Temperature is an average, not a total: a bathtub holds more energy than a cup and is not hotter

01The mistake

Two containers of the same gas sit at 300 K, one holding twice as many moles as the other. Ask which is hotter. A large share of a class says the bigger one, because it contains more energy. It does contain more energy, roughly twice as much, and the two are at exactly the same temperature.

The consequence shows up in mixing problems. Students combine two samples at 50 °C and predict 100 °C, having added the temperatures as though they were amounts of something. The result is 50 °C, and no arithmetic involving the two 50s produces it except an average.

It also produces a specific wrong answer on the ideal gas law. A student who thinks temperature tracks total energy expects $T$ to rise when $n$ rises at fixed volume and pressure, which $PV = nRT$ directly forbids. They have the formula and read it against their physical picture, and the picture wins.

The tell is the word “more” in an answer about temperature. “It has more heat so it is hotter” conflates the extensive quantity with the intensive one in a single clause, and a student who says it has not distinguished them even if they can define both.

02Why it makes sense to the student

Everyday language uses heat and temperature interchangeably, and both get treated as amounts. A room has more heat than a closet; a fire has a lot of heat. Nothing in ordinary usage marks temperature as a per-particle average, so the technical meaning has to displace a working one rather than fill a gap.

Temperature behaves like a total in the only case students have much experience with. Adding energy to a fixed sample does raise its temperature, so within one object the two move together. The distinction only becomes visible when the amount of substance changes, which is exactly the comparison everyday experience does not supply.

The intensive-extensive distinction is rarely named as such in a physics course, though students have met it in chemistry under density. Density does not double when you double the sample, and most students accept that readily. Nobody tells them temperature is the same kind of quantity.

And the formula $\frac{3}{2}k_BT$ has no $N$ in it, which is the piece students skip past. The absence of a particle count is the whole content of the equation: it is a statement about one particle, on average. Written beside $U = \frac{3}{2}Nk_BT$, the difference is visible, and the two are usually introduced pages apart.

03The correction

Put the two equations next to each other and read the symbols out loud. Average kinetic energy per particle is $\frac{3}{2}k_BT$ and total internal energy of a monatomic ideal gas is $\frac{3}{2}Nk_BT$. One has $N$ in it and one does not, and that is the entire distinction expressed in notation.

Then give the analogy that works, which is average rather than total in a familiar setting. The average height of a class does not change when more students of the same height walk in, while the total height does. Students accept that immediately, and temperature is the same structure.

Make the mixing prediction and then run it. Two identical samples of water at 50 °C, combined, give 50 °C. Students who have predicted 100 and watched a thermometer read 50 stop treating temperature as additive. This is one of the few thermodynamics misconceptions a two-minute demonstration settles.

Use the ideal gas law as the formal check. At fixed $P$ and $V$, doubling $n$ halves $T$. That is the opposite of the student's prediction and it comes from an equation they already trust, which makes the equation do the arguing.

Worth testing with a comparison where the energies and temperatures point different ways: a large sample at low temperature against a small sample at high temperature. The large cool sample can hold more total energy while being colder, and a student who can say that out loud has the distinction.

04A sample question

05What each wrong answer reveals

• A Temperature read as an amount. The dominant wrong answer, and its justification is a true statement attached to a false conclusion: X really does hold more total energy. Ask what the $\frac{3}{2}k_BT$ expression is the energy of. The answer is one particle, and the stem already gave both samples the same temperature.
• B Correct. Equal temperature means equal average kinetic energy per atom. Twice the atoms at that same average gives twice the total internal energy.
• C The extensive quantity made intensive. The mirror of A: this student has absorbed that temperature is per-particle and then applied the same reasoning to internal energy, which genuinely is a total. Encouraging, because the first half of the idea has landed. Point at the $N$ in $U = \frac{3}{2}Nk_BT$ and the repair is immediate.
• D Particle count read as particle speed. This student has transferred the factor of two onto the wrong variable entirely. The number of atoms does not affect how fast any one of them moves; at the same temperature the distributions of speed in X and Y are identical. Worth asking which quantity in the stem is different, since the answer names a quantity that is not.

A and C are the two halves of the same distinction failing in opposite directions, which means they need opposite sentences: A needs temperature made intensive, C needs internal energy kept extensive. D is a different error, about which quantity the factor of two belongs to.

06Try it in Mistake Master