§ 15.1 The mole
Key ideas
- The mole is the SI base unit for amount of substance; one mole contains the Avogadro number of particles.
- The number of molecules is N = n N_A.
Equations
N = n N_AN_A = 6.02 × 10²³ mol⁻¹no unit
Watch out: n (number of moles) and N (number of molecules) are different. pV = nRT uses moles; pV = NkT uses molecules. Mixing them is wrong by a factor of N_A.
§ 15.2 Equation of state
Key ideas
- An ideal gas obeys pV ∝ T for a fixed mass, leading to the equation of state.
- The molar form uses the gas constant R; the molecular form uses the Boltzmann constant k = R / N_A.
Equations
pV = n R TR = 8.31 J mol⁻¹ K⁻¹Pa, m³
pV = N k Tk = R / N_A = 1.38 × 10⁻²³ J K⁻¹Pa, m³
Watch out: T is in kelvin, p in pascals and V in cubic metres. A volume given in cm³ or litres must be converted (1 litre = 1 × 10⁻³ m³) before substituting.
§ 15.3 Kinetic theory of gases
Key ideas
- Assumptions: many identical molecules in random motion; negligible volume compared with the container; negligible forces except in collisions; collisions perfectly elastic and of negligible duration.
- Tracking the momentum change as molecules bounce off the walls gives pV = ⅓ N m ⟨c²⟩, where ⟨c²⟩ is the mean-square speed.
- Comparing with pV = NkT shows the average translational kinetic energy of a molecule is proportional to the absolute temperature.
- The root-mean-square speed is c_rms = √⟨c²⟩.
Equations
pV = ⅓ N m ⟨c²⟩kinetic theory result for pressurePa, m³
½ m ⟨c²⟩ = (3/2) k Tmean translational KE per moleculeJ
Fig. 1 · Pressure from collisions: each molecule reverses its momentum at the wall, and the total rate of momentum change over the wall area is the gas pressure.
Watch out: c_rms is the root of the mean of the squares, not the mean speed. Square each speed, average, then square-root: the order matters and gives a larger value than the plain average.