The sign convention of changes in free energy follows the general convention for thermodynamic measurements, in which a release of free energy from the system corresponds to a negative change in free energy, but to a positive change for the surroundings.
That is, the ΔS of the surroundings increases enough because of the exothermicity of the reaction so that it overcompensates for the negative ΔS of the system, and since the overall ΔS = ΔSsurroundings + ΔSsystem, the overall change in entropy is still positive.
Another way to view the fact that some spontaneous chemical reactions can lead to products with lower entropy is to realize that the second law states that entropy of an isolatedsystem must increase (or remain constant).
Since a negative enthalpy change in a reaction means that energy is being released to the surroundings, then the "isolated" system includes the chemical reaction plus its surroundings.
This means that the heat release of the chemical reaction sufficiently increases the entropy of the surroundings such that the overall entropy of the isolatedsystem increases in accordance with the second law of thermodynamics.
In classical thermodynamics, the concept of entropy is defined phenomenologically by the second law of thermodynamics, which states that the entropy of an isolatedsystem always increases or remains constant.
The more such microstates, the greater is the probability of the system being in the corresponding macrostate.
If we now redefine this as a single system (without actually mixing the two gases), then the entropy of the new system will be S = S1 + S2 but the number of microstates will be the product Ω1Ω2 because for each state of system 1, system 2 can be in any of Ω2 states.
Phase-change energy increases the entropy of a substance or system because it is energy that must be spread out in the system from the surroundings so that the substance can exist as a liquid or vapor at a temperature above its melting or boiling point.
When this process occurs in a 'universe' that consists of the surroundings plus the system, the total energy of the 'universe' becomes more dispersed or spread out as part of the greater energy that was only in the hotter surroundings transfers so that some is in the cooler system.
The more microstates available to the system, among which energy can be shared, the greater the entropy (disorder).
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