The micro/macro connection relates the macroscopic properties of a system to the motion and collisions of its atoms and molecules.

Pressure is due to the force of the molecules colliding with the walls:

The average translational kinetic energy of a molecule is ${ϵ}_{\text{avg}}=\frac{3}{2}{k}_{\text{B}}T$. The temperature of the gas $T=2{ϵ}_{\text{avg}}\text{/}3k_{\text{B}}$ measures the average translational kinetic energy.

Entropy S measures the number of ways in which a macrostate can be arranged microscopically:

where $\text{Ω}$, the multiplicity, is the number of microstates. Entropy is a state variable. A slow, reversible transfer of heat changes a system’s entropy by $dS=dQ/T$.

The thermal energy of a system is

$\begin{array}{lll}\hfill E_{\text{th}}& =\hfill & \text{translational kinetic energy }+\text{ rotational}\hfill \\ \hfill & \hfill & \text{kinetic energy }+\text{ vibrational energy}\hfill \end{array}$

• Monatomic gas $E_{\text{th}}=\frac{3}{2}N{k}_{\text{B}}T=\frac{3}{2}nRT$

• Diatomic gas  $E_{\text{th}}=\frac{5}{2}N{k}_{\text{B}}T=\frac{5}{2}nRT$

• Elemental solid $E_{\text{th}}=3N{k}_{\text{B}}T=3nRT$

Heat is energy transferred via collisions from more-energetic molecules on one side to less-energetic molecules on the other. Equilibrium is reached when ${\left({ϵ}_1\right)}_{\text{avg}}={\left({ϵ}_2\right)}_{\text{avg}}$, which implies $T_{1\text{f}}=T_{2\text{f}}$.

The root-mean-square speed $v_{\text{rms}}$ is the square root of the average of the squares of the molecular speeds:

For molecules of mass m at temperature T, $v_{\text{rms}}=\sqrt{{\displaystyle \frac{3k_{\text{B}}T}{m}}}$

Molar specific heats can be predicted from the thermal energy because $\text{Δ}{E}_{\text{th}}=nC\text{Δ}T$.

• Monatomic gas $C_{\text{V}}=\frac{3}{2}R$

• Diatomic gas  $C_{\text{V}}=\frac{5}{2}R$

• Elemental solid $C=3R$