Ideal Gasses
High-Yield Summary
- Four simple gas laws each hold two of {P, V, n, T} constant: Boyle's (P₁V₁=P₂V₂), Charles's (V₁/T₁=V₂/T₂), Avogadro's (V₁/n₁=V₂/n₂), Gay-Lussac's (P₁/T₁=P₂/T₂).
- Ideal gas law PV = nRT combines all four variables; R = 0.0821 L·atm/(mol·K) or 8.314 J/(mol·K). It's an empirical limiting law, most accurate at low P and high T.
- Combined gas law: P₁V₁/(n₁T₁) = P₂V₂/(n₂T₂), simplifying to P₁V₁/T₁ = P₂V₂/T₂ when moles are constant.
- Molar mass/density form: PM = dRT. Molar volume of any ideal gas at STP (273.15 K, 1 atm) = 22.4 L/mol.
- Dalton's Law: Ptotal = PA + PB + ...; partial pressure PA = XA × Ptotal (mole fraction × total pressure).
Key Terms
- Equation of state
- An equation describing the full physical condition of a gas at a given moment (e.g., PV = nRT); any three of P, V, n, T determine the fourth.
- STP
- Standard temperature and pressure: 273.15 K and 1.00 atm.
- Molar volume
- 22.4 L — the volume occupied by 1 mole of any ideal gas at STP.
- Partial pressure
- The pressure a gas in a mixture would exert if it alone occupied the same container at the same temperature.
- Mole fraction (X)
- Moles of one gas divided by total moles of gas in a mixture.
The Four Simple Gas Laws
| Law (held constant) | Relationship |
|---|---|
| Boyle's Law (constant T, n) | P inversely proportional to V — P₁V₁ = P₂V₂ |
| Charles's Law (constant P, n) | V directly proportional to T (Kelvin) — V₁/T₁ = V₂/T₂ |
| Avogadro's Law (constant T, P) | V directly proportional to n — V₁/n₁ = V₂/n₂ |
| Gay-Lussac's Law (constant V, n) | P directly proportional to T — P₁/T₁ = P₂/T₂ |
Ideal Gas Law
PV = nRT
- P = pressure
- V = volume
- n = moles
- R = universal gas constant — 0.0821 L·atm/(mol·K) or 8.314 J/(mol·K)
- T = absolute temperature (Kelvin)
- Worked example: 5.00 L container, 0.500 mol, 298 K → P = nRT/V = 2.45 atm.
- Best approximation at low pressure and high temperature — real gases deviate otherwise (see Real Gasses).
Combined Gas Law
P₁V₁/(n₁T₁) = P₂V₂/(n₂T₂)
- subscript 1 = initial state
- subscript 2 = final state
- Simplifies to P₁V₁/T₁ = P₂V₂/T₂ whenever moles don't change (n₁ = n₂).
Molar Mass / Density Form
PM = dRT
- M = molar mass
- d = density (mass/volume)
- Derived by substituting n = m/M into PV = nRT and rearranging for density d = m/V.
Dalton's Law of Partial Pressures
Ptotal = PA + PB + PC + ... | PA = XA × Ptotal
- XA = mole fraction of gas A = nA / ntotal
- Worked example: 3.00 mol N₂ + 1.00 mol O₂ at 5.00 atm total → P(O₂) = 1.25 atm, P(N₂) = 3.75 atm.
Must-Know Points
- At equal T and P, liter-to-liter gas ratios can substitute directly for mole-to-mole ratios in stoichiometry (a consequence of Avogadro's Law).
- 1 mole of ANY ideal gas at STP occupies 22.4 L, regardless of identity.
- R has two values depending on units — 0.0821 L·atm/(mol·K) for atm/L problems, 8.314 J/(mol·K) for SI/energy contexts.
Common MCAT Trap
- Always convert temperature to Kelvin before using any gas law — plugging in °C gives wrong answers.
- Don't confuse STP (273.15 K, 1 atm, molar volume 22.4 L) with standard state conditions used for ΔG°f/ΔH°f (298 K, 1 atm or 1 M) — different conventions, different chapters.
- In Dalton's Law, mole fraction must be calculated from TOTAL moles of gas in the mixture, not just the two gases being compared if a third is present.
Quick Recall
A gas at constant temperature and moles has its volume halved. What happens to pressure?
What is the molar volume of any ideal gas at STP?
A mixture is 60% N₂ and 40% O₂ by mole fraction at a total pressure of 2.0 atm. What's the partial pressure of O₂?