Gases

Gas Pressure: Origin and Units

Gas pressure results from the countless collisions of rapidly moving gas particles against the walls of their container; more frequent or more forceful collisions produce higher pressure. Pressure is measured with a barometer (atmospheric pressure, via a column of mercury balanced by air pressure) or a manometer (pressure of a contained gas, comparing mercury column heights). The many pressure units used in chemistry are all interconvertible and must be memorized as exact or precise equivalents for problem-solving.

• •1 atm = 760 mmHg = 760 torr exactly
• •1 atm = 101.325 kPa = 1.01325 bar = 14.7 psi
• •Barometer measures atmospheric pressure; manometer measures a contained gas sample's pressure
• •Higher temperature or more molecules → more frequent/forceful collisions → higher pressure

The Simple Gas Laws

Boyle's law states that at constant temperature and moles, pressure and volume are inversely proportional (P₁V₁ = P₂V₂) — compressing a gas raises its pressure. Charles's law states that at constant pressure and moles, volume and Kelvin temperature are directly proportional (V₁/T₁ = V₂/T₂) — heating a gas at constant pressure makes it expand. Avogadro's law states that at constant temperature and pressure, volume is directly proportional to the number of moles (V₁/n₁ = V₂/n₂), while Gay-Lussac's law states that at constant volume, pressure is directly proportional to Kelvin temperature (P₁/T₁ = P₂/T₂). All of these are special cases of the more general ideal gas law and all absolutely require temperature in Kelvin.

• •Boyle: P₁V₁ = P₂V₂ (inverse P-V relationship, constant n, T)
• •Charles: V₁/T₁ = V₂/T₂ (direct V-T relationship, constant n, P)
• •Avogadro: V₁/n₁ = V₂/n₂ (direct V-n relationship, constant T, P)
• •Gay-Lussac: P₁/T₁ = P₂/T₂ (direct P-T relationship, constant n, V)
• •Kelvin is mandatory in every gas-law calculation involving temperature

The Ideal Gas Law and Combined Gas Law

The ideal gas law PV = nRT unifies all the simple gas laws into a single equation relating pressure, volume, moles, and temperature, using the gas constant R = 0.08206 L·atm/(mol·K) when P is in atm and V in L, or R = 8.314 J/(mol·K) when working in SI/energy units. When comparing the same fixed amount of gas under two different sets of conditions, the combined gas law P₁V₁/T₁ = P₂V₂/T₂ (or including n if moles change: P₁V₁/n₁T₁ = P₂V₂/n₂T₂) is a convenient shortcut derived directly from the ideal gas law. Standard temperature and pressure (STP) is defined as 0°C (273.15 K) and 1 atm, at which one mole of any ideal gas occupies exactly 22.4 L (molar volume).

• •PV = nRT: R = 0.08206 L·atm/mol·K or 8.314 J/mol·K
• •Combined gas law: P₁V₁/n₁T₁ = P₂V₂/n₂T₂
• •STP = 0°C (273.15 K) and 1 atm; molar volume = 22.4 L/mol at STP
• •Always convert temperature to Kelvin before any gas-law calculation

Gas Density and Molar Mass

Because density is mass per volume and the ideal gas law relates n = m/M to P, V, and T, rearranging PV = (m/M)RT gives two very useful forms: density d = PM/RT (density increases with molar mass and pressure, decreases with temperature) and molar mass M = mRT/PV (or equivalently M = dRT/P). These relationships let you determine an unknown gas's molar mass experimentally by measuring the mass of a known volume of gas at known temperature and pressure, which is a classic identification technique.

• •d = PM/RT — gas density depends on molar mass, pressure, and temperature
• •M = mRT/PV = dRT/P — determine unknown molar mass from measured density
• •Denser gases (higher M) sink; used to explain why CO₂ can be poured like a fluid
• •Gas density is much lower than liquid/solid density due to large intermolecular spacing

Dalton's Law and Gases Collected Over Water

Dalton's law of partial pressures states that in a mixture of non-reacting gases, the total pressure equals the sum of the partial pressures of each component: P_total = P₁ + P₂ + P₃ + .... Each gas's partial pressure equals its mole fraction (χ) times the total pressure: P_i = χ_i × P_total, and gases behave independently as if the others were not present. A common lab application is collecting a gas over water, where the gas bubbles up and displaces water, but the collected gas is always mixed with water vapor; the partial pressure of the dry gas is found by subtracting the known vapor pressure of water at that temperature from the total measured pressure.

• •P_total = P₁ + P₂ + P₃ + ... (sum of partial pressures)
• •P_i = χ_i × P_total (mole fraction relationship)
• •Gas collected over water: P_dry gas = P_total − P_H₂O(vapor pressure at that T)
• •Mole fraction of a component = moles of component / total moles

Gas Stoichiometry

Gas stoichiometry problems combine balanced chemical equations with the ideal gas law, allowing conversions between the volume/pressure/temperature of one gaseous reactant or product and the moles (or mass, or volume) of another substance in the reaction. The general strategy is to convert given gas data to moles using PV = nRT (or 22.4 L/mol if at STP), use the mole ratio from the balanced equation to find moles of the target substance, then convert to whatever final units are requested (mass, volume, particles, or back to gas conditions using PV = nRT again).

• •Convert given quantity to moles (via PV=nRT, molar mass, or 22.4 L/mol at STP)
• •Use balanced equation mole ratio to move between substances
• •Convert moles of target back to requested units (volume at given T,P; mass; etc.)
• •Always check whether conditions are STP (use 22.4 L/mol) or non-standard (use PV=nRT)

Kinetic-Molecular Theory (KMT)

KMT models gas particles as point masses in constant, random, straight-line motion, with negligible volume compared to the container, no intermolecular attractions or repulsions between them, and average kinetic energy directly proportional to absolute (Kelvin) temperature — all collisions are perfectly elastic (no energy lost). This model explains each empirical gas law: raising temperature increases average kinetic energy and collision frequency/force, explaining Gay-Lussac's and Charles's laws; compressing volume increases collision frequency per unit area, explaining Boyle's law; and adding more particles at fixed volume/temperature increases collision frequency, explaining Avogadro's law and Dalton's law.

• •Gas particles have negligible volume and no IMFs (in the ideal model)
• •Average KE ∝ absolute temperature (Kelvin) for all gases at the same T
• •Collisions are perfectly elastic — no net energy loss
• •KMT provides the theoretical justification for all the empirical gas laws

Molecular Speeds, Effusion, and Diffusion

The root-mean-square speed u_rms = √(3RT/M) (using R = 8.314 J/mol·K and M in kg/mol) represents a kind of average molecular speed and shows that speed increases with temperature and decreases with molar mass — lighter gases move faster at the same temperature. The Maxwell-Boltzmann distribution is a graph of the fraction of molecules versus speed; increasing temperature broadens and shifts the curve toward higher speeds, while increasing molar mass narrows and shifts the curve toward lower speeds. Effusion is the escape of gas through a tiny pinhole into a vacuum, and diffusion is the gradual mixing of gases due to molecular motion; Graham's law states that the rate of effusion (or diffusion) is inversely proportional to the square root of molar mass, so lighter gases effuse/diffuse faster: rate₁/rate₂ = √(M₂/M₁).

• •u_rms = √(3RT/M); increases with T, decreases with M
• •Maxwell-Boltzmann distribution broadens/shifts right with higher T
• •Effusion: escape through a tiny hole into vacuum; diffusion: gradual mixing of gases
• •Graham's law: rate₁/rate₂ = √(M₂/M₁) — lighter gases effuse/diffuse faster

Real Gas Behavior and the van der Waals Equation

Real gases deviate most from ideal behavior at high pressure (molecules forced close together, so their finite volume becomes significant) and at low temperature (molecules move slowly enough for intermolecular attractions to matter, especially near condensation). The van der Waals equation, (P + an²/V²)(V − nb) = nRT, corrects the ideal gas law for these two effects: the constant 'a' corrects for intermolecular attractions (added to observed pressure, since attractions reduce measured pressure below ideal), and the constant 'b' corrects for the finite volume of gas molecules (subtracted from the container volume, since molecules aren't truly point masses). Larger, more polarizable, or more polar molecules typically have larger 'a' values, while larger molecules have larger 'b' values.

• •Real gases deviate most at high P and low T
• •van der Waals: (P + an²/V²)(V − nb) = nRT
• •'a' accounts for intermolecular attractions; larger a = stronger attractions
• •'b' accounts for finite molecular volume; larger b = larger molecules

A gas occupies 5.00 L at 1.20 atm and 25°C. What is its volume at 2.40 atm and 50°C?

1. 11. Convert temperatures to Kelvin: T₁ = 298 K, T₂ = 323 K.
2. 22. Use combined gas law: P₁V₁/T₁ = P₂V₂/T₂ (n constant).
3. 33. Rearrange for V₂: V₂ = P₁V₁T₂ / (T₁P₂).
4. 44. Substitute: V₂ = (1.20 atm × 5.00 L × 323 K) / (298 K × 2.40 atm).
5. 55. Calculate numerator: 1.20 × 5.00 × 323 = 1938; denominator: 298 × 2.40 = 715.2.
6. 66. V₂ = 1938 / 715.2 = 2.71 L.

V₂ ≈ 2.71 L

Calculate the molar mass of a gas if 2.50 g occupies 1.50 L at 1.00 atm and 300 K.

1. 11. Use ideal gas law rearranged for molar mass: M = mRT/PV.
2. 22. Identify values: m = 2.50 g, R = 0.08206 L·atm/mol·K, T = 300 K, P = 1.00 atm, V = 1.50 L.
3. 33. Substitute: M = (2.50 g × 0.08206 L·atm/mol·K × 300 K) / (1.00 atm × 1.50 L).
4. 44. Calculate numerator: 2.50 × 0.08206 × 300 = 61.545.
5. 55. Divide by denominator: 61.545 / 1.50 = 41.03 g/mol.

M ≈ 41.0 g/mol

A 2.0 L container holds a mixture of N₂ (0.40 mol) and O₂ (0.10 mol) at 25°C. Find the total pressure and the partial pressure of each gas.

1. 11. Total moles = 0.40 + 0.10 = 0.50 mol.
2. 22. Use ideal gas law for total pressure: P_total = nRT/V = (0.50 mol × 0.08206 L·atm/mol·K × 298 K) / 2.0 L.
3. 33. Calculate: 0.50 × 0.08206 × 298 = 12.23; divide by 2.0 = 6.11 atm.
4. 44. Mole fraction of N₂ = 0.40/0.50 = 0.80; mole fraction of O₂ = 0.10/0.50 = 0.20.
5. 55. P(N₂) = 0.80 × 6.11 atm = 4.89 atm; P(O₂) = 0.20 × 6.11 atm = 1.22 atm.

P_total ≈ 6.11 atm; P(N₂) ≈ 4.89 atm; P(O₂) ≈ 1.22 atm

How many liters of H₂ gas at STP are produced when 0.500 mol of Zn reacts completely with excess HCl? Zn(s) + 2HCl(aq) → ZnCl₂(aq) + H₂(g)

1. 11. Mole ratio from balanced equation: 1 mol Zn produces 1 mol H₂.
2. 22. Moles of H₂ produced = 0.500 mol Zn × (1 mol H₂ / 1 mol Zn) = 0.500 mol H₂.
3. 33. At STP, 1 mole of any ideal gas occupies 22.4 L (molar volume).
4. 44. Volume of H₂ = 0.500 mol × 22.4 L/mol.
5. 55. Calculate: 0.500 × 22.4 = 11.2 L.

11.2 L of H₂ gas at STP

Compare the root-mean-square speeds of He (M = 4.00 g/mol) and O₂ (M = 32.0 g/mol) at the same temperature. Which moves faster and by what factor?

1. 11. u_rms = √(3RT/M), so at the same T, the ratio of speeds depends only on molar masses.
2. 22. u_rms(He)/u_rms(O₂) = √(M(O₂)/M(He)).
3. 33. Substitute: √(32.0/4.00) = √8.00.
4. 44. Calculate: √8.00 ≈ 2.83.
5. 55. Since molar mass is in the denominator inside the square root, lighter He moves faster.

He moves about 2.83 times faster than O₂ at the same temperature.

Q1. What is the equivalent of 1 atm in mmHg?

Q2. According to Boyle's law, if the volume of a fixed amount of gas at constant temperature is halved, the pressure will:

Q3. Which gas law relates volume and Kelvin temperature at constant pressure and moles?

Q4. What is the molar volume of an ideal gas at STP?

Q5. A gas sample has a density of 1.96 g/L at STP. What is its approximate molar mass?

Q6. When collecting a gas over water, why must the vapor pressure of water be subtracted from the total pressure?

Q7. According to kinetic-molecular theory, average kinetic energy of gas particles is directly proportional to:

Q8. Which gas would have the highest root-mean-square speed at the same temperature?

Q9. Graham's law states that the rate of effusion of a gas is:

Q10. Real gases deviate most from ideal behavior under which conditions?

Q11. In the van der Waals equation, what does the constant 'b' correct for?

Q12. A mixture contains 2.0 mol N₂ and 3.0 mol O₂ at a total pressure of 5.0 atm. What is the partial pressure of O₂?

Q13. What value of R should be used when working with SI energy units (Joules)?

Q14. Which of the following is an assumption of the kinetic-molecular theory that breaks down under high pressure?

Q15. If temperature increases at constant volume and moles, what happens to the pressure of a gas according to Gay-Lussac's law?