How this instrument works
Pressure inside any vessel is momentum arriving at its walls. Nitrogen molecules at 0 °C average close to 490 metres per second, travel roughly 60 nanometres between collisions, and hammer every square millimetre billions of times a second; each rebound delivers a minute impulse, and their sum, smoothed over time, is what a gauge reads. PV = nRT says that drumming depends on only two things — how many particles share a space, and how hard warmth has them moving. Molecular identity drops straight out. Argon, methane and water vapour, equally crowded at equal temperature, push equally hard, which is why one constant serves every dilute gas rather than a table of them.
Daniel Bernoulli reached this by reasoning alone and was ignored for a century. Hydrodynamica, printed 1738, carries a section deriving exactly such pressure from corpuscles rattling inside a cylinder, decades before anyone would concede that matter came in particles. Rudolf Clausius revived that argument in 1857, James Clerk Maxwell supplied a distribution of speeds in 1860, and Josef Loschmidt turned it around in 1865 to count molecules and estimate their diameter, landing within a factor of two of today's 2.687 × 10²⁵ per cubic metre at ice point. R itself reads like bookkeeping but is not: 8.314 joules per mole per kelvin is precisely the work one mole performs shoving a piston outward while warming through one degree.
Ideality has a domain, and both its edges repay attention. Crowd molecules together and their own bulk plus mutual attraction start to register — methane at 200 bar strays by tens of percent, and any vapour nearing condensation strays long before that. Run the opposite way, into hard vacuum, and pressure stays perfectly well defined while gas quits behaving as a fluid: past about 10⁻³ Pa one molecule crosses a metre-wide chamber without meeting another, so pumping speed and heat transfer follow entirely separate rules. Subtlest trap of all is heat, because n must hold still. Above roughly 2000 K diatomic molecules begin splitting apart, every dissociation raises particle count, and pressure behind a re-entry shock or inside a rocket nozzle outruns anything a fixed amount predicts.
- Put your mole count into Amount of gas (mol). Working from a mass? Divide grams by molar mass first — 32 g of oxygen is 1 mol, not 32.
- Convert from Celsius before filling Absolute temperature (K) — 25 °C becomes 298.15 K. Values at or under zero get rejected outright.
- Set Volume to whatever space your gas occupies — ml, litres, m³ or ft³ on its unit menu. Vessel volume, not volume the sample would fill at STP.
- Read Pressure in pascals, switching its unit menu to kPa, bar, atm or psi as your trade prefers. Output is absolute; subtract 101325 Pa before trusting it against any gauge.
Worked example — one mole at ice point, 22.414 litres
Take molar volume as chemistry teaches it: one mole of an ideal gas at 0 °C filling 22.414 litres. Amount of gas 1 mol, Absolute temperature 273.15 K, Volume 0.022414 m³. Numerator first — 1 × 8.314462618 × 273.15 = 2271.0955 joules — then divide by that space: 2271.0955 ⁄ 0.022414 = 101324.8623 Pa.
Which sits 0.14 Pa beneath one standard atmosphere, defined as 101325 Pa exactly. A shortfall of 1.4 parts per million, and it lives in your input rather than in physics: CODATA puts molar volume at those conditions at 22.413 969 5 litres, so rounding to 22.414 is what costs you those final digits. Sixth-figure precision in Volume would close it.
Behind such tidy figures sits one real box of air roughly 28 cm on each side, holding 6.022 × 10²³ molecules and weighing about 29 grams — a small handful of paperclips. Students meet it as a stoichiometry shortcut. Anaesthetists and welders meet it whenever cylinder contents must be turned from moles into litres somebody can actually breathe or burn.
Questions
What belongs in the Amount of gas field?
Moles — never grams, never litres. That slip accounts for more wrong answers here than everything else combined. Convert first by dividing mass in grams by molar mass in grams per mole: 32 g of oxygen is 1 mol, 32 g of helium is 8 mol, and 32 g of butane is 0.55 mol. Identical weight on the scales, three wildly different pressures. Counting molecules instead? Divide by 6.02214076 × 10²³. For mixtures, total up everything present, since this law tallies particles and ignores what they happen to be.
Which units keep R = 8.314 valid?
Strict SI: pascals, cubic metres, moles, kelvin. That value of R carries joules per mole per kelvin, and one joule is one pascal times one cubic metre, so only those cancel cleanly. Hand it litres and your answer lands a thousandfold out. Chemistry texts sidestep this by keeping a second constant, 0.082057 L·atm/(mol·K), which pairs with atmospheres and litres — same physics, different clothing. Unit menus on Volume and Pressure convert for you, so entering 22.414 l is safe; typing 22.414 into a field still set to m³ is not.
Does it hold for a mixture such as air?
Yes, and more exactly than most people expect. Pressure answers to total particle count, so 1 mol of air behaves like 1 mol of anything: 0.78 mol nitrogen, 0.21 mol oxygen and 0.01 mol argon sharing one vessel push just as 1 mol of pure argon would. John Dalton spelled out the consequence in 1801 — each component contributes a partial pressure proportional to its share, and those partials add up to the whole. Enter total moles, read total pressure, then multiply by 0.21 if oxygen's partial pressure is what you actually want.
Is 22.414 litres per mole still the standard molar volume?
Only under older standard conditions. That figure belongs to 273.15 K with 101.325 kPa. IUPAC shifted standard pressure to exactly 100 kPa in 1982, and molar volume moved with it, to 22.711 litres per mole at ice point. Room-temperature work usually quotes 24.79 L/mol, meaning 298.15 K at 100 kPa. Textbooks printed either side of 1982 therefore disagree, which snarls surprising amounts of homework. This sheet holds no opinion — hand it the temperature and volume genuinely in front of you, and it returns the matching pressure.
Where does PV = nRT stop being accurate?
Wherever molecules quit being lonely. Under about 10 bar at room temperature, air, nitrogen, oxygen and helium stay within a fraction of one percent, covering nearly all workshop and laboratory use. Press toward 100 bar and attraction plus finite molecular bulk drag real behaviour several percent off. Approach any condensation line — steam near 100 °C, propane near 8 bar — and errors reach tens of percent, since part of your sample would rather be liquid. Fierce heat spoils it from the far side, dissociation quietly raising n past about 2000 K.
What happens if I heat a sealed rigid vessel?
Pressure climbs in direct proportion to absolute temperature, because amount and volume are both pinned. Leave a sealed jar of air at 20 °C, which is 293.15 K, inside a car reaching 60 °C: the ratio 333.15 ⁄ 293.15 takes 101325 Pa up to 115151 Pa, some 14 kPa of gauge pressure straining a lid. Aerosol cans wear their fire warning for that same arithmetic run much further. Hold Amount of gas and Volume fixed, vary Absolute temperature, and watch Pressure follow it step for step.