Question

For hydrogen peroxide and chlorine gas that are in the same amount, phase, temperature, and volume, why is it that hydrogen peroxide has greater entropy?

Answer 1

Because polyatomic molecules have more entropy than diatomic molecules. They have more ways to vibrate, and more rotational angles.

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Entropy has many formulations, but one of them is in terms of microstates:

`S = k_Bln Omega`

where `ln Omega` is a term that accounts for the distribution of microstates in a system. The more complex the molecule, the more microstates it can have.

In this answer, I give an explicit example on how to find `ln Omega` for methane.

Roughly speaking, `ln Omega` primarily depends on the number of ways the molecule can move in terms of translation, rotation, and vibration.

• Obviously, both `"H"_2"O"_2` and `"Cl"_2` move translationally in three dimensions, so they are said to have three translational degrees of freedom.

• `"Cl"_2` is linear and only has two angles of rotation (`theta, phi`), while `"H"_2"O"_2` is polyatomic, and has three angles of rotation (`theta,phi,gamma`). These are said to have two or three rotational degrees of freedom, respectively.

• `"Cl"_2` has only one way to vibrate (symmetrically). `"H"_2"O"_2` has `3(4) - 6 = bb6` ways to vibrate, and as such, has more vibrational degrees of freedom.

Thus, `H_2O_2` is more entropic mainly due to more rotational and vibrational degrees of freedom.

This is summarized here: