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Lesson 36 · Combined and ideal
R, and the state it referees
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R, and the state it referees

Stack all three laws and they collapse into one: P1V1T1=P2V2T2\dfrac{P_1 V_1}{T_1} = \dfrac{P_2 V_2}{T_2}, the combined gas law — P-one V-one over T-one equals P-two V-two over T-two. Same fixed amount of gas, same subscript convention (1 before, 2 after), and now any of the three knobs may move. Pin any one of them and the matching pair cancels, handing you Boyle, Charles or Gay-Lussac back. Learn this one and you own all four.

What the combined law cannot tell you is HOW MUCH gas is in the vessel — it compares two states and the amount cancels itself out. For that you need PV=nRTPV = nRT, the ideal gas law, read aloud P V equals n R T. PP is the absolute pressure, VV the volume, nn the amount of gas in moles, TT the absolute temperature, and RR the universal gas constant — the exchange rate between the mechanical side and the molecular side. On this paper R=8.314 kPa⋅L/(mol⋅K)R = 8.314\ \mathrm{kPa \cdot L/(mol \cdot K)}, so pressures arrive in kPa and volumes in L, and every unit cancels: kPa·L on the left, and mol times kPa·L/(mol·K) times K on the right. Push the units through your rearrangement every time. If they refuse to cancel into the answer's units, the rearrangement is wrong, no appeal — and units that do work out never prove you right. The check is a one-way street.

One shortcut is worth memorising. At STP — 101.325 kPa and 273.15 K — PV=nRTPV = nRT has already been solved for you: one mole of any ideal gas fills Vm=22.4 LV_m = 22.4\ \mathrm{L}, so V=n VmV = n\,V_m. Two cautions. It is STP only: at SATP (25 °C, 100 kPa) the molar volume is 24.8 L/mol, about 10 % larger, and swapping them is a favourite exam trap. And VmV_m does not care WHICH gas it is — Avogadro's insight, and the reason a mole of hydrogen and a mole of sulfur dioxide fill the same balloon.

P1V1T1P2T2V2

P1V1T1=P2V2T2\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}

  • P1P_1= Initial pressure (pressure)
  • V1V_1= Initial volume (volume)
  • T1T_1= Initial absolute temperature (temperature)
  • P2P_2= Final pressure (pressure)
  • V2V_2= Final volume (volume)
  • T2T_2= Final absolute temperature (temperature)
Combined Gas Law solver →
PVTn

PV=nRTP V = n R T

  • PP= Pressure (pressure)
  • VV= Volume (volume)
  • nn= Amount (amount of substance)
  • TT= Temperature (temperature)
Ideal Gas Law solver →
Vn

V=n VmV = n\,V_m

  • VV= Gas volume at STP (volume)
  • nn= Amount of gas (amount of substance)
Gas Volume at STP solver →