Specific Heat and Thermal Conductivity - AP Physics 2

1. Two rods, U and V, are made of different materials but have the same length and area. Their ends are held at $80^\circ\text{C}$ and $20^\circ\text{C}$. Rod U has thermal conductivity $k_U=10\,\text{W/(m\,K)}$ and rod V has $k_V=40\,\text{W/(m\,K)}$. (Conductivity sets heat-flow rate; specific heat affects temperature change when energy is stored.) Which rod has the larger steady heat current?
   Answer: Rod V This question tests understanding of specific heat and thermal conductivity. Thermal conductivity determines the steady-state heat flow rate through a material under a temperature gradient, while specific heat affects temperature changes during energy storage or release. The steady heat current through a rod is P = kA(ΔT)/L. Since both rods have the same geometry and temperature difference, the heat current is directly proportional to thermal conductivity. Rod V with $k_V$ = 40 W/(m·K) has four times the thermal conductivity of rod U with $k_U$ = 10 W/(m·K), so rod V has four times the heat current. Choice C incorrectly suggests specific heat affects steady heat flow, but specific heat only matters when materials are changing temperature, not during steady conduction. Remember: specific heat affects temperature change; conductivity affects transfer rate.
2. A $10\,\text{cm}$-long rod of Material X and an identical rod of Material Y connect a $100^\circ\text{C}$ plate to a $0^\circ\text{C}$ plate. Both rods have the same cross-sectional area. Thermal conductivity determines the steady heat current; specific heat affects transient warming only. Which material has the larger thermal conductivity?
   Answer: The one with the larger steady heat current through it This question tests understanding of specific heat and thermal conductivity. In steady-state heat conduction, thermal conductivity determines the rate of heat flow through a material according to Fourier's law: Q/t = kA(ΔT)/L. Specific heat only affects how quickly a material reaches steady state but not the final heat current. Since both rods have identical geometry and temperature differences, the rod carrying the larger heat current must have the higher thermal conductivity. Choice D incorrectly suggests that specific heat affects conduction rate, revealing the misconception that heat capacity properties influence steady-state heat transfer. To identify which property matters, ask: Is this about temperature change (specific heat) or heat flow rate (conductivity)?
3. Two identical metal spoons start at $20^\circ\text{C}$ and are placed into the same cup of hot tea at $80^\circ\text{C}$. Spoon A is stainless steel; Spoon B is aluminum. Thermal conductivity affects how fast heat flows from the tea into the spoon; specific heat affects how much energy is needed per degree. Which spoon transfers thermal energy from the tea faster at first?
   Answer: Aluminum, because its higher thermal conductivity allows faster heat flow This question tests understanding of specific heat and thermal conductivity. When objects are placed in contact with a heat source, thermal conductivity determines how quickly heat flows into the material. Aluminum has much higher thermal conductivity (237 W/m·K) than stainless steel (16 W/m·K), so heat flows from the tea into the aluminum spoon much faster initially. Specific heat affects how much the spoon's temperature rises for a given amount of absorbed energy but not the rate of energy transfer. Choice A incorrectly suggests that specific heat affects the rate of heat transfer, revealing the misconception that heat capacity influences conduction rate. To determine heat transfer rates, focus on thermal conductivity; for temperature changes, consider specific heat.
4. Equal masses of metals X and Y absorb the same heat $Q$. X has larger specific heat; Y has higher thermal conductivity. Which changes temperature least?
   Answer: Metal X, because larger specific heat gives smaller $\Delta T$ for the same $Q$ This question tests understanding of specific heat and thermal conductivity. Specific heat determines how much a material's temperature changes when it absorbs or releases thermal energy, following the relationship Q = mcΔT. Thermal conductivity, on the other hand, determines how quickly heat flows through a material but does not affect the equilibrium temperature change when a fixed amount of heat is absorbed. Since both metals absorb the same heat Q and have equal masses, the temperature change depends only on specific heat: ΔT = Q/(mc). Metal X has larger specific heat, so it experiences a smaller temperature change. Choice B incorrectly assumes thermal conductivity affects the final temperature change for a fixed amount of absorbed heat. When comparing temperature changes for fixed heat absorption, focus on specific heat capacity, not conductivity.
5. Equal masses ($m$) of water and cooking oil start at $25^\circ\text{C}$ in identical cups. Each receives the same thermal energy $Q$ from a hot plate. Specific heat determines the temperature change for a given $Q$; thermal conductivity mainly affects how quickly the liquid becomes uniform in temperature. Which liquid's temperature increases less?
   Answer: Water, because its higher specific heat makes $\Delta T$ smaller for the same $Q$ This question tests understanding of specific heat and thermal conductivity. When equal masses of different materials absorb the same amount of thermal energy Q, their temperature changes are inversely proportional to their specific heats: ΔT = Q/(mc). Water has a much higher specific heat (4186 J/kg·K) than cooking oil (approximately 2000 J/kg·K), so water's temperature will increase less. Thermal conductivity affects how quickly the heat distributes within each liquid but not the final average temperature. Choice C incorrectly states that smaller specific heat leads to smaller temperature change, revealing the misconception of confusing the inverse relationship in the heat capacity equation. Remember: higher specific heat means smaller temperature change for the same energy input.
6. Two identical cups each contain $0.30\,\text{kg}$ of liquid at $10^\circ\text{C}$. Cup 1 has liquid with $c_1=4200\,\text{J/(kg\,K)}$; cup 2 has liquid with $c_2=2100\,\text{J/(kg\,K)}$. Each cup receives $Q=2.52\,\text{kJ}$ from the same heater. (Specific heat sets $\Delta T$; conductivity affects internal temperature uniformity.) Which cup's liquid reaches the higher final temperature?
   Answer: Cup 2 This question tests understanding of specific heat and thermal conductivity. Specific heat determines how much a liquid's temperature increases for a given amount of absorbed thermal energy, while thermal conductivity affects how uniformly the temperature distributes within the liquid. Using Q = mcΔT, we find: Δ$T_1$ = 2520/(0.30 × 4200) = 2.0°C and Δ$T_2$ = 2520/(0.30 × 2100) = 4.0°C. Cup 2, with the lower specific heat liquid, experiences twice the temperature increase, reaching 14°C compared to cup 1's 12°C. Choice C incorrectly focuses on thermal conductivity, which affects how quickly heat spreads internally but doesn't change the final equilibrium temperature for a fixed energy input. Remember: specific heat affects temperature change; conductivity affects transfer rate.
7. Two identical cups contain equal masses of liquid A and liquid B. Both receive the same heater power for $60\text{ s}$. Which statement is correct?
   Answer: The liquid with higher specific heat has the smaller temperature rise This question tests understanding of specific heat and thermal conductivity. When liquids receive the same power (energy per second) for the same time, they absorb equal amounts of heat Q. The temperature rise is given by Q = mcΔT, so ΔT = Q/(mc). The liquid with higher specific heat will have a smaller temperature rise for the same heat input. Thermal conductivity doesn't affect this result since it only determines heat flow rate, not temperature change for absorbed heat. Choice D incorrectly claims conductivity affects the amount of heat absorbed, but power × time determines Q regardless of material properties. Remember: specific heat determines temperature change; conductivity determines transfer rate.
8. A hot reservoir touches one end of two identical rods, A and B. Same length and area; $k_A>k_B$. Which transfers thermal energy fastest initially?
   Answer: Rod A, because higher thermal conductivity gives larger heat current This question tests understanding of specific heat and thermal conductivity. Thermal conductivity k determines the rate of heat transfer through a material according to Fourier's law: heat current = kA(ΔT)/L. Specific heat affects how much a material's temperature changes when absorbing energy, but does not directly determine the rate of heat flow through the material. Since rods A and B have identical geometry and experience the same temperature difference initially, the rod with higher thermal conductivity (rod A) will transfer heat faster. Choice C incorrectly claims that specific heat increases heat transfer rate, confusing the role of specific heat with thermal conductivity. To determine heat transfer rates through materials, use thermal conductivity; specific heat only affects temperature changes.
9. A $1.0\,\text{m}$ slab of insulation and a $1.0\,\text{m}$ slab of glass have the same area. One side is kept at $40^\circ\text{C}$ and the other at $20^\circ\text{C}$. The insulation has much smaller thermal conductivity than glass. (Conductivity controls steady heat current; specific heat does not determine steady heat flow.) Which material has the smaller heat current through it?
   Answer: The insulation This question tests understanding of specific heat and thermal conductivity. Thermal conductivity determines the rate of steady-state heat flow through a material under a temperature gradient, while specific heat affects temperature changes during transient processes. For steady heat flow through a slab, the heat current is P = kA(ΔT)/L, where k is thermal conductivity. Since both slabs have the same area, thickness, and temperature difference, the heat current is directly proportional to thermal conductivity. The insulation, with much smaller thermal conductivity than glass, has a much smaller heat current flowing through it. Choice C incorrectly suggests specific heat matters for steady heat flow, but specific heat only affects how materials respond to energy changes, not steady-state conduction. Remember: specific heat affects temperature change; conductivity affects transfer rate.
10. Two spheres, A and B, have the same material (same $c$ and $k$) but $m_B=2m_A$. Each absorbs the same heat $Q$. Which statement about temperature change is correct?
    Answer: Sphere B has smaller $\Delta T$ because $\Delta T=Q/(mc)$ and $m$ is larger This question tests understanding of specific heat and thermal conductivity. The temperature change when absorbing heat is given by Q = mcΔT, which rearranges to ΔT = Q/(mc). Since both spheres are made of the same material, they have identical specific heat c and thermal conductivity k. With sphere B having twice the mass of sphere A (mB = 2mA) and both absorbing the same heat Q, sphere B's temperature change is ΔTB = Q/(2mA·c) while sphere A's is ΔTA = Q/(mA·c). Therefore, sphere B has half the temperature change of sphere A. Choice D incorrectly claims thermal conductivity determines temperature change for heat absorption, confusing it with specific heat's role. For temperature changes from heat absorption, use Q = mcΔT; conductivity only affects heat flow rates.
11. Two blocks, P and Q, have equal mass and start at $20^\circ$C. Each absorbs $600$ J. $c_P=900$ J/(kg·K), $c_Q=450$ J/(kg·K). Which ends cooler?
    Answer: Block P, because higher specific heat means a smaller temperature increase This question tests understanding of specific heat and thermal conductivity. Specific heat c determines the temperature change when a material absorbs thermal energy: Q = mcΔT, which rearranges to ΔT = Q/(mc). Thermal conductivity is irrelevant here since we're not considering heat flow through the blocks, only their temperature change from absorbed energy. With equal masses and equal heat absorbed (600 J), the block with higher specific heat will have a smaller temperature increase. Block P has c = 900 J/(kg·K) while block Q has c = 450 J/(kg·K), so block P experiences half the temperature rise of block Q and ends cooler. Choice A incorrectly claims lower specific heat leads to smaller temperature increase, reversing the relationship. Remember: higher specific heat means smaller temperature change for the same heat absorbed.
12. A wall section is replaced with either material M or N of equal thickness and area. Same indoor–outdoor $\Delta T$. $k_M>k_N$ but $c_M=c_N$. Which reduces heat loss rate?
    Answer: Material N, because lower conductivity reduces the heat current This question tests understanding of specific heat and thermal conductivity. Thermal conductivity k determines the rate of heat flow through a material: heat current = kA(ΔT)/L. Specific heat affects temperature changes when materials absorb or release energy but doesn't influence steady-state heat flow rates through walls. For the same temperature difference, area, and thickness, the material with lower thermal conductivity (material N) will have a lower heat current and thus reduce heat loss rate more effectively. Choice A incorrectly claims higher conductivity reduces heat current, which contradicts the direct proportionality in Fourier's law. For insulation purposes, choose materials with low thermal conductivity to minimize heat transfer rates.
13. Identical aluminum and steel blocks (same mass) cool in the same room. Which statement is correct about the initial cooling rate?
    Answer: The block with higher thermal conductivity cools faster This question tests understanding of specific heat and thermal conductivity. When objects cool in air, the rate of heat loss depends on thermal conductivity at the surface, not specific heat. Aluminum has higher thermal conductivity than steel (237 W/m·K vs 50 W/m·K), so it transfers heat to the surrounding air more quickly. The initial cooling rate is determined by how fast heat can flow from the interior to the surface and then to the air. Choice B incorrectly suggests specific heat affects cooling rate, but specific heat only determines how much the temperature drops for a given heat loss. The strategy is: conductivity determines heat transfer rate; specific heat determines temperature change for given heat transfer.
14. Two liquids, L1 and L2, each have mass $0.20\,\text{kg}$ and start at $25^\circ\text{C}$. Each receives $Q=1.6\,\text{kJ}$ from an immersion heater. Their specific heats are $c_1=2000\,\text{J/(kg\,K)}$ and $c_2=4000\,\text{J/(kg\,K)}$. (Specific heat affects final temperature; conductivity affects how quickly heating spreads.) Which liquid ends at the higher temperature?
    Answer: Liquid L1 This question tests understanding of specific heat and thermal conductivity. Specific heat determines how much a substance's temperature changes when it gains or loses thermal energy, while thermal conductivity affects how quickly heat spreads through a material. Using Q = mcΔT, we can find the temperature changes: Δ$T_1$ = 1600/(0.20 × 2000) = 4.0°C and Δ$T_2$ = 1600/(0.20 × 4000) = 2.0°C. Liquid L1 with the lower specific heat experiences the larger temperature increase, reaching 29°C compared to L2's 27°C. Choice C incorrectly focuses on thermal conductivity, which affects how quickly the heat spreads through the liquid but doesn't determine the final equilibrium temperature for a given amount of absorbed energy. Remember: specific heat affects temperature change; conductivity affects transfer rate.
15. Two insulating cups each contain $0.20$ kg of liquid at $80^\circ$C. Both lose $400$ J to the room. Liquid R has larger specific heat than S. Which cools less?
    Answer: Liquid R, because larger specific heat means smaller temperature change for the same heat loss This question tests understanding of specific heat and thermal conductivity. When a fixed amount of heat is lost, the temperature change is determined by Q = mcΔT, giving ΔT = Q/(mc). Thermal conductivity would affect the rate of heat loss but not the temperature change for a given amount of heat already lost. Since both liquids have the same mass (0.20 kg) and lose the same heat (400 J), the liquid with larger specific heat will experience a smaller temperature drop. Liquid R, with larger specific heat, cools less than liquid S. Choice A incorrectly claims smaller specific heat leads to smaller temperature change, reversing the correct relationship. For fixed heat transfer, remember: larger specific heat means smaller temperature change.
16. Two identical cups contain equal masses of liquid A and liquid B, both initially at $10^\circ\text{C}$. A $500\,\text{W}$ immersion heater runs for $40\,\text{s}$ in each cup. Specific heat sets $\Delta T=Q/(mc)$; thermal conductivity affects how uniform the temperature is during heating. Which liquid has the smaller specific heat?
    Answer: The one with the larger temperature increase after $40\,\text{s}$ This question tests understanding of specific heat and thermal conductivity. When equal masses of different liquids receive the same amount of energy (500 W × 40 s = 20,000 J), their temperature changes are inversely proportional to their specific heats: ΔT = Q/(mc). The liquid with the larger temperature increase must have the smaller specific heat. Thermal conductivity affects how uniformly the temperature rises throughout the liquid but not the average temperature change. Choice B incorrectly links uniform warming to specific heat, revealing the misconception that conductivity and heat capacity are related. To compare specific heats, measure temperature changes for equal energy inputs to equal masses.
17. Two bars, M and N, have equal length and are clamped between plates at $T_h$ and $T_c$. Bar M has twice the cross-sectional area of bar N, and both are the same material. (Thermal conductivity with geometry sets heat-flow rate; specific heat does not set steady heat current.) Which bar transfers thermal energy at the greater rate?
    Answer: Bar M This question tests understanding of specific heat and thermal conductivity. Thermal conductivity combined with geometry determines the rate of heat flow through a material, while specific heat affects temperature changes when energy is stored or released. For steady heat conduction, the heat flow rate is P = kA($T_h$ - $T_c$)/L, where A is cross-sectional area. Since bar M has twice the cross-sectional area of bar N and both have the same material (same k), length, and temperature difference, bar M transfers heat at twice the rate of bar N. Choice C incorrectly suggests specific heat matters for steady heat flow, but specific heat only affects transient temperature changes, not steady-state heat transfer rates. Remember: specific heat affects temperature change; conductivity affects transfer rate.
18. A $1.0\,\text{cm}$-thick slab of foam and a $1.0\,\text{cm}$-thick slab of glass, same area, separate a $60^\circ\text{C}$ surface from $20^\circ\text{C}$ air. Thermal conductivity sets the rate of heat transfer through the slab; specific heat only affects how long the slab takes to warm up. Which slab allows the greater heat transfer rate?
    Answer: Glass, because its higher thermal conductivity gives a larger heat current This question tests understanding of specific heat and thermal conductivity. In steady-state heat conduction through a slab, the heat transfer rate depends on thermal conductivity according to Q/t = kA(ΔT)/L. Glass has much higher thermal conductivity than foam (glass ~1 W/m·K, foam ~0.03 W/m·K), so glass allows a much greater heat transfer rate. Specific heat only affects how long the materials take to reach steady state, not the steady-state heat flow. Choice A incorrectly attributes heat transfer rate to specific heat, revealing the misconception that heat capacity affects conduction rate rather than just temperature change. For steady heat flow problems, thermal conductivity is the key property, not specific heat.
19. A $100\,\text{g}$ metal sample at $200^\circ\text{C}$ is placed into $200\,\text{g}$ of water at $20^\circ\text{C}$ in an insulated cup. Specific heat determines the equilibrium temperature via energy conservation; thermal conductivity affects only how quickly equilibrium is reached. Which change would make the final equilibrium temperature higher?
    Answer: Use a metal with smaller specific heat, keeping masses and initial temperatures the same This question tests understanding of specific heat and thermal conductivity. In thermal equilibrium problems, the final temperature depends on the specific heats and masses of all materials involved through energy conservation: m₁c₁(T₁ᵢ - Tf) = m₂c₂(Tf - T₂ᵢ). To increase the final temperature, you need the hot object to lose less temperature or the cold object to gain more, which happens when the metal has a smaller specific heat. Thermal conductivity only affects how quickly equilibrium is reached, not the final temperature. Choice A incorrectly suggests that conductivity affects equilibrium temperature, revealing the misconception that heat transfer rate properties influence final energy distribution. For equilibrium temperature problems, use specific heat in energy balance equations, not conductivity.
20. Two rods, X and Y, have the same length and cross-sectional area. Their ends are held at $100^\circ\text{C}$ and $0^\circ\text{C}$. Rod X has thermal conductivity $k_X=200\,\text{W/(m\,K)}$ and rod Y has $k_Y=50\,\text{W/(m\,K)}$. (Conductivity sets heat-flow rate; specific heat sets temperature change for given $Q$.) Which rod transfers thermal energy fastest?
    Answer: Rod X This question tests understanding of specific heat and thermal conductivity. Thermal conductivity determines the rate at which heat flows through a material under a temperature gradient, while specific heat determines temperature changes when energy is absorbed or released. The heat flow rate through a rod is given by P = kA(ΔT)/L, where k is thermal conductivity, A is area, ΔT is temperature difference, and L is length. Since both rods have the same geometry and temperature difference, the heat flow rate is directly proportional to thermal conductivity. Rod X with $k_X$ = 200 W/(m·K) transfers heat four times faster than rod Y with $k_Y$ = 50 W/(m·K). Choice C incorrectly suggests specific heat matters for steady-state heat flow, but specific heat only affects transient temperature changes, not steady heat current. Remember: specific heat affects temperature change; conductivity affects transfer rate.

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