The first law of thermodynamics, ΔU = q + w, conserves energy. It does not, however, give any information about whether a reaction can or cannot occur. For example, at room temperature and pressure, solid H₂O melts spontaneously. If you take an ice cube from your freezer and place it on the countertop, it shouldn't surprise you to find it as a puddle of water a couple of hours later. Unless your house is very cold, ≤ 273 K, you shouldn't have ever observed liquid water freeze into ice under the same conditions, nor should you expect to see this happen in the future. We say that the phase transition

ice ---> water

is thermodynamically spontaneous under these conditions. It also happens to be endothermic; it takes about 6.0 kJ to melt a mole of ice at 273 K and another 1.8 kJ to warm it up to 298 K, for a total of ΔH = 7.8 kJ/mole.

As you might imagine, the reverse transition

water ---> ice

is not spontaneous under these same conditions and, in fact, is thermodynamically forbidden.You can still calculate an enthalpy change associated with the phase transition in this direction; at 298 K and1 bar ΔH = - 7.8 kJ/mole for water turning into ice.

Because we are familiar with water, it is not hard for us to predict the conditions under which ice will melt or liquid water will freeze. However, what about the reaction:

At 298 K and 1 bar, the reaction is exothermic, ΔH = -311 kJ, as it is written and it is also spontaneous. Obviously both endothermic and exothermic reactions can be spontaneous, or can be thermodynamically forbidden. The first law tells us that we must conserve energy, but gives us no information about whether a phase transition or chemical reaction can occur under the conditions at hand.

To find out which phase transitions or chemical reactions are thermodynamically allowed, we define a new quantity,the entropy, S

In this equation, dq_rev represents a differential change in reversible heat transfer and both q_rev and T are variables. A "reversible" change is one in which everything occurs infinitely slowly so that even though the system is changing, each incremental change is so small as to be only infinitesimally different than the step before it. At each step in the change, reactions going in the forward and reverse direction in must be in equilibrium and dS must equal zero throughout.

All other changes are irreversible. For irreversible processes, pressure and temperature gradients develop during the change, leading to inefficiencies. While only one possible reversible pathway connects a change from state 1 (where the system begins, e.g. the reactants) to state 2 (where the system ends up, e.g. the products), there are infinite numbers of irreversible pathways linking the two states. For all of these pathways ΔU must be the same since U is a state function and the initial and final states are the same. However, different values of q and w can be transferred in each of the different pathways as long as ΔU = q + w.

If we concentrate on macroscopic changes, we can state the second law of thermodynamics:

In words, the second law states that for a process to be thermodynamically allowed, the entropy of an adiabatic (thermally isolated) system must either remain unchanged or it must increase. It cannot decrease; this is thermodynamically forbidden. For systems that can communicate with their surroundings - ones that can exchange information in the form of heat - the sum of the changes in entropy of the system plus the surroundings must either remain unchanged or increase

We will gain a deeper appreciation for what entropy is and what happens when a system gains or loses entropy in the next section. But for now, we need to consider the implications of the inequality : ΔS ≥ 0. The "Δ" implies a change from an initial state to a final equilibrium state. However ΔS = 0 does not necessarily mean that nothing has happened. There can still be a phase change or a chemical reaction, as long as the change has occurred reversibly. And just because that ΔS_universe ≥ 0 is a requirement for change to occur, it does not mean that it is impossible to have a spontaneous reaction in which ΔS for the reaction is negative. The system (the reaction), itself, can gain or lose entropy as long as it is balanced out by an equal or larger positive change in entropy of the surroundings (the laboratory).

Since entropy is a state function, we can use Hess's law to find ΔS for a phase transition or reaction by subtracting the initial state (reactants) entropies from the final state (products) entropies. Unlike enthalpy we will be able to find absolute entropy values and will not need to rely on entropies of formation. A tutorial on entropy calculations, including an abbreviated thermodynamic table, can be found by clicking here.