Heat & Thermal Equilibrium
Two bodies always end at one temperature, and almost nobody guesses which one, or why melting ice takes in heat without warming up.
Introduction
Drop a hot block of copper into a beaker of cool water and wait. The two end at one temperature. The interesting question is which temperature.
1 kg of copper at 200 °C goes into 0.5 kg of water at 20 °C, in a perfectly insulated flask. What temperature do the two end up at?
Heat and temperature are different quantities
Temperature is a property of a body: how hot it is, the block's colour and its marker on the strip. Heat is energy in transit, from the hotter body to the colder one: the arrow between the blocks. A body never has heat, and heat stops flowing when the two temperatures match.
Heat flows at a rate set by the temperature difference, through a contact of conductance :
Where they settle
Whatever one body gives up, the other receives, so both end at a weighted mean of their starting temperatures. The weights are the heat capacities :
Why metal feels colder than wood
A metal railing and a wooden bench have stood side by side in the same cold park all night. You touch each of them. The metal feels much colder. Why?
The plateau: heat with no temperature change
If heat and temperature were the same quantity, this would be impossible: pour energy into a body for a long time and watch its temperature stay put.
A block of ice sitting at exactly 0 °C is put in warm surroundings, so heat flows steadily into it. What does its temperature do?
When the heat runs out mid-melt, the equilibrium temperature is the melting point itself, and no weighted mean would have told you that.
The ledger closes, always
Across the bottom of the simulation runs one line of arithmetic:
It reads zero at every instant of every run, because the model subtracts one number from one body and adds the same number to the other: conservation is the update rule itself.
A kg copper block at °C goes into 0.5 kg of water at 20 °C in an insulated flask. Where do they settle?
What this leaves open
Each body here had one temperature at a time, a good approximation for a small block. And heat was the only way energy entered a body: let a gas push a piston and it can also do work, which is the first law in full. Both questions run through the motion of molecules, where the next lesson starts.
Ideas this lesson took on
Each of these is a common intuition the lesson put to the test. If one still feels right, the section that tests it is worth another look.
- “Two bodies in contact settle halfway between their temperatures”
- “Metal really is colder than wood”
- “Heat going in always means temperature going up”
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Research behind this lesson
- Erickson 1979: One of the earliest systematic accounts of how children explain heating and cooling: heat is understood as a substance-like, caloric thing that objects contain and that moves into and out of them, a framework that survives well past the age at which it is first taught otherwise.
- Harrison et al. 1999: Students arrive with heat and temperature poorly differentiated, routinely using "heat" for the internal energy a body contains; this study follows one grade-11 student through a unit designed to separate them, and shows how much deliberate substitution of one concept for the other the separation takes.
- Lawson & McDermott 1987: Students who could state both theorems often could not apply them, and frequently treated energy as a substance that gets used up rather than a quantity that is transferred.
- Wiser & Carey 1983: A historical case study of the seventeenth-century Florentine Experimenters, whose theory of heat had no separation between how hot a thing is and how much heat is in it: the same undifferentiated concept students bring to the topic today, and one that took the science itself about a century to pull apart.