20.7.3 Reasoning with Entropy
Heating a system increases its thermal energy. By increasing the thermal energy, we also increase the ability of the system to disperse thermal energy, so we increase the entropy. Similarly, removing heat energy from a system reduces its ability to disperse thermal energy, so the entropy is reduced.
The entropy can also change when there is no change in thermal energy. FIGURE 20.18 shows an insulated container of a gas with the molecules kept on one side by a thin membrane. The container is an isolated system; no heat is transferred to or from the environment. If the membrane breaks, the gas expands to fill the container—what we call a free expansion. The volume has changed and the pressure will change. But if we think about energy, we can see that this expansion does not change the gas temperature. No work is done because the expanding gas isn’t pushing against anything, and no heat is transferred from the environment, so the first law of thermodynamics tells us there’s no change in thermal energy and hence no change in temperature. But the system’s entropy has increased! The thermal energy may not have changed in value, but it’s more spatially dispersed because each molecule has more room to move around. And a larger dispersal of energy corresponds to greater entropy. The fact that entropy increases makes the free expansion an irreversible process; the reverse process of all the gas molecules moving into the left half of the container never occurs spontaneously.
FIGURE 20.18 The free expansion of a gas increases its entropy.
Something else that changes is the ability of the gas to do work. Instead of breaking the membrane, we could have allowed the compressed gas to do work by pushing on a piston. This would have transferred some of the thermal energy to the environment by doing mechanical work. After expanding freely, the gas has the same thermal energy but less ability to do work. In general, our ability to extract useful work from thermal energy depends not only on the quantity of thermal energy but also on the system’s entropy. Increased entropy, with the energy more dispersed, means less ability to do work.
EXAMPLE 20.7 Does entropy change when ice melts?
A phase change, such as melting or freezing, occurs at a fixed temperature. What happens to the entropy of the water if a block of frozen water at 0°C becomes liquid water at 0°C?
SOLVE The entropy can change, even at the same temperature, if one of the phases of matter—solid or liquid in this case—is able to spread out the energy more. Ice is a solid, so water molecules are frozen into a lattice. Molecules jiggle around—that’s what thermal energy is—but their energy is confined to small volumes around the lattice positions. In contrast, the molecules in liquid water can move freely throughout the volume of the liquid. This allows the thermal energy to be more spatially dispersed, just as it was for the free expansion of a gas. Thus the entropy increases as ice melts.
REVIEW This result makes sense. You know that you need to heat ice to melt it, so you expect this change to increase the entropy.