Entropy and Second Law of Thermodynamics
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AP Physics 2 › Entropy and Second Law of Thermodynamics
Two identical copper blocks, one at $350,\text{K}$ and the other at $300,\text{K}$, are placed in contact and isolated from the environment. They reach $325,\text{K}$; the process is spontaneous. Which statement best accounts for the sign of the total entropy change?
The total entropy increases because the magnitude of $\Delta S$ for the cooler block exceeds that of the warmer block.
The total entropy is zero because the energy lost by one block equals the energy gained by the other.
The total entropy decreases because energy flows from hot to cold, reducing temperature differences.
The total entropy increases only if the blocks melt, since entropy change requires a phase change.
Explanation
This question tests understanding of entropy and the second law of thermodynamics. When two blocks at different temperatures equilibrate in isolation, heat flows from hot to cold spontaneously. The second law requires total entropy to increase for this spontaneous process. The cooler block gains heat Q at lower temperature (300 K), while the hotter block loses the same heat Q at higher temperature (350 K), so |ΔS_cold| = Q/300 > |ΔS_hot| = Q/350, making the total entropy change positive. Choice A incorrectly assumes reducing temperature differences decreases entropy, confusing uniformity with total entropy. The principle is: when equal amounts of heat transfer occur, the entropy gain at lower temperature exceeds the entropy loss at higher temperature.
A proposed device operates in a cycle, absorbing $500,\text{J}$ from a single $400,\text{K}$ reservoir and producing $500,\text{J}$ of work each cycle (claimed spontaneous). Which conclusion is consistent with the second law?
The device is possible if its internal entropy decreases each cycle.
The device is impossible because a cyclic engine cannot convert heat from one reservoir entirely into work.
The device is possible because it conserves energy and cycles back to its initial state.
The device is possible if the reservoir is large enough to supply constant temperature.
Explanation
This question tests understanding of entropy and the second law of thermodynamics. The proposed device violates the Kelvin-Planck statement of the second law: no cyclic engine can convert heat from a single reservoir entirely into work. Such a device would decrease the entropy of the universe (removing heat/entropy from the reservoir without adding it elsewhere), making it impossible. All cyclic heat engines require at least two reservoirs at different temperatures, rejecting some heat to the cold reservoir. The Carnot efficiency limit shows that even ideal engines cannot achieve 100% conversion when operating between finite temperatures. Choice A incorrectly focuses only on energy conservation, ignoring entropy constraints—the second law imposes additional restrictions beyond the first law. When evaluating proposed devices, check both energy conservation and entropy requirements.
A student proposes a cyclic device that absorbs $100,\text{J}$ from a single thermal reservoir at $300,\text{K}$ and converts all of it into $100,\text{J}$ of work each cycle. The student claims the process is spontaneous once started. Which statement best explains why this proposal fails?
It fails because work is a form of heat, so $Q$ cannot be fully converted into $W$.
It fails because the reservoir’s temperature would have to increase as it loses heat.
It fails because it would require the entropy of the universe to decrease for a complete cycle.
It fails because the efficiency of any engine must exceed $100%$ to be useful.
Explanation
This question tests understanding of entropy and the second law of thermodynamics. The proposed device attempts to convert heat entirely into work using a single reservoir, which violates the Kelvin-Planck statement of the second law. For a cyclic process extracting heat Q from a single reservoir, the entropy change would be ΔS_universe = -Q/T < 0, decreasing the total entropy of the universe, which is forbidden for spontaneous processes. Choice B incorrectly claims work is a form of heat, confusing different energy transfer mechanisms. The key principle is: no cyclic device can convert heat entirely to work while operating with a single thermal reservoir.
A $1.0,\text{kg}$ ice–water mixture at $0^\circ\text{C}$ is left in a $20^\circ\text{C}$ room. Heat flows from the room into the mixture and eventually all the ice melts; this process is spontaneous. Which conclusion is consistent with the second law for the room+mixture system?
The total entropy is zero because melting is a phase change at constant temperature.
The total entropy increases only if the ice appears more disordered than liquid water.
The total entropy decreases because the system becomes more uniform after melting.
The total entropy increases because heat transfer from warmer air to colder ice-water yields net positive $\Delta S$.
Explanation
This question tests understanding of entropy and the second law of thermodynamics. When ice melts in a warm room, heat flows spontaneously from the warmer air to the colder ice-water mixture. The second law requires that total entropy increases for this spontaneous process. The room loses heat at 293 K while the ice-water gains heat at 273 K, so the entropy increase of the cold system exceeds the entropy decrease of the warm room, yielding net positive ΔS. Choice C incorrectly assumes phase changes have zero entropy change, ignoring that melting increases entropy due to increased molecular freedom. The strategy is: for spontaneous heat flow, entropy gain of the cold object exceeds entropy loss of the hot object.
A cup of hot coffee cools from $70^\circ\text{C}$ to room temperature $22^\circ\text{C}$ in still air (spontaneous). Which statement best describes the entropy change of the coffee itself?
The coffee entropy increases because its molecules become more disordered as it cools.
The coffee entropy must increase because the total entropy of the universe increases.
The coffee entropy decreases because it loses thermal energy while cooling.
The coffee entropy remains constant because cooling is spontaneous.
Explanation
This question tests understanding of entropy and the second law of thermodynamics. As hot coffee cools to room temperature, it loses thermal energy to its surroundings, and its entropy decreases. The entropy of any system decreases when it loses heat: ΔS = Q/T, where Q is negative for heat loss. However, the room air gains this heat at a lower temperature, experiencing a larger entropy increase than the coffee's entropy decrease, ensuring total entropy increases. This makes the cooling process spontaneous according to the second law. Choice A incorrectly claims cooling increases molecular disorder—actually, lower temperature corresponds to less thermal motion and lower entropy. Remember: individual system components can decrease in entropy during spontaneous processes, but total entropy must increase.
A $0.50,\text{kg}$ metal block at $400,\text{K}$ is placed in thermal contact with a $2.0,\text{kg}$ water bath at $300,\text{K}$ inside an insulated container. The block cools and the water warms until they reach a common final temperature; the process is spontaneous. Which statement best explains why the process occurs in the direction observed?
The process occurs because the thermal energy transferred to the water exceeds the energy lost by the block.
The entropy of the isolated block–water system must decrease because the final temperature is uniform.
The total entropy of the isolated block–water system increases as energy disperses from hot to cold.
The process occurs because entropy measures disorder, and the water becomes more disordered than the metal.
Explanation
This question tests understanding of entropy and the second law of thermodynamics. When a hot metal block contacts cooler water in an isolated system, heat flows spontaneously from hot to cold until thermal equilibrium is reached. The second law states that the total entropy of an isolated system must increase for any spontaneous process. During this heat transfer, the entropy decrease of the cooling block is smaller in magnitude than the entropy increase of the warming water (since ΔS = Q/T and the water receives heat at a lower temperature). Choice B incorrectly assumes uniform temperature means lower entropy, confusing equilibrium with disorder. The key strategy is: for spontaneous processes in isolated systems, total entropy always increases.
A $0.20,\text{kg}$ metal block at $80^\circ\text{C}$ is placed in contact with a $0.80,\text{kg}$ water bath at $20^\circ\text{C}$ inside a rigid, insulated container. After some time, they reach a common temperature (a spontaneous process). Which conclusion is consistent with the second law?
The total entropy of the isolated block–water system decreases because the system becomes more ordered.
The total entropy of the isolated block–water system must decrease if the final temperature is uniform.
The total entropy of the isolated block–water system increases during the heat transfer.
The total entropy of the isolated block–water system remains zero because energy is conserved.
Explanation
This question tests understanding of entropy and the second law of thermodynamics. When a hot metal block contacts cold water in an isolated system, heat flows spontaneously from hot to cold until thermal equilibrium is reached. During this irreversible process, the total entropy of the isolated system must increase according to the second law. The entropy decrease of the cooling block is smaller in magnitude than the entropy increase of the warming water because entropy changes are inversely proportional to temperature (ΔS = Q/T). Choice B incorrectly assumes that uniform temperature means more order and less entropy, but this confuses microscopic disorder with macroscopic uniformity. Remember: in any spontaneous process in an isolated system, total entropy always increases or remains constant (reversible case only).
A perfectly insulated box contains two equal copper blocks at 350 K and 250 K that are placed in thermal contact; the process is spontaneous. Which statement best describes the entropy change?
The total entropy of the two-block system remains constant because it is isolated.
The total entropy of the two-block system increases until thermal equilibrium is reached.
The total entropy of the two-block system becomes negative because temperature differences create order.
The total entropy of the two-block system decreases because energy flows out of the hot block.
Explanation
This question tests understanding of entropy and the second law of thermodynamics. When two blocks at different temperatures (350 K and 250 K) are placed in thermal contact within an isolated system, heat flows spontaneously from hot to cold until thermal equilibrium is reached. During this process, the hot block loses entropy (ΔS_hot < 0) while the cold block gains entropy (ΔS_cold > 0), but the magnitude of the cold block's entropy gain exceeds the hot block's entropy loss because |ΔS| = |Q|/T is larger for lower temperatures. The total entropy increases throughout the process until equilibrium is reached at 300 K. Choice C incorrectly assumes that isolation means constant entropy—isolation prevents energy exchange with surroundings but doesn't prevent internal entropy increases. For any spontaneous thermal equilibration in an isolated system, total entropy always increases.
A $0.50,\text{kg}$ metal block at $400,\text{K}$ is placed in thermal contact with a $1.0,\text{kg}$ water bath at $300,\text{K}$ inside an insulated container. The block cools and the water warms until they reach a common final temperature; the process is spontaneous. Which conclusion is consistent with the second law?
The total entropy of the block-plus-water system increases because energy flows from higher to lower temperature.
The total entropy of the block-plus-water system must decrease so that the final temperature is between $300,\text{K}$ and $400,\text{K}$.
The total entropy of the block-plus-water system decreases because the system becomes more ordered as it equilibrates.
The total entropy of the block-plus-water system remains constant because the container is insulated.
Explanation
This question tests understanding of entropy and the second law of thermodynamics. When a hot metal block is placed in contact with cooler water in an insulated container, heat flows spontaneously from the hot block to the cold water until thermal equilibrium is reached. According to the second law, the total entropy of an isolated system must increase for any spontaneous process. The entropy increase occurs because energy disperses from the concentrated high-temperature region to the lower-temperature region, creating more accessible microstates overall. Choice B incorrectly claims entropy decreases due to "ordering," confusing macroscopic uniformity with microscopic disorder. Remember: in any spontaneous heat transfer process, total entropy increases even though energy is conserved.
A refrigerator removes 900 J of heat from a 270 K compartment and exhausts heat to a 300 K room; it operates non-spontaneously using electrical work. Which statement best explains why work is required?
Work is required because the room’s entropy must always decrease during cooling.
Work is required because entropy must decrease in any cyclic device.
Moving heat from cold to hot conserves energy but violates the first law unless work is added.
Moving heat from cold to hot would decrease total entropy unless work is added.
Explanation
This question tests understanding of entropy and the second law of thermodynamics. A refrigerator moves heat from a cold space (270 K) to a warmer room (300 K), which is the opposite of spontaneous heat flow. This process would decrease total entropy if done alone: removing heat from the cold compartment decreases its entropy more than adding the same heat to the warm room increases the room's entropy (since |ΔS| = |Q|/T is larger at lower T). To make this process possible while still increasing total entropy, work must be added to the system, which ultimately gets converted to additional heat expelled to the room. Choice B incorrectly claims this violates the first law—energy is conserved, but the second law requires work input. Remember that moving heat from cold to hot requires work input to ensure total entropy increases.