The second law (but not the frst law) allows one to predict the direction of a particular reaction. But to make it useful for this purpose, one needs a convenient measure of the probability or, equivalently, the degree of disorder of a state. The entropy (S) is such a measure. It is a logarithmic function of the probability such that the change in entropy (˜S) that occurs when the reaction A
B converts one mole of A into one mole of B is where pA and pB are the probabilities of the two states A and B, R is the gas constant (8.31 J K–1 mole_1), and ˜S is measured in entropy units (eu). In our initial example of 1000 coins, the relative probability of all heads (state A) versus half heads and half tails (state B) is equal to the ratio of the number of different ways that the two results can be obtained. One can calculate that pA = 1 and pB = 1000!(500! x 500!) = 10299. Therefore, the entropy change for the reorientation of the coins when their container is vigorously shaken and an equal mixture of heads and tails is obtained is R In (10298), or about 1370 eu per mole of such containers (6 x 1023 containers). We see that, because ˜S defned above is positive for the transition from state A to state B (pB /pA > 1), reactions with a large increase in S (that is, for which ˜S > 0) are favored and will occur spontaneously.
As discussed in Chapter 2, heat energy causes the random commotion of molecules. Because the transfer of heat from an enclosed system to its surroundings increases the number of different arrangements that the molecules in the outside world can have, it increases their entropy. It can be shown that the release of a fxed quantity of heat energy has a greater disordering effect at low temperature than at high temperature, and that the value of ˜S for the surroundings, as defned above (˜Ssea), is precisely equal to h, the amount of heat transferred to the surroundings from the system, divided by the absolute temperature (T ):