# What causes gas molecules to spread through air?

May 18, 2017

Mostly entropy. The process of mixing gases, particularly ideal gases, is favored by entropy, and for ideal gases, it is always spontaneous.

Below is a thermodynamics proof of why entropy and Gibbs' free energy of mixing are positive and negative, respectively, for ideal gases.

The entropy of mixing is given by the difference in entropy between the unmixed (${\overline{S}}^{\text{*}}$) and mixed ($\overline{S}$) substances:

$\Delta {\overline{S}}_{m i x} = \overline{S} - {\overline{S}}^{\text{*}}$

$= {\sum}_{i} {n}_{i} {\overline{S}}_{i} - {\sum}_{i} {n}_{i} {\overline{S}}_{i}^{\text{*}}$

$= {\sum}_{i} {n}_{i} \left({\overline{S}}_{i} - {\overline{S}}_{i}^{\text{*}}\right)$ $\text{ "" } \boldsymbol{\left(1\right)}$

where:

• ${n}_{i}$ is the mols of gas $i$.
• ${\overline{S}}_{i}$ is the molar entropy, ${S}_{i} / {n}_{i}$, of gas $i$.
• $\text{*}$ indicates the unmixed substance.

The entropy is related to the Gibbs' free energy $G$ and enthalpy $H$:

$\Delta G = \Delta H - T \Delta S$$\text{ "" } \boldsymbol{\left(2\right)}$

These have their own thermodynamic mixing expressions:

$\Delta {\overline{G}}_{m i x} = {\sum}_{i} {n}_{i} \left({\overline{G}}_{i} - {\overline{G}}_{i}^{\text{*}}\right)$$\text{ "" } \boldsymbol{\left(3\right)}$

$\Delta {\overline{H}}_{m i x} = {\sum}_{i} {n}_{i} \left({\overline{H}}_{i} - {\overline{H}}_{i}^{\text{*}}\right)$$\text{ "" } \boldsymbol{\left(4\right)}$

For ideal gases, there is zero heat released or absorbed after mixing (no intermolecular forces exchanged), so ${\overline{H}}_{i} \approx {\overline{H}}_{i}^{\text{*}}$, and from $\left(4\right)$, $\Delta {\overline{H}}_{m i x} \approx 0$.

From $\left(2\right)$, $\Delta {\overline{G}}_{m i x} \approx - T \Delta {\overline{S}}_{m i x}$. We can write for $\overline{G}$:

${\overline{G}}_{i} = {\overline{G}}_{i}^{\text{*}} + R T \ln {\chi}_{i}$$\text{ "" } \boldsymbol{\left(5\right)}$

where ${\chi}_{i}$ is the mol fraction of gas $i$.

As a result, plugging $\left(5\right)$ into $\left(3\right)$:

$\textcolor{b l u e}{\Delta {\overline{G}}_{m i x} = R T {\sum}_{i} {n}_{i} \ln {\chi}_{i}} \approx - T \Delta {\overline{S}}_{m i x}$

or, comparing $\left(2\right)$ with $\left(3\right)$:

$\textcolor{b l u e}{\Delta {\overline{S}}_{m i x} \approx - R {\sum}_{i} {n}_{i} \ln {\chi}_{i}}$

But since mol fractions are always less than or equal to $1$, $\ln {\chi}_{i} < 0$, and we have two conclusions:

• $\Delta {\overline{G}}_{m i x} < 0$. Therefore, mixing ideal gases is spontaneous.
• $\Delta {\overline{S}}_{m i x} > 0$. Therefore, mixing ideal gases increases the entropy of the system.