Physics › Thermal physics › Ideal gases and the gas laws
Ideal gases and the gas laws
Three empirical laws, discovered by squeezing and warming real gases, extrapolate to a shared absolute zero and merge into one equation of state, written once for moles and once for molecules.
Builds on Thermal energy transfer and specific heat capacity.
IN THIS TOPIC
- Use the empirical gas laws and explain how extrapolation locates absolute zero.
- Apply pV = nRT and pV = NkT, choosing moles or molecules as the question demands.
- Calculate work done at constant pressure with pΔV, and convert between molar and molecular mass.
WHAT YOU PROBABLY THINK
Zero degrees means zero heat.
Three empirical laws
The gas laws were found by experiment, squeezing, warming and measuring, which is what empirical means, and each holds one variable fixed. Boyle's law: at constant temperature, pV is constant.
Charles's law: at constant pressure, V/T is constant. The pressure law: at constant volume, p/T is constant. Both temperature laws demand kelvin; in Celsius they are simply false. Required practical 8 investigates Boyle's and Charles's laws directly.
Absolute zero
Plot volume against Celsius temperature for any gas and the line is straight; extrapolate it backwards and it reaches zero volume at −273 °C, and, remarkably, every gas's line meets the axis at the same place. That shared intercept defines absolute zero, 0 K, the temperature at which particle energy is at its minimum. Which corrects the lie: 0 °C is merely where water freezes, a parochial landmark 273 degrees above the real floor.
One equation of state
The three laws merge into the ideal gas equation, written in two currencies. For n moles:
with R = 8.31 J K−1 mol−1, the molar gas constant. For N molecules:
with k = 1.38 × 10−23 J K−1, the Boltzmann constant.
The Avogadro constant NA = 6.02 × 1023 mol−1 is the exchange rate between the two: N = nNA, and k = R/NA. The same bridge links molar mass (kilograms per mole) to molecular mass (kilograms per molecule): M = NAm. Choosing the wrong currency is the unit's most reliable mark-loser.
One more tool: when a gas expands or is compressed at constant pressure, the work done is
which is not on the data sheet, and is how “work” from mechanics buys its way into gas problems.
THE EXAM BIT
- Kelvin, always: T(K) = θ(°C) + 273. A single Celsius temperature inside any gas-law calculation poisons the whole answer.
- Choose the currency before the constant: moles go with R, molecules with k. Mixing n with k is the planted error.
- Extrapolation questions want the sentence: all gases' lines meet the temperature axis at the same point, −273 °C, which defines absolute zero.
- W = pΔV is not printed on the data sheet: learn it, and quote the condition, constant pressure.
- Molar mass in kg mol−1: nitrogen is 0.028, not 28. The stray factor of a thousand is the examiners' favourite harvest.
CHECK YOURSELF
A cylinder of volume 0.020 m3 holds gas at 250 kPa and 300 K. How many moles does it contain, and roughly how many molecules is that?
Show a hint
One equation for n, then cross the bridge.
Show the answer
n = pV/RT = (250 × 103 × 0.020)/(8.31 × 300) = 2.0 mol.
Molecules: N = nNA = 2.0 × 6.02 × 1023 = 1.2 × 1024.
Sense check: about two grams' worth of hydrogen or fifty-six of nitrogen, a perfectly ordinary cylinderful, and already a trillion trillion particles.
Kelvin in every gas law, no exceptions.
Moles ride with R; molecules ride with k.
No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.