Entropy, Enthalpy and Free Energy. What is a Spontaneous Process?

Entropy is a measure of possibilities. A high entropy system has many accessible states. A low entropy system has few accessible states. Adding heat will increase the number of accessible molecular rotational, translational and vibrational states. So heating a system will increase its entropy. For a gas, increasing the volume will increase the number of accessible states. Increasing the volume of a gas will increase its entropy.

There are many analogies to these phenomena in our everyday lives. A messy room has high entropy because there are many possibly ways to be messy. Toys and clothes can be laying around anywhere within the room. A neat room has low entropy because there are few ways to be neat. All toys and clothes must be in their proper places.

A process can be driven, or forced to happen, only by increasing entropy. A driven process is said to be "spontaneous". But there is an important accounting issue: entropy can increase in one place somewhere and increase in another. A process is spontaneous only if there is a net increase in entropy. So we say a process is spontaneous if it increases the entropy of the universe. By involking the universe, we are saying that we are accounting for all of the entropy change assocated with that process. A few spontaneous processes are a falling brick, a bouncing ball, a crashing car, a melting ice cube on a warm day, freezing water on a cold day, or the chemical transformations of a burning match. These processes go in one direction only. You can reverse them only by coupling them to some other process that increases in entropy.

The entropy of the universe increases during each of these spontaneous processes:

ΔSuniv > 0 (equation 1)

The change in entropy of the universe is separable into two parts. Part one is the change in the entropy of the "system". Part two is the change in entropy of the "surroundings".

ΔSuniv = ΔSsys + ΔSsur (equation 2)

The change in entropy of the surroundings (ΔSsur) is caused by dissipation of heat. Dissipation of heat drives many processes in our daily lives. A basketball might spontaneously roll off a table, bounce a couple of times, and come to rest on the floor. During this process the initial potential energy of the basketball is converted to kinetic energy, then to thermal energy (heat). The heat is dissipated. That heat dissipation is what drives the process in one direction, and keeps it from going in the reverse direction. Once a basket ball is at rest on the floor, it will not spontaneously gather heat from the surroundings, make the conversion from thermal to kinetic energy and bounce back onto the table.

Temperature matters. The change in entropy of the surroundings (ΔSsur) is the heat evolved (q) divided by the temperature (T).

ΔSsur = q/T (equation 3)

This equation says that the entropy gain when heat is dissipated is greater at low temperature than at high temperature. To help understand this concept use money is an analogy for heat. Dissipating ten dollars (by spending it) buys a lot when money is dear (before inflation) but buys very little when money is cheap (after inflation). In the same way, dissipating a given amount of heat generates a lot of entropy when heat is dear (when it is cold) but generates less entropy when heat is cheap (when it is hot).

The temperature dependence of the entropy gain during heat dissipation is subtle but extremely important. It explains why the direction of a spontaneous process can change. At high temperature duplex DNA spontaneously melts, proteins spontaneously unfold, ice spontaneously melts. At low temperature DNA spontaneously anneals, proteins spontaneously fold, water spontaneously freezes. Let's think carefully about liquid water and ice. At all temperatures, the entropy of water molecules (the system) decreases upon freezing because water molecules are more ordered in the crystalline state than in the liquid. Water molecules have more rotational and translational freedom in liquid than in the solid. So ΔSsys always pushes ice to water. But when water freezes, the heat of fusion is released to the surroundings. So ΔSsur always pushes water to ice. The ΔSsur is greater at low temperature (T down => q/T up) than at high temperature (T up => q/T down) . (Here we assume that the heat of fusion doesn't change much with temperature.) So the increase in the entropy of the surroundings is greater at low temperature than at high temperature. At low temperature (<O°C) ΔSsur > -ΔSsys and water spontaneously freezes. At high temperature (<O°C) ΔSsur < -ΔSsys and ice spontaneously melts.

At constant pressure the heat evolved is equivalent to the negative of the enthalpy change of the system.

q = -ΔHsys (equation 4)

combining equations 3 and 4 gives

ΔSsur = -ΔHsys/T (equation 5)

combining equations 2 and 5 gives

ΔSuniv = ΔSsys + -ΔH/T (equation 6)

-TΔSuniv = ΔHsys - TΔSsys (equation 7, rearranged 6)

The condition for a spontaneous process is -TΔSuniv < 0.

Let's invent a new function, the Gibbs Free Energy, G: ΔG=-TΔSuniv

ΔG = ΔHsys - TΔSsys (equation 8)

The condition for a spontaneous process is ΔG < 0, still an increasing entropy of the universe. In general we leave off the sys and write

ΔG = ΔH - TΔS (equation 9)