Lesson 9.8 · Unit 9 · CED topic 9.6
Entropy and the second law of thermodynamics
Film a glass falling off a table and shattering, then run the film backward. You know at once which is real, though the backward film does not break the first law of thermodynamics: every joule of energy is accounted for either way. Something else tells the two directions apart. Hot coffee cools to room temperature but never warms itself up by cooling the room; a drop of ink spreads through water but never gathers itself back into a drop. The first law says energy is conserved. The second law says which processes actually occur. This lesson introduces entropy, the quantity that tracks that one-way tendency, and uses it to explain why heat engines cannot be perfectly efficient. The treatment is qualitative.
Describe the change in entropy for a given system over time.
- The second law of thermodynamics states that the total entropy of an isolated system can never decrease and is constant only when all processes the system undergoes are reversible.
- Entropy can be qualitatively described as the tendency of energy to spread or the unavailability of some of the system’s energy to do work. Localized energy will tend to disperse and spread out. Entropy is a state function and therefore depends only on the current state or configuration of a system, not on how the system reached that state. Maximum entropy occurs when a system is in thermodynamic equilibrium.
- The change in a system’s entropy is determined by the system’s interactions with its surroundings. Isolated systems spontaneously move toward thermodynamic equilibrium. The entropy of an isolated system never decreases, but the entropy of a closed system can decrease because energy can be transferred into or out of the system.
- Boundary: only a qualitative treatment of the second law is within the scope of AP Physics 2.
Why some processes never run backward
Take 10 gas atoms in a box, with a removable divider down the middle. Place all 10 on the left and remove the divider. Each atom now wanders at random, and at any instant it is equally likely to be on either side. How many arrangements put all 10 on the left? Exactly one. How many put 5 on each side? 252. Counting the possible ways to arrange the atoms (called microstates), the near-even splits dominate, and the lopsided ones are rare. The gas does not “want” to spread out; it ends up spread out because the spread-out arrangements are overwhelmingly more numerous.
With 10 atoms, the chance of finding all of them back on the left is about 1 in 1000. With a mole of atoms (\(6\times10^{23}\)), it is far smaller than one in \(10^{10^{20}}\), a number with so many digits that it is never seen in the lifetime of the universe. This is why free expansion of a gas is irreversible even though each atom’s motion is reversible in principle.
Entropy: the spreading of energy
Entropy measures how spread out the energy of a system is, equivalently, how many microscopic arrangements are consistent with its macroscopic state. At the level of this course it is enough to say it this way: entropy increases when energy that was localized becomes dispersed. A hot object next to a cold one holds energy concentrated in one place; after contact, the energy is shared evenly, with a higher entropy. A compressed gas spreads out into a larger volume, with higher entropy. A pot of boiling water cooling in a room spreads its energy into the room.
- State function. Entropy depends only on the state of the system (its \(P\), \(V\), \(T\), composition), not on the path. If a gas goes from state A to state B, the entropy change is the same by any process.
- Maximum at equilibrium. The entropy of an isolated system rises until the system reaches thermodynamic equilibrium, where it is as large as it can be. Nothing further happens spontaneously.
- Unavailable energy. Energy that has dispersed evenly cannot be gathered to do work. A lake of warm water holds enormous energy but you cannot run a heat engine from it without a colder place to dump energy.
The second law
The second law of thermodynamics states that the total entropy of an isolated system never decreases; it remains constant only if every process in the system is reversible (idealized, frictionless and infinitely slow). All real processes, including those with friction, heat flow across a temperature difference, mixing, and free expansion, are irreversible, and they increase entropy.
An important distinction: the law applies to an isolated system. A closed system can exchange energy with its surroundings, and then its entropy can decrease. When water freezes in a freezer, the water’s entropy decreases (it becomes more ordered), but the freezer expels energy into the kitchen, and the entropy of the kitchen increases by more than the water’s entropy decreases. The total, water plus kitchen plus freezer, increases. Living things also decrease their own entropy while increasing the entropy of their surroundings.
Heat engines and the second law
A heat engine takes energy \(Q_H\) by heating from a hot reservoir, converts part of it to useful work \(W_{out}\), and must dump the rest, \(Q_C\), to a cold reservoir. Over a complete cycle the engine returns to its initial state, so \(\Delta U = 0\), and conservation of energy gives \[ Q_H = W_{out} + Q_C. \] The efficiency is \(e = W_{out}/Q_H = 1 - Q_C/Q_H\). The second law says \(Q_C\) cannot be zero. If it were, the engine would convert all of the energy from a single reservoir into work, and entropy would fall in the hot reservoir without any increase elsewhere. The energy you must dump into the cold reservoir spreads out and raises entropy, which pays for the entropy removed from the hot reservoir. So no engine can be 100% efficient. The cycle of lesson 9.6 had \(Q_H = 18000\) J and \(W_{out} = 4000\) J, so \(e = 22\%\) and 14000 J had to be rejected.
A refrigerator runs the cycle backward (counterclockwise on the PV diagram): work is done on the system and energy is carried from the cold interior to the warm room. That does not violate the second law, because the work supplied produces extra entropy elsewhere: the room receives more energy than was removed from the food. Energy does not flow from cold to hot by itself, only when something else changes.
A heat engine absorbs 600 J from a hot reservoir in each cycle and rejects 450 J to the cold reservoir. Find the work output and the efficiency, and decide if the engine violates the first or second law if it claimed to produce 200 J.
Work. \(W_{out} = Q_H - Q_C = 600 - 450 = 150\) J, so \(e = 150/600 = 0.25\) (25%).
Claim of 200 J. \(Q_H - W_{out} = 400\) J would have to be rejected, not 450 J, so the energy would not balance; the claim as stated violates the first law (conservation of energy) for the given \(Q_C\). A claim of 600 J of work with \(Q_C = 0\) would conserve energy but violate the second law.
An isolated box has a gas on the left half and a vacuum on the right. A divider is removed. A student claims the gas will later (a) fill the box evenly and (b) after a while, all go back to the left half by itself. Which is consistent with the second law?
(a) The gas spreads out: the entropy increases and the system approaches equilibrium. Allowed and observed. (b) The gas returning to the left half would decrease the entropy of an isolated system. Not allowed in practice (the probability is astronomically small), even though energy would be conserved.
An ice cube melts in a glass of warm water inside an insulated container. Explain in terms of entropy why the final state has the temperature of both parts equal, and why the reverse (a warm cube and a cooler glass reappearing) does not happen.
Argument. The glass and cube form an isolated system. At first the energy is concentrated in the warmer water. When energy spreads from the water to the ice, the system moves toward a state with the same temperature everywhere, in which the energy is as dispersed as possible; that is thermodynamic equilibrium, the state of maximum entropy. The reverse process would take the system to a state with less dispersed energy and a lower entropy, which the second law forbids for an isolated system.
- Second law. Total entropy of an isolated system never decreases; constant only for reversible processes.
- Meaning. Entropy measures dispersal of energy; spreading raises entropy; equilibrium is maximum entropy.
- State function. Depends only on the current state, not the path.
- Closed versus isolated. A closed system can lose entropy if energy flows out, as with freezing water; the surroundings gain more.
- Engines. \(Q_H = W + Q_C\); \(e = 1 - Q_C/Q_H \lt 1\). Some energy must be rejected to a cold reservoir.
- A heat engine takes in 800 J per cycle and does 200 J of work. How much energy is rejected, and what is its efficiency?
Show answer
\(Q_C = 800 - 200 = 600\) J; \(e = 200/800 = 25\%\).
- Does the entropy of a hot cup of coffee in a room increase or decrease as it cools? What about the entropy of the coffee plus the room?
Show answer
The coffee’s entropy decreases as it loses energy, but the room’s entropy increases by more. The total (coffee plus room) increases.
- True or false: entropy depends on the path that takes a gas from one state to another. Explain.
Show answer
False. Entropy is a state function, so the change depends only on the initial and final states.
- A student claims a device removes 500 J from a lake and converts it entirely to work, with no other change. Which law does the claim violate?
Show answer
The second law. A cyclic device cannot convert energy from a single reservoir entirely to work; some energy must go to a colder reservoir. Energy is conserved, so the first law is not violated.
- Why can entropy decrease inside a refrigerator while the second law remains valid?
Show answer
The refrigerator is not isolated: work is supplied, and more energy is released into the room than is removed from the food. The entropy of the room increases by more than the entropy of the contents decreases, so the total increases.
- Four atoms are in a box with a divider removed. Estimate whether it is more probable to find all four on the left, or two on each side. Explain.
Show answer
Two on each side is far more probable: there are 6 arrangements giving 2 and 2, but only 1 giving all four on the left. Equal spreading corresponds to more microstates, which is why spread-out states have higher entropy.