the amount of substance containing 6.02 × 1023 particles (the Avogadro constant number of particles).
Avogadro constant (NA)
the number of particles in one mole of a substance: 6.02 × 1023 mol-1.
Molar mass (M)
the mass of one mole of a substance.
Key points
Number of molecules N = nNA; mass of gas = nM.
Molar mass in g mol-1 equals the nucleon number: convert to kg mol-1 before use.
Mass of one molecule m = M / NA.
One mole of any substance has the same number of particles, whatever its mass.
Converting between moles, number of particles and mass.
Watch out!
n (moles) and N (molecules) differ by a factor of NA: pV = nRT uses n, pV = NkT uses N.
Molar mass of helium is 4 g mol-1 = 0.004 kg mol-1.
15.2Equation of state
Definitions
Ideal gas
a gas that obeys pV ∝ T (or pV = nRT) at all pressures, volumes and temperatures.
Boyle's law
for a fixed mass of gas at constant temperature, pressure is inversely proportional to volume.
Charles's law
for a fixed mass of gas at constant pressure, volume is proportional to thermodynamic temperature.
Pressure law
for a fixed mass of gas at constant volume, pressure is proportional to thermodynamic temperature.
Key points
The three gas laws combine to pV / T = constant for a fixed mass of ideal gas.
Molar form uses the molar gas constant R; molecular form uses the Boltzmann constant k = R / NA.
Real gases behave most ideally at low pressure and high temperature (well above boiling point).
Isotherms on a p-V graph are hyperbolas; a higher temperature gives a curve further from the axes.
A graph of pV against T is a straight line through the origin with gradient nR.
Boyle's law isotherms: p ∝ 1/V at constant T; the hotter isotherm lies further out.
pV against T (kelvin): straight line through the origin, gradient nR = Nk.
Watch out!
Use p in Pa, V in m3 (1 cm3 = 10-6 m3, 1 litre = 10-3 m3) and T in K.
The gas laws need a fixed mass of gas: if gas leaks, n changes.
15.3aKinetic theory: assumptions and pressure
Definitions
Kinetic theory assumptions
a large number of identical molecules in random motion; negligible volume of molecules compared with the container; negligible intermolecular forces except during collisions; perfectly elastic collisions; collision time negligible compared with time between collisions.
Pressure (of a gas)
the force per unit area exerted on the walls due to the rate of change of momentum of molecules colliding with them.
Key points
Molecule of mass m hits a wall with velocity component cx: momentum change = 2mcx.
In a cube of side L it returns to the same wall after time 2L / cx.
Force = rate of change of momentum = 2mcx / (2L / cx) = mcx2 / L.
For N molecules, p = F / L2 = Nm⟨cx2⟩ / V.
Random motion: ⟨cx2⟩ = ⟨cy2⟩ = ⟨cz2⟩, so ⟨cx2⟩ = ⅓⟨c2⟩.
Since density ρ = Nm / V, the result can also be written p = ⅓ρ⟨c2⟩.
Derivation set-up: one molecule in a cube of side L bouncing between opposite faces.
At the wall the molecule's momentum reverses: change 2mcx, and the wall feels an equal and opposite push.
Watch out!
The momentum change at the wall is 2mcx, not mcx: the velocity reverses.
⟨c2⟩ is the mean of the squared speeds, not the square of the mean speed.
15.3bKinetic energy of a molecule and temperature
Definitions
Root-mean-square speed (cr.m.s.)
the square root of the mean of the squares of the speeds of the molecules.
Boltzmann constant (k)
the molar gas constant divided by the Avogadro constant, k = R / NA.
Key points
Equate pV = ⅓Nm⟨c2⟩ with pV = NkT to get ½m⟨c2⟩ = (3/2)kT.
Mean translational kinetic energy of a molecule is proportional to thermodynamic temperature.
At the same temperature all ideal gases have the same mean kinetic energy per molecule; lighter molecules move faster.
cr.m.s. = √(3kT / m): doubling T multiplies cr.m.s. by √2.
r.m.s. speed: square each speed, take the mean, then the square root.
At higher T the speed distribution spreads out and its peak shifts to higher speed.
Molecular speed distributions: at higher T the peak moves right and flattens; cr.m.s. ∝ √T.
Mean kinetic energy per molecule against T: straight line through the origin, gradient 3k/2.
Watch out!
cr.m.s. is larger than the mean speed; do not average speeds and then square.
Use the mass of one molecule (M / NA, in kg) in ½m⟨c2⟩ = (3/2)kT.