15. Ideal gases

Amount of substance n is measured in moles, allowing microscopic entities to be counted through a macroscopic quantity.

Name the entities being counted—atoms, molecules, ions or formula units—before converting between mass, moles and particles.

One mole of water means one mole of H₂O molecules, not one mole of hydrogen atoms alone.

Moles are not a mass unit; different substances have different masses per mole.

One mole contains N_A≈6.02×10²³ specified particles or entities. The number is a count, not a mass.

Multiply moles by N_A to find particles, or divide particle count by N_A to find amount. State whether entities are atoms, molecules or ions.

0.50 mol of electrons contains about 3.01×10²³ electrons; 1 mol of NaCl contains that many formula units.

Avogadro’s constant is not the number of grams in a mole and does not change with substance.

At fixed amount and volume, an ideal gas obeys p∝T, so p/T is constant when T is measured in kelvin.

Use absolute temperature and keep volume and amount fixed. A pressure change can be compared through p₁/T₁=p₂/T₂.

Heating a sealed rigid container from 300 K to 450 K raises ideal-gas pressure by a factor of 1.5.

The proportionality fails if volume or amount changes, and Celsius temperature cannot be used in the ratio.

For an ideal gas, pressure p, volume V, amount n and thermodynamic temperature T satisfy pV=nRT.

Use pascals, cubic metres, moles and kelvin with R=8.31 J mol⁻¹ K⁻¹. Check whether the gas is dilute enough for the ideal model.

One mole at 300 K in 0.0249 m³ has pressure about 1.00×10⁵ Pa.

pV=nRT is not a universal exact law for real gases under all conditions.

Boltzmann constant k=R/N_A, so kT is the thermal energy scale per particle while RT is the corresponding molar scale.

Use k when counting individual particles and R when working with moles; the two forms describe the same thermal physics at different scales.

The average translational energy scale of one molecule uses kT, whereas one mole uses RT.

k and R are not interchangeable numbers; their units and particle-count basis differ.

The ideal-gas kinetic model assumes particles are point-like compared with container volume, move randomly, collide elastically and exert negligible forces except during collisions.

These assumptions explain pressure and temperature while marking the limits of the model for dense or strongly interacting gases.

At lower density, the average spacing increases and intermolecular attractions become less important, so ideal behaviour improves.

“No forces” applies between collisions in the ideal model; collisions still transfer momentum and create pressure.

Molecules colliding with a wall change momentum; the rate of momentum transfer per unit area produces macroscopic pressure.

More frequent or harder collisions raise pressure. Increasing temperature raises average molecular kinetic energy and therefore collision impulses and rates.

Compressing a gas at fixed temperature increases wall-collision frequency per unit area and raises pressure.

Pressure is not caused by molecules “pushing continuously” between collisions; it is the aggregate effect of impacts.

The root-mean-square speed c_rms is defined by c_rms=√⟨c²⟩ and for an ideal gas c_rms=√(3RT/M), where M is molar mass.

Use absolute temperature and molar mass in kg mol⁻¹; lighter gases move faster at the same temperature.

At fixed T, helium has a larger rms speed than oxygen because its molar mass is smaller.

c_rms is not the arithmetic mean speed and does not mean every molecule travels at that speed.

For N molecules of mass m, pV=⅓Nm⟨c²⟩. Combining with pV=NkT links pressure to mean translational kinetic energy per particle.

Use ⟨c²⟩ as the mean square speed, not the square of the mean speed, and keep N as particle count.

At fixed volume, raising T raises mean molecular kinetic energy and therefore pressure in proportion.

The equation describes a statistical average; not every molecule has the same speed or energy.