Ideal Gasses

Four variables describe the physical state of any gas: pressure, temperature, volume, and number of moles. The relationships between them were first worked out one at a time — as Boyle's, Charles's, Avogadro's, and Gay-Lussac's laws — before being combined into a single equation, the ideal gas law. This article covers the four simple gas laws, how they combine into PV = nRT, the combined gas law for a gas changing states, the molar-mass/density form of the ideal gas law, gas stoichiometry, and Dalton's Law of partial pressures.

Key Takeaways

• Four variables describe a gas's state: pressure, temperature, volume, and moles. Boyle's Law (P₁V₁ = P₂V₂), Charles's Law (V₁/T₁ = V₂/T₂), Avogadro's Law (V₁/n₁ = V₂/n₂), and Gay-Lussac's Law (P₁/T₁ = P₂/T₂) each relate two of these variables while holding the other two constant.

• The ideal gas law, PV = nRT, combines all four variables using the universal gas constant R (0.0821 L·atm/(mol·K), or 8.314 J/(mol·K) in SI units). It's an empirical, limiting-law equation of state — most gases obey it closely enough below 1 atm to treat as ideal.

• The combined gas law, P₁V₁/(n₁T₁) = P₂V₂/(n₂T₂), relates a gas's initial and final states; when moles don't change it simplifies to P₁V₁/T₁ = P₂V₂/T₂.

• The molar-mass/density form of the ideal gas law is PM = dRT.

• Avogadro's Law enables liter-to-liter stoichiometric ratios at equal T/P, and gives the molar volume at STP (273.15 K, 1.00 atm) as 22.4 L/mol.

• Dalton's Law: total pressure equals the sum of partial pressures (Ptotal = PA + PB + ...); a gas's partial pressure equals its mole fraction times the total pressure (PA = XA × Ptotal).

The Four Simple Gas Laws

Each of these laws holds two of the four state variables constant and relates the other two:

• Boyle's Law: at constant temperature and moles, pressure is inversely proportional to volume. Reducing a gas's volume increases its pressure, and vice versa. On a pressure-vs-volume graph this traces a hyperbolic curve. Mathematically: P₁V₁ = P₂V₂.

• Charles's Law: at constant pressure and moles, volume is directly proportional to absolute temperature. As temperature (in Kelvin) increases, volume increases in the same proportion, producing a straight line on a volume-vs-temperature plot. Mathematically: V₁/T₁ = V₂/T₂. Charles's own research is also the basis of the Kelvin scale — it showed that −273.15°C corresponds to 0 K.

• Avogadro's Law: at constant temperature and pressure, volume is directly proportional to the number of moles of gas. Mathematically: V₁/n₁ = V₂/n₂.

• Gay-Lussac's Law: at constant volume and moles, pressure is directly proportional to absolute temperature. Mathematically: P₁/T₁ = P₂/T₂.

The Ideal Gas Law

Boyle's, Charles's, and Avogadro's Laws can be combined into a single equation relating all four state variables at once — the ideal gas law:

PV = nRT

where P is pressure, V is volume, n is the number of moles, T is absolute temperature, and R is the universal gas constant, the proportionality constant that ties the three simple laws together. R has two commonly used numeric values, depending on the units in play:

• R = 0.0821 L·atm/(mol·K) — used when pressure is in atmospheres and volume is in liters (the most common form for PV = nRT problems).

• R = 8.314 J/(mol·K) — used in SI-unit or energy contexts.

Because the ideal gas law is an equation of state — it describes the full physical condition of a gas at a given moment — knowing any three of {P, V, n, T} is enough to solve for the fourth.

The ideal gas law is an empirical equation: it's based on experimental measurements of gas behavior, not derived from first principles. A gas that obeys it is said to behave ideally. It's best understood as a limiting law — it describes the behavior real gases approach at low pressure and high temperature, and an "ideal gas" is a hypothetical substance no real gas perfectly matches. Even so, most real gases obey the ideal gas law closely enough below 1 atm that assuming ideal behavior introduces only minimal error — which is why, unless a problem states otherwise, ideal behavior is the standard assumption.

Worked example — a 5.00 L rigid container holds 0.500 mol of an ideal gas at 25°C (298 K). What's the pressure?

• Rearranging PV = nRT for pressure: P = nRT/V

• P = (0.500 mol)(0.0821 L·atm/(mol·K))(298 K) / (5.00 L)

• P = (0.500 × 0.0821 × 298) / 5.00 = 12.23 / 5.00 = 2.45 atm

The Combined Gas Law

Suppose a gas moves from an initial state (P₁, V₁, n₁, T₁) to a final state (P₂, V₂, n₂, T₂). Because the ideal gas law holds at both states, PV/(nT) equals the same constant R in each case — so the two states can be set equal to each other:

P₁V₁ / (n₁T₁) = P₂V₂ / (n₂T₂)

This is the general form of the combined gas law, useful for any problem involving a gas undergoing a change in conditions where you need to solve for an unknown initial or final variable. Many textbook treatments present a special case that holds moles constant (n₁ = n₂), which cancels n out of both sides and simplifies to P₁V₁/T₁ = P₂V₂/T₂ — that version applies whenever the amount of gas itself doesn't change between the two states.

Molar Mass and Density Form of the Ideal Gas Law

The ideal gas law can also be rewritten in terms of a gas's molar mass and density. Starting from molar mass M = mass/moles, moles can be rewritten as n = m/M (where m is mass). Substituting into PV = nRT gives PV = (m/M)RT. Rearranging so that mass over volume — density, d = m/V — sits on one side produces:

PM = dRT

This form is useful whenever a problem gives (or asks for) a gas's density or molar mass directly, rather than its number of moles.

Gas Stoichiometry

Avogadro's Law — equal volumes of gas at the same temperature and pressure contain equal numbers of moles — has two important consequences for stoichiometry problems:

• When all gases in a reaction are measured at the same temperature and pressure, liter-to-liter ratios can be used directly in place of mole-to-mole ratios.

• The molar volume of any gas at STP (standard temperature and pressure — 273.15 K and 1.00 atm) is 22.4 L. That is, one mole of any ideal gas at STP occupies 22.4 liters, regardless of its identity.

Both shortcuts sit alongside the mole-to-mole stoichiometric ratios covered elsewhere in this course, giving several equivalent ways to work through the same gas stoichiometry problem.

Dalton's Law of Partial Pressures

When two or more gases that don't chemically react with each other share a container, each gas behaves independently — as if it were the only gas present — because gas molecules are spread far enough apart that they don't significantly affect one another. The pressure exerted by each individual gas is called its partial pressure: the pressure that gas would exert if it alone occupied the same container at the same temperature.

In 1801, John Dalton showed that the total pressure of a gas mixture is the sum of the partial pressures of its components — Dalton's Law of partial pressures:

Ptotal = PA + PB + PC + ...

A gas's partial pressure is also related to its mole fraction — the moles of that gas divided by the total moles of gas in the mixture (XA = nA / ntotal) — by:

PA = XA × Ptotal

Worked example — a container holds 3.00 mol N₂ and 1.00 mol O₂ at a total pressure of 5.00 atm. What's the partial pressure of each gas?

• Total moles = 3.00 + 1.00 = 4.00 mol

• Mole fraction of O₂: X(O₂) = 1.00/4.00 = 0.250 → P(O₂) = 0.250 × 5.00 atm = 1.25 atm

• Mole fraction of N₂: X(N₂) = 3.00/4.00 = 0.750 → P(N₂) = 0.750 × 5.00 atm = 3.75 atm

• Check: 1.25 atm + 3.75 atm = 5.00 atm, matching the given total pressure.

Common MCAT Mistakes

• Forgetting to convert temperature to Kelvin. PV = nRT requires absolute temperature. Plugging in Celsius directly gives a wrong answer, sometimes even a negative or zero pressure/volume.

• Using the wrong value of R. R = 0.0821 L·atm/(mol·K) only works when pressure is in atm and volume in liters; mixing that R with SI units (Pa, m³) gives a nonsense answer. Match R's units to the problem's units.

• Applying the simplified P₁V₁/T₁ = P₂V₂/T₂ form when moles actually change. That shortcut only holds when n₁ = n₂. If gas is added, removed, or reacts, the full combined gas law — with n on both sides — is required.

• Confusing mole fraction with partial pressure directly. Mole fraction (XA) is a ratio between 0 and 1; partial pressure (PA) is XA multiplied by the total pressure. Reporting the mole fraction as if it were the partial pressure skips a step.

MCAT-Style Concept Check

Question: A rigid 2.00 L container holds an ideal gas at 1.00 atm and 273 K. If the temperature is raised to 546 K at constant volume and constant moles, what is the new pressure?

• A) 0.50 atm

• B) 1.00 atm

• C) 2.00 atm

• D) 4.00 atm

Answer: C

Explanation: At constant volume and moles, this is Gay-Lussac's Law: P₁/T₁ = P₂/T₂. Solving for P₂: P₂ = P₁ × (T₂/T₁) = 1.00 atm × (546 K/273 K) = 1.00 atm × 2 = 2.00 atm. Doubling absolute temperature at constant volume doubles the pressure.

FAQ

What is the ideal gas law?

The ideal gas law is PV = nRT, where P is pressure, V is volume, n is moles, T is absolute temperature, and R is the universal gas constant. It combines Boyle's, Charles's, and Avogadro's Laws into a single equation relating all four state variables of a gas.

What value of R should I use?

Use R = 0.0821 L·atm/(mol·K) when pressure is in atmospheres and volume is in liters. Use R = 8.314 J/(mol·K) when working in SI units or with energy quantities.

What is STP, and why does molar volume matter at STP?

STP (standard temperature and pressure) is 273.15 K and 1.00 atm. At STP, one mole of any ideal gas occupies 22.4 L — this molar volume lets you convert directly between a gas's volume and its number of moles without needing the full ideal gas law.

How is partial pressure related to mole fraction?

A gas's partial pressure equals its mole fraction (its moles divided by total moles in the mixture) multiplied by the total pressure: PA = XA × Ptotal. Summing all partial pressures in a mixture gives the total pressure, per Dalton's Law.