Question

What causes gas molecules to spread through air?

Answer 1

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.

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The entropy of mixing is given by the difference in entropy between the unmixed (`barS^"*"`) and mixed (`barS`) substances:

`DeltabarS_(mix) = barS - barS^"*"`

`= sum_i n_ibarS_i - sum_i n_ibarS_i^"*"`

`= sum_i n_i(barS_i - barS_i^"*")` `" "" "bb((1))`

where:

• `n_i` is the mols of gas `i`.
• `barS_i` is the molar entropy, `S_i/n_i`, of gas `i`.
• `"*"` indicates the unmixed substance.

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

`DeltaG = DeltaH - TDeltaS` `" "" "bb((2))`

These have their own thermodynamic mixing expressions:

`DeltabarG_(mix) = sum_i n_i(barG_i - barG_i^"*")` `" "" "bb((3))`

`DeltabarH_(mix) = sum_i n_i(barH_i - barH_i^"*")` `" "" "bb((4))`

For ideal gases, there is zero heat released or absorbed after mixing (no intermolecular forces exchanged), so `barH_i ~~ barH_i^"*"`, and from `(4)`, `DeltabarH_(mix) ~~ 0`.

From `(2)`, `DeltabarG_(mix) ~~ -TDeltabarS_(mix)`. We can write for `barG`:

`barG_i = barG_i^"*" + RTlnchi_i` `" "" "bb((5))`

where `chi_i` is the mol fraction of gas `i`.

As a result, plugging `(5)` into `(3)`:

`color(blue)(DeltabarG_(mix) = RTsum_i n_ilnchi_i) ~~ -TDeltabarS_(mix)`

or, comparing `(2)` with `(3)`:

`color(blue)(DeltabarS_(mix) ~~ -Rsum_i n_ilnchi_i)`

But since mol fractions are always less than or equal to `1`, `lnchi_i < 0`, and we have two conclusions:

• `DeltabarG_(mix) < 0`. Therefore, mixing ideal gases is spontaneous.
• `DeltabarS_(mix) > 0`. Therefore, mixing ideal gases increases the entropy of the system.