Entropy and Second Law of Thermodynamics - AP Physics 2

1. An ideal gas undergoes a reversible isothermal expansion at 350 K while absorbing 500 J of heat from a reservoir; the process can be made quasi-static. Which statement is consistent with the second law?
   Answer: The gas's entropy change is $\Delta S=\frac{Q_{rev}}{T}=\frac{500}{350}\,\text{J/K}$. This question tests understanding of entropy and the second law of thermodynamics. For a reversible isothermal process, the entropy change is calculated using ΔS = Q_rev/T, where Q_rev is the heat absorbed reversibly and T is the constant temperature. When the ideal gas expands isothermally at 350 K while absorbing 500 J of heat, its entropy increases by ΔS = 500/350 = 10/7 ≈ 1.43 J/K. This positive entropy change reflects the increased volume and thus increased number of accessible microstates for the gas molecules. Choice D incorrectly inverts the formula as T/Q_rev—entropy has units of J/K, which requires Q/T not T/Q. For any reversible isothermal process, use ΔS = Q_rev/T to calculate entropy change, with positive Q for heat absorbed.
2. Two identical copper blocks, one at $400\,\text{K}$ and one at $300\,\text{K}$, are placed in thermal contact inside an insulated box until they reach equilibrium at $350\,\text{K}$. The process is spontaneous. Which statement best supports this outcome?
   Answer: Total entropy increases because heat transfer between different temperatures is irreversible. This question tests understanding of entropy and the second law of thermodynamics. When two blocks at different temperatures come into thermal contact within an insulated system, heat flows spontaneously from hot to cold until they reach the same temperature. This irreversible heat transfer increases the total entropy because the cold block gains more entropy than the hot block loses (since ΔS = Q/T and the same Q is transferred at different temperatures). The final uniform temperature represents a more probable macrostate with higher entropy than the initial state of temperature difference. Choice D incorrectly claims entropy decreases when temperatures equalize, misunderstanding that uniform temperature actually represents higher entropy. To analyze thermal equilibration, calculate entropy changes using ΔS = Q/T and verify total entropy increases.
3. A refrigerator removes $Q_c$ from $4^\circ\text{C}$ food and exhausts $Q_h$ to a $25^\circ\text{C}$ kitchen while consuming work $W$ (spontaneous only when powered). Which statement best matches the second-law constraint?
   Answer: It must satisfy $Q_h=Q_c+W$, so $Q_h>Q_c$ when $W>0$. This question tests understanding of entropy and the second law of thermodynamics. A refrigerator transfers heat from a cold region to a hot region, which is non-spontaneous and requires work input. By energy conservation (first law), the heat expelled to the hot side equals the heat absorbed from the cold side plus the work input: Qh = Qc + W. Since work is positive, Qh must be greater than Qc. The second law prohibits transferring heat from cold to hot without work input (W cannot be zero for continuous operation). Choice B incorrectly claims heat can be destroyed, violating energy conservation—heat is energy and cannot be created or destroyed. For any refrigeration cycle, always verify that energy is conserved and that work is required to move heat against its natural direction.
4. Two identical insulated chambers contain the same ideal gas, one at $4.0\,\text{atm}$ and the other at $1.0\,\text{atm}$, separated by a valve. The valve is opened briefly and the gases mix until pressures equalize; the process is spontaneous. Which conclusion is consistent with the second law?
   Answer: The total entropy increases because the gas undergoes free expansion into a larger accessible volume. This question tests understanding of entropy and the second law of thermodynamics. When two gas chambers at different pressures are connected, the gas spontaneously expands from high pressure to low pressure until equilibrium is reached. This free expansion increases the total entropy because gas molecules gain access to a larger volume, increasing the number of available microstates. The entropy increase occurs regardless of temperature changes—it's driven by the expansion into previously inaccessible space. Choice C incorrectly assumes entropy remains constant due to insulation, confusing heat transfer with entropy changes from expansion. Remember: free expansion of gases always increases entropy, even in isolated systems.
5. A sealed, rigid container is divided by a removable partition. One side contains ideal gas; the other side is a vacuum. The partition is removed and the gas expands to fill the entire container; the process is spontaneous. Which statement best describes the entropy change of the gas?
   Answer: The gas entropy increases because more microstates are accessible after expansion into a larger volume. This question tests understanding of entropy and the second law of thermodynamics. When an ideal gas expands into a vacuum (free expansion), its entropy increases because the molecules can access a larger volume, creating more possible microstates. This entropy increase occurs even though the temperature remains constant for an ideal gas in free expansion (no work done, no heat exchanged). The process is irreversible and spontaneous, driven by the statistical tendency toward more probable macrostates. Choice B incorrectly claims entropy remains constant, confusing constant internal energy with constant entropy. Remember: entropy increases when gases expand into larger volumes, regardless of energy changes.
6. 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?
   Answer: The total entropy of the two-block system increases until thermal equilibrium is reached. 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.
7. A sealed, rigid container is divided by a removable partition. Side A contains an ideal gas; side B is vacuum. The partition is removed and the gas freely expands to fill the whole container; no heat enters and no work is done. The expansion is spontaneous. Which statement best explains why it occurs?
   Answer: The gas's entropy increases because more microstates are available after expansion. This question tests understanding of entropy and the second law of thermodynamics. When a gas freely expands into a vacuum, no work is done (no external pressure to push against) and no heat is transferred (adiabatic process), yet the expansion occurs spontaneously. The driving force is the increase in entropy: with more volume available, the gas molecules have access to many more possible spatial configurations (microstates), leading to higher entropy. This irreversible process increases the entropy of the isolated system, consistent with the second law. Choice B incorrectly claims entropy decreases in an adiabatic free expansion, confusing it with reversible adiabatic processes. To predict spontaneous processes, identify whether the number of accessible microstates increases.
8. A closed, insulated container holds two different ideal gases initially separated by a partition at the same temperature. The partition is removed and the gases mix; the process is spontaneous. Which statement best explains why mixing occurs without violating the second law?
   Answer: Mixing is spontaneous because the total entropy of the isolated container increases when the gases spread into more accessible microstates. This question tests understanding of entropy and the second law of thermodynamics. When different gases mix spontaneously, the total entropy increases because each gas expands into the full container volume, accessing more microstates. This mixing entropy arises from the increased positional disorder as molecules of each type spread throughout the available space. The process is irreversible and occurs without external intervention in isolated systems. Choice B incorrectly claims entropy must decrease to conserve energy, confusing energy conservation with entropy increase. Remember: spontaneous mixing always increases entropy due to increased molecular disorder.
9. 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?
   Answer: The total entropy of the block-plus-water system increases because energy flows from higher to lower temperature. 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.
10. A 300 K metal block is placed in contact with a 500 K reservoir and absorbs 600 J of heat; the process is spontaneous. Which conclusion is consistent with the second law?
    Answer: The total entropy change is positive because heat flows from hot to cold. This question tests understanding of entropy and the second law of thermodynamics. When heat flows spontaneously from a hot reservoir (500 K) to a cooler block (300 K), the total entropy of the universe must increase according to the second law. The entropy decrease of the hot reservoir (ΔS = -Q/T_hot = -600/500 = -1.2 J/K) is smaller in magnitude than the entropy increase of the cold block (ΔS = +Q/T_cold = +600/300 = +2.0 J/K), resulting in a net positive entropy change of +0.8 J/K. Choice B incorrectly confuses energy conservation with entropy conservation—while energy is conserved, entropy increases in spontaneous processes. For any spontaneous heat transfer, always check that the total entropy change is positive by calculating ΔS_total = Q(1/T_cold - 1/T_hot).
11. A student claims a heat engine operating between $T_h=600\,\text{K}$ and $T_c=300\,\text{K}$ can convert all absorbed heat into work in a cycle; such a device would be non-spontaneous. Which statement best explains why this violates the second law?
    Answer: A cyclic engine must reject some heat to the cold reservoir, so $Q_c>0$ and $\eta<1$. This question tests understanding of entropy and the second law of thermodynamics. The second law states that no heat engine operating in a cycle can convert all absorbed heat into work—some heat must be rejected to a cold reservoir. This fundamental limitation exists because converting all heat to work would decrease the total entropy of the universe, violating the second law. For any cyclic engine, the efficiency η = W/Qh must be less than 1, meaning Qc > 0 (some heat is always rejected). Choice C incorrectly suggests efficiency can exceed 100% with high enough temperature, misunderstanding that efficiency is fundamentally limited by η ≤ 1 - Tc/Th. Remember: cyclic engines must reject heat to maintain entropy balance in the universe.
12. A perfectly insulated container is divided into two equal volumes by a removable barrier. One side contains an ideal gas at $2.0\,\text{atm}$, the other side is vacuum. The barrier is removed and the gas fills the container; the process is spontaneous. Which statement is consistent with the second law?
    Answer: The entropy of the gas increases because expansion into vacuum increases the number of accessible microstates. This question tests understanding of entropy and the second law of thermodynamics. When a gas expands into vacuum (free expansion), its molecules can occupy twice the original volume, doubling the number of possible spatial configurations and increasing entropy. The second law requires entropy to increase in this irreversible process because the gas molecules spontaneously spread to fill the available space, increasing disorder. Since the container is perfectly insulated (Q = 0) and no work is done against vacuum (W = 0), the internal energy and temperature remain constant, but entropy still increases due to the volume change. Choice C incorrectly assumes insulation means ΔS = 0, confusing heat transfer with the fundamental increase in microstates during expansion. The strategy is to remember that for ideal gas free expansion into vacuum, entropy always increases by ΔS = nR ln(V_final/V_initial) > 0.
13. A student proposes a device that, in a complete cycle, absorbs $200\,\text{J}$ from a single $300\,\text{K}$ reservoir and converts it entirely into $200\,\text{J}$ of work, with no other effect. The proposal is non-spontaneous in reality. Which statement best explains why?
    Answer: It violates the second law because a cyclic device cannot convert heat from one reservoir entirely into work. This question tests understanding of entropy and the second law of thermodynamics. The proposed device would convert heat entirely to work in a cycle with no other effect, violating the Kelvin-Planck statement of the second law. Any cyclic device that converts heat to work must reject some heat to a cold reservoir, ensuring positive entropy generation to compensate for the entropy decrease when removing heat from the hot reservoir. Complete conversion would mean ΔS_universe = -Q/T < 0, violating the second law's requirement that total entropy never decreases. Choice D incorrectly invokes the first law; energy is conserved in the proposal, but the second law forbids the process. The strategy is to recognize that no cyclic device can convert heat from a single reservoir entirely to work—some heat must be rejected to maintain ΔS_universe ≥ 0.
14. An insulated box contains a battery connected to a resistor. When the circuit is closed, electrical energy is converted to thermal energy in the resistor and warms the air in the box; the conversion occurs spontaneously. Which statement is consistent with the second law?
    Answer: The total entropy of the contents of the insulated box increases during the energy conversion. This question tests understanding of entropy and the second law of thermodynamics. When electrical energy is converted to thermal energy in a resistor, the organized energy of moving charges becomes randomized thermal motion of molecules, increasing entropy. This irreversible process increases the total entropy of the insulated system, as required by the second law for any spontaneous process in an isolated system. The conversion from ordered electrical energy to disordered thermal energy represents a transition to a more probable macrostate with more accessible microstates. Choice D incorrectly claims electrical energy is more disordered than thermal energy, reversing the actual relationship. When analyzing energy conversions in isolated systems, verify that entropy increases as energy becomes less organized.
15. 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?
    Answer: The coffee entropy decreases because it loses thermal energy while cooling. 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.
16. A refrigerator removes $Q_c=200\,\text{J}$ from the cold space each cycle while consuming $W=50\,\text{J}$ of electrical work; the process is non-spontaneous and requires input work. Which statement is consistent with the second law for this device?
    Answer: It is consistent because external work allows heat to be transferred from cold to hot in a cycle. This question tests understanding of entropy and the second law of thermodynamics. A refrigerator moves heat from cold to hot, which doesn't occur spontaneously, but the second law allows this when external work is supplied. The device must reject Qh = Qc + W = 250 J to the hot side while consuming 50 J of work, maintaining positive entropy production overall. The coefficient of performance (COP = Qc/W = 4) being greater than 1 is perfectly allowed and doesn't violate any laws. Choice B incorrectly claims heat can never move from cold to hot, misunderstanding that work input makes this possible. Remember: non-spontaneous processes can occur when external work compensates for entropy decrease.
17. 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?
    Answer: It fails because it would require the entropy of the universe to decrease for a complete cycle. 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.
18. A $500\,\text{J}$ quantity of heat is transferred from a $600\,\text{K}$ reservoir to a $300\,\text{K}$ reservoir through a metal rod. No work is done; the transfer is spontaneous. Which statement best accounts for why this direction occurs?
    Answer: The total entropy change is positive because $\Delta S=\frac{Q}{300\,\text{K}}-\frac{Q}{600\,\text{K}}>0$. This question tests understanding of entropy and the second law of thermodynamics. When heat Q flows from a hot reservoir to a cold reservoir, the hot reservoir loses entropy ΔS_hot = -Q/T_hot while the cold reservoir gains entropy ΔS_cold = +Q/T_cold. Since T_cold < T_hot, we have |ΔS_cold| > |ΔS_hot|, making the total entropy change positive: ΔS_total = Q/300 - Q/600 = Q/600 > 0. This positive entropy change explains why heat spontaneously flows from hot to cold, never the reverse. Choice B incorrectly claims the total entropy decreases, misunderstanding that while the hot reservoir's entropy decreases, the cold reservoir's entropy increases by a larger amount. The key principle is that for any spontaneous heat transfer between reservoirs, calculate ΔS_total = Q(1/T_cold - 1/T_hot) and verify it's positive.
19. A heat engine operates between a hot reservoir at $500\,\text{K}$ and a cold reservoir at $300\,\text{K}$. In one cycle it absorbs $Q_H=1000\,\text{J}$ and produces $W=700\,\text{J}$. The cycle is proposed to be spontaneous once started. Which statement is consistent with the second law?
    Answer: The claim violates the second law because it implies $Q_C=300\,\text{J}$ and an efficiency exceeding the Carnot limit. This question tests understanding of entropy and the second law of thermodynamics. The claimed engine absorbs $Q_H$ = 1000 J and produces W = 700 J, implying $Q_C$ = $Q_H$ - W = 300 J by energy conservation. This gives an efficiency η = W/$Q_H$ = 700/1000 = 0.70 = 70%, which exceeds the Carnot limit η_Carnot = 1 - $T_C$/$T_H$ = 1 - 300/500 = 0.40 = 40%. The second law states no heat engine can exceed the Carnot efficiency between given temperatures, as this would decrease total entropy. Choice B incorrectly claims efficiency can exceed 100%, violating both energy conservation and the second law. The key strategy is to always check if a proposed engine's efficiency exceeds the Carnot limit (1 - $T_C$/$T_H$) for the given temperatures.
20. A sealed container holds liquid water and water vapor at equilibrium at $300\,\text{K}$. The container is placed in a freezer at $260\,\text{K}$, and over time the water freezes; this is spontaneous. Which statement best explains why freezing can occur spontaneously here?
    Answer: Total entropy of the water plus surroundings increases because the surroundings gain enough entropy from released heat. This question tests understanding of entropy and the second law of thermodynamics. Although water's entropy decreases when it freezes (solid has fewer microstates than liquid), the process can be spontaneous if the surroundings gain enough entropy to make the total change positive. When water freezes, it releases latent heat to the colder surroundings; since the surroundings are at lower temperature (260 K), they gain more entropy from this heat than the water loses by ordering into ice. The second law requires only that total entropy increases, not that every component's entropy increases. Choice C incorrectly claims solids have higher entropy than liquids, reversing the actual relationship. To analyze phase transitions, consider entropy changes of both the substance and its surroundings.

Start this quiz