Specific Heat and Thermal Conductivity
▶︎ Watch it animatedinteractive step-through · ~3 min · optional ⚙︎ Open the appletMixing and Wall Bench · pour equal energy into five materials, mix two samples against the split-the-difference answer, and separate a wall's rate from the energy a clock lets through itSpecific heat sets how far a temperature moves for a given energy input, $\Delta T = Q/(mc)$, so a large $c$ produces a small change and water is slow to heat and slow to cool. In an insulated mixture the energy one substance loses is the energy the other gains, $m_1c_1(T_f - T_1) + m_2c_2(T_f - T_2) = 0$, and the final temperature is the $mc$-weighted average rather than the midpoint. Thermal conductivity is a separate constant answering a separate question: $Q/\Delta t = kA\,\Delta T/L$ gives a rate in watts, rising with conductivity, area and temperature difference and falling with thickness.
Three errors dominate. Reading a large specific heat as a large temperature response, which predicts that water heats faster than the pan it sits in. Averaging the two starting temperatures in a mixing problem, which is right only when both sides happen to have the same $mc$, and which often lands outside the starting range once a sign slips. And converting a conduction rate into a temperature, so a metal bench is reported as colder than the wooden one beside it, or quoting a wattage as though it were an amount of energy.
The work
3 ways in · any order
Lesson
Specific Heat and Thermal Conductivity
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Puts specific heat in the denominator where the temperature change actually finds it, weights mixing problems by mc, and keeps a conduction rate from being read as a temperature.
Diagnostic
10-item topic check
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Ten items spanning the failure modes of this topic: expecting the material with the larger specific heat to warm faster, averaging the starting temperatures of a mixture, and reading a fast conduction rate as a lower temperature or quoting it as an amount of energy. Take it cold to find which one is yours, or after the lesson to confirm it is not.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.