Kelvin—a mismatch in any is an instant wrong answer.

How it works

Every gas problem on the MCAT follows a predictable decision tree. First, identify whether the gas is ideal (point particles, no forces, perfect collisions) or real (high P, low T, strong IMFs). If ideal, write PV = nRT and identify which three variables are known—the fourth is your solve-for. If conditions change and n is fixed, use the combined gas law: P₁V₁/T₁ = P₂V₂/T₂. If the gas is part of a mixture, apply Dalton's law: P_total = ΣP_i, and each P_i = X_i × P_total. If the question involves rates of effusion or diffusion, reach for Graham's law: rate ∝ 1/√M. If the question asks about molecular speed, v_rms = √(3RT/M). If the gas is real, check whether P is high enough or T is low enough to cause deviation and apply van der Waals conceptually—the MCAT rarely requires solving the full cubic equation, but it expects you to know the direction of the a and b corrections. Finally, if the passage involves respiration, diving, or gas solubility, Henry's law bridges partial pressure to dissolved concentration. The exam rewards fluent algebraic manipulation and unit awareness far more than raw memorization. Always check your units: R = 0.0821 L·atm/mol·K forces you to use atm, liters, and Kelvin—a mismatch in any is an instant wrong answer.

Comparisons

• C/P (Gas laws): PV = nRT manipulations, combined gas law, STP molar volume (22.4 L/mol), density from PM/RT, unit conversions between atm, mmHg, kPa, and torr.
• C/P (Kinetic theory): v_rms = √(3RT/M), Maxwell-Boltzmann distribution shape, temperature dependence of KE_avg, relationship between molecular speed and molar mass.
• C/P (Real gases): van der Waals equation, Z = PV/nRT deviations, identifying which gases deviate most (strong IMFs, large molecular volume), conditions favoring ideality.
• B/B (Respiratory physiology): Dalton's law for alveolar gas pressures (PO₂, PCO₂), Henry's law for dissolved blood gases, O₂ and CO₂ transport mechanisms, oxyhemoglobin dissociation curve.
• B/B (Diving physiology): Decompression sickness as Henry's law application, nitrogen narcosis, hyperbaric oxygen therapy, gas solubility and bubble formation during ascent.
• P/S (Research methods): Gas collection over water (subtracting water vapor pressure), manometer and barometer measurements, experimental determination of gas constant R.

Common confusions

• Forgetting to convert Celsius to Kelvin. The MCAT will deliberately place a temperature in °C in the problem statement and °C in the denominator of a gas law equation is always wrong—add 273 first. Every gas-law variable that scales with T scales with absolute temperature.
• Using the wrong value of R. R = 0.0821 L·atm/mol·K pairs with pressure in atm and volume in L. R = 8.314 J/mol·K pairs with energy in joules (gas expansion work, kinetic energy). Never mix units across the two R values.
• Confusing STP (0°C, 1 atm; 22.4 L/mol) with standard state (25°C, 1 M, 1 atm—used in thermodynamics). STP is for gas molar volume; standard state is for ΔG°, ΔH°, and electrochemistry. The MCAT loves mixing these up in passage distractors.
• Applying the ideal gas law when the gas is clearly non-ideal. At very high pressure (>100 atm) or very low temperature (near boiling point), real gas behavior dominates. A passage about liquifying gases or deep-sea diving is your cue to consider van der Waals corrections.
• Miscomputing partial pressure when water vapor is present. For gas collected over water: P_dry_gas = P_total − P_H₂O. Similarly, in the lungs: inspired PO₂ = (P_atm − P_H₂O) × FiO₂, not simply P_atm × 0.21. Forgetting the water vapor correction (47 mmHg at 37°C) is a classic trap.
• Treating Graham's law as a linear relationship. Rate ∝ 1/√M means that to double the effusion rate, you need one-fourth the molar mass (the square of the inverse ratio). Students often mistakenly treat rate ∝ 1/M instead of 1/√M.
• Assuming O₂ solubility in blood is sufficient without hemoglobin. Henry's law says dissolved [O₂] = k_H × PO₂. At alveolar PO₂ ≈ 100 mmHg, plasma O₂ is only ~0.3 mL O₂/100 mL—far below the ~5 mL O₂/100 mL that tissues extract. Hemoglobin carries ~98% of O₂; Henry's law alone cannot sustain life.
• Misreading Maxwell-Boltzmann distributions: at a given temperature, heavier gases have a tighter, left-shifted distribution (lower most probable speed) but the area under all curves at the same temperature is equal (same number of particles). The distribution is NOT Gaussian—it's asymmetric with a long right tail.

Quick review

• PV = nRT. R = 0.0821 L·atm/mol·K. STP: 273 K, 1 atm, 22.4 L/mol. Combined: P₁V₁/T₁ = P₂V₂/T₂.
• Dalton's law: P_total = ΣP_i. P_A = X_A × P_total. Subtract water vapor pressure for gas collected over water.
• KMT assumptions: negligible volume, no IMFs, elastic collisions, KE_avg ∝ T, KE_avg = (3/2) k_B T.
• v_rms = √(3RT/M). Lighter gas at same T = faster. Higher T = faster. Maxwell-Boltzmann: asymmetric, right-skewed.
• Graham's law: rate ∝ 1/√M. t_A/t_B = √(M_A/M_B). Effusion is through pinhole; diffusion is spreading through space.
• Real gas deviation: high P (volume matters), low T (IMFs matter). van der Waals: [P + a(n/V)²][V − nb] = nRT.
• Large a = strong IMFs (H₂O, polar gases). Large b = big molecules (hydrocarbons). He most ideal. Z = PV/nRT: Z < 1 attraction dominates; Z > 1 repulsion/volume dominates.
• Henry's law: C = k_H × P_gas. CO₂ ~20× more soluble than O₂ in blood. Hb carries ~98% of O₂; dissolved fraction alone is insufficient.
• Alveolar PO₂ = (P_atm − 47 mmHg) × 0.21 ≈ 150 mmHg, drops to ~100 mmHg in alveoli due to O₂ uptake and CO₂ mixing.
• Decompression sickness: N₂ dissolves at depth (high P) → bubbles on ascent (Henry's law). Hyperbaric O₂ increases dissolved O₂ in plasma.
• Gas expansion work: w = −P_ext ΔV. Units check: if P in atm, ΔV in L, convert to J using 1 L·atm = 101.3 J.
• Water vapor pressure: 47 mmHg at 37°C (body temp). Always subtract from total pressure in humidified air calculations.