Engineer's Notebook §1.1

Ideal gas law

For most gases near everyday conditions, pressure, volume, amount and absolute temperature are tied by one equation:

pV = nRT
p
absolute pressure
Pa
V
volume
m³
n
amount of gas
mol
R
molar gas constant, 8.314
J/(mol·K)
T
absolute temperature
K
m
mass
kg
Rs
specific gas constant, R/M
J/(kg·K)
M
molar mass
kg/mol
ρ
density
kg/m³
A pressure–volume graph. A curve shows every state at the chosen temperature, and a marked point shows the current volume and pressure. VOLUME V, LPRESSURE p, kPa p = 102 kPa 500 100 T = 293.2 K
Fig. 1.1-1The curve is an isotherm: every state of this gas at the chosen temperature.
T in kelvin293.15 K
nRT2,440 J
p = nRT/V102 kPa

Worked example

Uses the values in Fig. 1.1-1. Change them and the working updates.

Given
  • namount of gas1 mol
  • Ttemperature20 °C
  • Vvolume24 L
Find

The pressure p of the gas.

Solution
T = 20 + 273.15 = 293.15 K p = nRT/V = 1 mol × 8.314 J/(mol·K) × 293.15 K / 0.024 m³ = 1.02 × 10⁵ Pa
Answer
p = 102 kPa

That is 1.02 bar, or 1 times standard atmospheric pressure (101.325 kPa).

Related forms

For dry air, Rs ≈ 287 J/(kg·K).

  • Per unit mass

    pV = mRsTRs = RM

  • Fixed amount, two states

    p1V1T1 = p2V2T2

  • Density

    ρ = pRsT

Assumptions
  • The molecules take up negligible volume and do not attract each other.
  • This holds well for air and most gases near room conditions, and less well at high pressure or close to condensing.
  • Pressure and temperature are absolute: add 273.15 to °C, and use absolute rather than gauge pressure.
  • The gas is in equilibrium, with one pressure and temperature throughout.

Derivation

Three experimental laws each hold two quantities fixed and relate the other two.

  1. Boyle: at fixed n and T, pV is constant.
  2. Charles: at fixed n and p, V ∝ T.
  3. Avogadro: at fixed p and T, V ∝ n.
  4. Together, pV ∝ nT. The constant is the same for every ideal gas, R = 8.314 J/(mol·K), so pV = nRT.

Kinetic theory gives the same result from molecules bouncing off the walls: pressure comes from their impacts, and temperature measures their mean kinetic energy.