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General Chemistry I · Gases

Combined and Ideal Gas Laws

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On this page 7 sections
  1. In 30 seconds
  2. Why this matters
  3. The college version
  4. Eli explains
  5. Key takeaway
  6. Study tools
  7. Sources & references

In 30 seconds

The combined gas law (P₁V₁/T₁ = P₂V₂/T₂) relates pressure, volume, and temperature for a fixed amount of gas. The ideal gas law (PV = nRT) goes one step further and also includes the amount of gas, tying all four quantities together through the gas constant R.

Why this matters

The ideal gas law is the workhorse of gas calculations: it links the amount of a gas to its measurable pressure, volume, and temperature. It powers gas stoichiometry, balloon and airbag sizing, respiratory-gas calculations, and the determination of molar mass of unknown vapors in the lab.

The college version

Key Ideas

  • Combined gas law merges Boyle's, Charles's, and Gay-Lussac's laws when n is constant.
  • Ideal gas law adds the amount: PV = nRT.
  • R has two common values: 0.08206 L·atm/(mol·K) and 8.314 J/(mol·K) — pick the one that matches your pressure units.
  • An ideal gas is a hypothetical gas of point particles with no attractions; real gases approximate it at low pressure and high temperature.
  • Density and molar mass follow by rearranging: d = PM/RT and M = (mass)RT/(PV).

Equations and Variables

  • Combined gas law: P₁V₁/T₁ = P₂V₂/T₂ — P = pressure, V = volume, T = absolute temperature (K); n constant.
  • Ideal gas law: PV = nRT — n = amount (mol), R = gas constant, T = kelvin temperature.
  • R = 0.08206 L·atm/(mol·K) when P is in atm and V in L; R = 8.314 J/(mol·K) when using SI units.
  • Density: d = m/V = PM/RT (M = molar mass, g/mol).
  • Molar mass: M = mRT/(PV) (m = mass in grams).

How It Works

  1. The four simple laws each hold two variables fixed and let the other two change. The combined gas law holds only n fixed and lets P, V, and T all change together.
  2. Solve any combined-law problem by isolating the unknown: V₂ = P₁V₁T₂/(T₁P₂).
  3. The ideal gas law adds the amount (n), so it can also answer "how much gas" questions.
  4. For density, note n = m/M, so PV = (m/M)RT → m/V = PM/RT.
  5. For molar mass, solve the same equation for M: M = mRT/(PV).

Worked Example

Combined gas law: A gas occupies 3.0 L at 1.2 atm and 300 K. What is its volume at 2.0 atm and 350 K? V₂ = P₁V₁T₂/(T₁P₂) = (1.2 atm)(3.0 L)(350 K) / [(300 K)(2.0 atm)] = 2.1 L.

Ideal gas law: What pressure does 0.50 mol of gas exert in a 5.0 L container at 25.0 °C? T = 25.0 + 273.15 = 298.15 K. P = nRT/V = (0.50 mol)(0.08206 L·atm/mol·K)(298.15 K)/(5.0 L) = 2.4 atm.

Molar mass: An unknown gas with mass 3.15 g occupies 2.00 L at 0.950 atm and 25.0 °C. Find its molar mass. n = PV/RT = (0.950 atm)(2.00 L) / [(0.08206 L·atm/mol·K)(298.15 K)] = 0.0777 mol. M = mass/n = 3.15 g / 0.0777 mol = 40.5 g/mol.

Common Confusions

  • "Use °C in PV = nRT" — never; temperature must be in kelvin, or R won't cancel correctly.
  • "R is always 0.08206" — that value only works with L·atm; with joules you use 8.314 J/(mol·K).
  • "The combined gas law lets n change" — no; the combined gas law requires a constant amount of gas.
  • "Density has nothing to do with the gas laws" — density of a gas is d = PM/RT, derived directly from the ideal gas law.
Eli, the EliExplains learning guide

Eli explains

The same idea, in plain words

Explain it like I’m 10

The gas laws are like four dials — pressure, volume, temperature, and amount — that are connected. The combined and ideal gas laws are just the instruction manual that says how all the dials move together at once. If you know three of the dials and want the fourth, PV = nRT finds it for you. The analogy's limit: this "manual" is only exact for an imaginary "ideal" gas; real gases need small corrections (the van der Waals equation) when squeezed hard or cooled down.

Key takeaways

  • Combined gas law: P₁V₁/T₁ = P₂V₂/T₂ (n constant).
  • Ideal gas law: PV = nRT.
  • R = 0.08206 L·atm/(mol·K); also R = 8.314 J/(mol·K).
  • Always use kelvin for T.
  • d = PM/RT; M = mRT/(PV).
  • Real gases approach ideal behavior at low P and high T.
  • Combined gas law: P₁V₁/T₁ = P₂V₂/T₂ (constant n).
  • Ideal gas law: PV = nRT.
  • R = 0.08206 L·atm/(mol·K) or 8.314 J/(mol·K).
  • Kelvin only; match R to your units.
  • d = PM/RT; M = mRT/(PV).

Keep learning

Ready to build on this? Continue to the next lesson.

Practice General Chemistry I

This lesson has no separate scored set. Practice draws from the subject’s question bank.

Study tools & related lessonsYou’ll learn to · Related

You’ll learn to

  • Combine Boyle's, Charles's, and Gay-Lussac's laws into a single equation.
  • State the ideal gas law and each of its variables.
  • Solve for pressure, volume, temperature, or amount using PV = nRT.
  • Use the ideal gas law to find gas density and molar mass.

Sources & references

  1. OpenStax, "9.2 Relating Pressure, Volume, Amount, and Temperature," Chemistry 2e.
  2. Petrucci et al., "6.3 Combining the Gas Laws: The Ideal Gas Equation and the General Gas Equation," Chemistry LibreTexts.
  3. NIST CODATA, "molar gas constant."

This lesson was adapted from the open educational references above; their licenses and attributions are preserved. See Copyright & Licensing.

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