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?

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

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: