9.6 Entropy and the Second Law of Thermodynamics

1. AIt increases, because energy spreads from the spoon into the water. Correct
   Energy transfer from a hot object to a cooler one is irreversible, and the energy becomes more spread out, so by the second law the total entropy of the isolated system increases. It stops increasing when the spoon and the water reach the same temperature.
2. BIt stays the same, because no energy enters or leaves the container.
   A student who thinks entropy can change only when energy crosses the system's boundary picks this. No energy leaves the container, but energy spreads within it, from the spoon into the water, and that increases the total entropy.
3. CIt decreases, because the temperature of the spoon decreases.
   A student who judges the whole system by the spoon alone picks this. The spoon's entropy does decrease, but the water's increases by more, so the total entropy of the isolated system increases.
4. DIt increases at first, then decreases as the temperatures become equal.
   A student who thinks entropy is greatest while energy is moving fastest picks this. The total entropy of an isolated system never decreases; it rises until the spoon and the water share one temperature and then stays at that maximum.

CED 9.6.A.1 · Read this in Fix

1. AThe total energy is less, because some was used up in stopping the car.
   A student who thinks energy is used up picks this. The kinetic energy is not destroyed; it becomes internal energy of the brakes, the road and the air, so the total energy of the isolated system is unchanged.
2. BThe total energy is the same, but less of it is available to do work. Correct
   Energy is conserved in the isolated system, so the total is unchanged. But the car's kinetic energy, which could have done work, has spread out as internal energy of the brakes, the road and the air, and it can no longer all be used to do work: the entropy of the system has increased.
3. CThe total energy is the same, and all of it is still available to do work.
   A student who thinks conservation of energy keeps all the energy usable picks this. The energy is still there, but spread out as internal energy of many objects it cannot all be gathered back to set the car moving; less of it is available to do work.
4. DThe total entropy is the same, because the total energy is the same.
   A student who thinks entropy is conserved like energy picks this. Braking is irreversible and spreads the energy out, so the total entropy of the isolated system increases while its total energy stays the same.

CED 9.6.A.2 · Read this in Fix

1. AThe system was not isolated during the process. Correct
   The entropy of an isolated system never decreases. A decrease is possible only if the system interacts with its surroundings, with energy transferred out of it, as when a gas is cooled or is compressed while giving energy to its surroundings.
2. BThe second law of thermodynamics was violated.
   A student who thinks the second law forbids any system's entropy from decreasing picks this. The law applies to isolated systems; a system that exchanges energy with its surroundings can decrease in entropy while the total entropy of system and surroundings does not decrease.
3. CThe temperature of the system became lower.
   A student who thinks entropy depends only on temperature picks this. The temperature need not fall: a gas compressed slowly at constant temperature, giving energy to its surroundings, decreases in entropy.
4. DThe system moved closer to thermodynamic equilibrium.
   A student who thinks equilibrium is an orderly, low-entropy state picks this. Nothing requires it: a gas can be compressed so slowly that it is in equilibrium at every stage, and its entropy still decreases. What a decrease does require is an interaction with the surroundings.

CED 9.6.A.3 · Read this in Fix

1. AGraph 1
   A student who thinks the energy of the hot section is used up picks the graph in which the whole rod returns to its starting temperature. The rod is insulated, so the energy stays in it and spreads out; the parts near the middle must end up warmer than they started.
2. BGraph 2
   A student who thinks heat carries its temperature with it picks the graph in which the whole rod reaches the temperature of the hot section. The same energy spread over more of the rod raises each part less; bringing the whole rod to the peak temperature would need far more energy than was supplied.
3. CGraph 3
   A student who thinks nothing changes inside an isolated object picks the graph in which the temperature profile stays the same. No energy crosses the insulation, but energy still spreads within the rod from the hot section to the cooler ones.
4. DGraph 4 Correct
   Energy concentrated in the heated section spreads along the rod: the middle cools and the neighboring sections warm. The rod is insulated, so no energy is lost, and the hump becomes lower and wider rather than disappearing.

Working The rod is insulated, so its total internal energy is constant; energy spreads from the heated section into the cooler neighboring sections. The peak falls and the hump widens, with the area between the curve and the starting-temperature level (proportional to the energy added) unchanged. The correct graph shows a lower, wider hump.

CED 9.6.A.2.i · Read this in Fix

1. ANegative, because net energy is transferred out of the gas by cooling
   A student who ties the gas's entropy change to the energy transferred by heating and cooling picks this. For this counterclockwise process the net work done on the gas is positive, so the gas does lose net energy by cooling, but its entropy depends only on its state, which is the same at the end of the cycle as at the start.
2. BPositive, because the entropy of any system increases in a real process
   A student who applies 'entropy always increases' to any system picks this. The gas exchanges energy with its surroundings, so it is not isolated; the second law limits the total entropy of the gas and its surroundings, and the gas itself returns to its starting entropy.
3. CZero, because the gas ends the process in the same state it started in Correct
   Entropy is a state function: it depends only on the state of the gas. The diagram shows the process ending back at state a, with the same P, V and T, so its entropy is the same as at the start, whatever energy was transferred on the way.
4. DPositive if the process is irreversible, and zero only if it is reversible
   A student who thinks irreversibility adds to the system's own entropy change picks this. An irreversible cycle increases the entropy of the surroundings, but the gas itself returns to state a, so its entropy change is zero either way.

CED 9.6.A.2.ii · Read this in Fix

1. ASample 2's is greater, because its expansion is irreversible.
   A student who thinks an irreversible process gives the system a larger entropy change picks this. The free expansion does increase the total entropy of gas and surroundings more, but each gas's own entropy change is fixed by its end states, which are the same.
2. BThe two are equal, and both are greater than zero. Correct
   Both samples end in the same state: the same amount of gas at the same temperature in twice the volume. Entropy is a state function, so the two entropy changes are equal, and they are positive because each gas's energy is spread through a larger volume.
3. CSample 1's is greater, because only sample 1 is heated.
   A student who ties entropy change to heating picks this. Sample 1 is heated as it expands and sample 2 is not, but entropy depends only on the state, and the two samples end in the same state.
4. DBoth are zero, because neither sample changes temperature.
   A student who thinks entropy depends only on temperature picks this. Each gas occupies twice the volume, so its energy is more spread out and its entropy is greater, even though its temperature is unchanged.

CED 9.6.A.2.ii · Read this in Fix

1. ABefore the partition is removed, while the gas is on one side
   A student who thinks an even spread is orderly, and so low in entropy, picks this. The confined gas has its energy and matter in a smaller region; spreading through the whole box increases the entropy.
2. BWhile the gas is rushing into the empty side most rapidly
   A student who thinks entropy is greatest while things change fastest picks this. The entropy is still increasing while the gas rushes in; it is greatest once the gas has spread evenly and stopped changing.
3. CAt every time equally, since no energy crosses the walls
   A student who thinks entropy changes only when energy crosses the boundary picks this. No energy enters or leaves the box, but the gas spreads within it, and that increases its entropy.
4. DAfter the gas has spread out evenly through all of the box Correct
   The gas is an isolated system that moves toward equilibrium, and its entropy increases as it spreads. The entropy reaches its maximum when the gas is in equilibrium, spread evenly through the whole box, and stays there.

CED 9.6.A.2.iii · Read this in Fix

1. AIt is right: heat rises, so energy will gradually collect near the top of the flask.
   A student who thinks heat rises picks this. Warmer fluid rises only when it is surrounded by cooler, denser fluid; in water at a uniform temperature there is no such difference, and energy does not collect at the top.
2. BIt is right, as long as the energy gained by the top equals the energy lost by the bottom.
   A student who thinks conservation of energy is the only rule picks this. Such a change could conserve energy, but it would decrease the entropy of an isolated system, so it does not happen.
3. CIt is wrong: the water is already in equilibrium, which an isolated system never leaves by itself. Correct
   Water at one uniform temperature is in thermodynamic equilibrium, with its maximum entropy. Separating into warmer and cooler regions would concentrate energy and lower the entropy of an isolated system, which the second law rules out.
4. DIt is wrong: energy cannot move from one part of an isolated system to another.
   A student who thinks nothing can change inside an isolated system picks this. Energy can move within an isolated system; that is how hot and cold regions inside one come to equilibrium. The prediction fails because the water is already in equilibrium.

CED 9.6.A.3.i · Read this in Fix

1. AGraph 1
   A student who thinks the entropy of every system increases picks the graph in which S rises. The coffee is not isolated; energy leaves it, so its entropy falls, while the entropy of the room increases by more.
2. BGraph 2
   A student who thinks entropy is conserved picks the graph in which S stays constant. The coffee loses energy by cooling, so its entropy decreases.
3. CGraph 3
   A student who thinks the coffee cools at a steady rate until it reaches room temperature picks this. The coffee's entropy does decrease, but energy leaves more slowly as the temperature difference shrinks, so the graph flattens gradually rather than stopping abruptly.
4. DGraph 4 Correct
   The coffee is a closed system: energy leaves it by cooling, so its entropy decreases. The coffee cools more and more slowly as it approaches room temperature, so its entropy decreases more and more slowly and levels off.

Working The coffee (covered, so no matter leaves) is a closed system. Energy leaves it by cooling, so its entropy decreases; the rate of cooling falls as its temperature approaches room temperature, so S decreases ever more slowly and levels off; a straight-line decrease is ruled out because the rate of energy transfer falls with the temperature difference. The correct graph shows S decreasing and leveling off.

CED 9.6.A.3.ii · Read this in Fix

1. AThe claim is right: the second law forbids any decrease in the entropy of a system.
   A student who applies the second law to every system picks this. The law forbids a decrease in the total entropy of an isolated system; a closed system such as the water can decrease in entropy while energy is transferred out of it.
2. BThe claim is wrong: cold flows into the water from the freezer, so the water is not left to itself.
   A student who thinks of cold as a substance picks this. Nothing called cold enters the water; the freezer removes energy from it, which is why the water's entropy decreases and why the water is not an isolated system.
3. CThe claim is wrong: the water is not isolated, and the energy it loses spreads into its surroundings. Correct
   The second law applies to an isolated system, and the water is not one: energy is transferred out of it into the freezer and on into the room. The entropy of the surroundings increases by more than the water's entropy decreases, so the total entropy does not decrease.
4. DThe claim is wrong: the energy the water loses is conserved, and that is all the law requires.
   A student who treats the second law as a statement about energy conservation picks this. Conservation of energy is the first law; the second law concerns entropy, and it is satisfied here because the surroundings gain more entropy than the water loses.

CED 9.6.A.3.ii · Read this in Fix