The different phases exist for different conditions of temperature $T$ and pressure $p$. The boundaries separating the regions of a phase diagram are lines of phase equilibrium. Any amounts of the two phases can coexist in equilibrium. The triple point is the one value of temperature and pressure at which all three phases can coexist in equilibrium.

• Atoms and molecules are small, hard spheres that travel freely through space except for occasional collisions with each other or the walls.

• The model is valid when the density is low and the temperature well above the condensation point.

The state variables of an ideal gas are related by the ideal-gas law

where $R=8.31\text{J}\text{/}\text{mol}\text{K}$ is the universal gas constant and $k_{\text{B}}=1.38\times {10}^{-23}\text{J}\text{/}\text{K}$ is Boltzmann’s constant. $p$, $V$, and T must be in SI units of Pa, ${\text{m}}^3$, and K.

For a gas in a sealed container, with constant $n$:

A macroscopic sample of matter consists of $N$ atoms (or molecules), each of mass $m$ (the atomic or molecular mass):

Alternatively, we can state that the sample consists of $n$ moles:

where $N_{\text{A}}=6.02\times {10}^{23}{\text{mol}}^{-1}$ is Avogadro’s number.

The molar mass $M_{\text{mol}}$, in kg/mol, is the numerical value of the atomic or molecular mass in u divided by 1000. The atomic or molecular mass, in atomic mass units u, is well approximated by the atomic mass number A. The atomic mass unit is

The number density of the sample is ${\displaystyle \frac{N}{V}}$.

The Kelvin scale has absolute zero at $T_0=0\text{K.}$

For a temperature change $\text{Δ}T$,

For a solid, $\beta =3\alpha$.

Ideal-gas processes are shown as trajectories through the $pV$ diagram.

1. Isochoric, or constant volume

2. Isobaric, or constant pressure

3. Isothermal, or constant temperature