Entropy, given the symbol S, is a state function which is a measure of the disorder of a system.

i.e. the larger the value of S of a system, the more disordered it is. Entropy is given a thermodynamic definition in terms of the change in entropy dS that occurs as a result of some process.

The definition is based upon the idea that a change in the distribution of energy in a system must depend upon the quantity of energy, dq , transferred as heat. (Heat stimulates disorderly motion of particles, resulting in redistribution of energy. Work, on the other hand, stimulates orderly motion of particles. It thus does not change the relative distribution of energy in the system, and hence does not affect the degree of disorder or entropy of the system.)

Further, we consider that transfer of a given quantity of heat is likely to have more effect at lower temperatures. This is most simply understood by considering that the heat transferred to a system will be much smaller relative to the thermal motion of particles already present in a hot system than in a cold one where the thermal motion of particles is much less to start with. (As an analogy, a pebble produces more noticeable ripples when thrown into a still pond than when thrown into a turbulent river.) This suggests an inverse relationship between the entropy and the temperature. We thus give entropy the thermodynamic definition:

which integrates to give the following expression for a measurable change:

where the integral is evaluated between the final and initial states of the system.

Technically, the above expression can apply to either a system or its surroundings. However, it can be used to derive an expression specifically for ΔS_sur. We consider an infinitesimally small transfer of heat to the surroundings dq_sur. Usually the surroundings (commonly the rest of the universe) are at constant volume, allowing us to equate dq_sur with the change in the internal energy of the surroundings, dU_sur. U is a state function, meaning that dU_sur is an exact differential, and from this it follows that the value of dU_sur is independent of how the change occurred, in particular whether the heat transfer occurred reversibly or not. Since dq_sur is equal to dU_sur, it follows that dq_sur must also be independent of the manner in which the transfer occurred, allowing us to write:

Since the temperature of the surroundings is constant whatever the change (due to their huge size compared to the amount of heat being supplied to them) , we may integrate this expression to give, for any measurable change:

An alternative way to derive this expression is to consider the surroundings as being at constant pressure. dq_sur may then be equated with dH_sur. H, like U, is a state function, and the argument is then totally analogous. The use of this method gives precisely the same result, but along the way throws up a relation which sometimes proves useful in thermodynamic calculations at constant pressure: