States of matter and gases

Why this matters

Gases are everywhere in technical work: compressed air in a plant, carbon dioxide in a cell-culture incubator, nitrogen purges in a chemistry lab, natural gas in a building. Engineers size tanks and pipes with gas calculations, and lab staff convert gas volumes to amounts every day. The ideal gas law is the one equation that ties pressure, volume, temperature and amount together.

The three everyday states

Matter is made of particles that are always moving. The state depends on how strongly they attract each other compared with how much energy they have.

Adding heat gives particles more energy, so substances melt (solid to liquid) and boil or evaporate (liquid to gas). Removing heat reverses this: condensation and freezing. Some solids go straight to gas (sublimation), like dry ice (solid CO₂). During a change of state the temperature stays constant while the energy goes into breaking attractions: an ice-water mix stays at 0 °C until the ice is gone.

Temperature must be in kelvin

Gas laws only work with an absolute temperature scale, where zero means no thermal motion:

T(K) = T(°C) + 273.15
25 °C  -> 298.15 K
0 °C   -> 273.15 K

If you plug in Celsius, 0 °C would make the volume of every gas zero, which is clearly wrong.

The ideal gas law

P V = n R T

• P pressure, V volume, n amount in moles, T temperature in kelvin
• R the gas constant. Its value depends on the units you use:

Pick one row and keep every quantity in those units. Unit mismatches are the most common source of wrong answers.

Worked example 1: amount of gas in a balloon

A balloon holds 5.0 L of helium at 100 kPa and 25 °C. How many moles?

T = 25 + 273.15 = 298.15 K
n = PV / (RT) = (100 × 5.0) / (8.314 × 298.15)
  = 500 / 2478.8 ≈ 0.202 mol

Worked example 2: pressure in a sealed container

2.0 mol of nitrogen is sealed in a 10.0 L rigid container at 25 °C.

P = nRT / V = (2.0 × 8.314 × 298.15) / 10.0 ≈ 496 kPa

That is almost five times normal atmospheric pressure (101.325 kPa), which is why gas cylinders are thick steel and are handled only after training.

Worked example 3: the molar volume

At 0 °C and 1 atm (101.325 kPa), one mole of any ideal gas occupies

V = nRT / P = (1 × 8.314 × 273.15) / 101.325 ≈ 22.4 L

You may also see 22.7 L: that is for the IUPAC standard of 0 °C and 100 kPa. Always check which "standard conditions" a source means.

The simpler gas laws are special cases

Hold two of the four quantities fixed and PV = nRT gives the classic laws:

• Boyle: at constant n and T, P₁V₁ = P₂V₂ (squeeze a gas and its pressure rises).
• Charles: at constant n and P, V₁/T₁ = V₂/T₂ (warm a balloon and it expands).
• Gay-Lussac: at constant n and V, P₁/T₁ = P₂/T₂ (a heated sealed can builds pressure).

When "ideal" stops being true

The ideal gas law assumes the particles take up no space and do not attract each other. That is a good approximation for most gases near room temperature and atmospheric pressure. It breaks down at high pressure and low temperature, near the point where the gas would condense. Engineers then use real-gas equations (such as van der Waals) or tables.

Safety note

Compressed-gas cylinders, cryogenic liquids and pressure vessels are real hazards. This lesson is awareness level only: handling them requires your employer's or institution's training and supervision. OSHA's compressed gases page is a starting point. Checked on 2026-10-23.

Common mistakes

• Celsius in the equation. Always convert to kelvin first.
• Mixing units. kPa with the atm value of R gives an answer off by about 100 times.
• Forgetting mL to L. 250 mL is 0.250 L.
• Applying it to liquids or solids. PV = nRT is for gases only.

Quick recap

• Solids, liquids and gases differ in particle spacing and motion.
• PV = nRT links pressure, volume, amount and absolute temperature.
• Match R to your units; one mole of gas is about 22.4 L at 0 °C and 1 atm.

Try it

Open the Do tab: you will finish a Python function that calculates moles of gas from pressure, volume and temperature, then answer the quiz.