Heat vs Temperature: What Is the Difference?

The distinction, stated once

One of these is a state of a system and the other is a transfer between systems. That is the whole difference, and everything below is a consequence of it.

Temperature is a property. AP Physics 2 essential knowledge 9.1.B.1 says the temperature of a system is characterized by the average kinetic energy of the atoms within that system, and 9.1.B.1.ii ties it to the root-mean-square speed through $K_{\text{avg}} = \frac{3}{2}k_B T = \frac{1}{2}mv_{\text{rms}}^2$. Every atom in a glass of water has its own speed; the temperature reports the average of the kinetic energies, not the total.

Heat is a transfer. The CED's own appendix on vocabulary is unusually blunt about this. It says that in physics the term heat has a very specific definition that is a thermodynamic analog to work, and that similar to the amount of work done, heat is the amount of energy transferred into or out of a thermodynamic system through thermal processes such as conduction, convection, or radiation. Then it draws the conclusion: using the physics definition of heat, an object cannot have heat any more than an object may have work.

The course framework itself avoids the noun almost entirely and names the processes instead. 9.3.A.1.i defines heating as the transfer of energy into a system by thermal processes and 9.3.A.1.ii defines cooling as the transfer of energy out of a system by thermal processes. Even the equation sheet follows suit: in its Thermal Physics panel, the symbol $Q$ is defined not as "heat" but as the energy transferred to a system by heating.

The appendix explains why the CED writes it that way. Because the word heat is so commonly misused and is hard to use naturally in a sentence, heating and cooling are used to help emphasize the processes by which thermal energy is transferred, and it notes that this misuse leads students to confuse heat with temperature and to believe objects and systems can have an amount of heat. That is this page's whole subject, named in the source.

Heat against temperature, row by row

The row that does the most work is the eighth. A temperature difference is what causes a transfer, so the two quantities are not rivals: one is the cause and the other is the effect. 9.3.A.3 states the direction, that energy is transferred through thermal processes spontaneously from a higher-temperature system to a lower-temperature system, and 9.3.A.4 states the stopping condition, that thermal equilibrium results when no net energy is transferred by thermal processes between two systems in thermal contact with each other.

Read the last row alongside it. You never measure $Q$; you measure temperatures and infer $Q$ from $Q = mc\Delta T$. That is why the two are so easily fused together in a student's head: the only evidence for one is a change in the other.

The case that separates them: the same energy, four times the temperature rise

Put $2.0\ \text{kg}$ of water in one beaker and $0.50\ \text{kg}$ in another, and deliver exactly $8.0\ \text{kJ}$ to each with an identical immersion heater. The transfer is the same in both. The temperatures do something completely different.

Rearrange the printed $Q = mc\Delta T$ from essential knowledge 9.5.A.1 to $\Delta T = Q/(mc)$. Same $Q$, same substance so same $c$, so the temperature rise goes inversely with the mass:

The small beaker warms four times as much as the large one, from an identical $8.0\ \text{kJ}$. Notice that the ratio needed no value for $c$ at all: only the definition of $Q$ as a transfer, plus the fact that a temperature is a per-atom average and therefore does not care how many atoms there are.

Now run the comparison the other way. Take two containers of the same monatomic ideal gas. Container A holds $2.0\ \text{mol}$ at $300\ \text{K}$; container B holds $0.20\ \text{mol}$ at $400\ \text{K}$. From 9.4.A.1.ii's $U = \frac{3}{2}nRT$, container A holds $7479\ \text{J}$ of internal energy and container B holds $997\ \text{J}$, so A holds seven and a half times as much energy while sitting at the lower temperature.

Connect them through a conducting wall and, by 9.3.A.3, energy flows from B to A: from the smaller, hotter, lower-energy system into the larger, cooler, higher-energy one. Temperature sets the direction. Total energy has nothing to do with it.

That is the distinction with consequences. "Which one has more energy?" and "which one is hotter?" are different questions with different answers, and only the second one predicts which way the transfer goes. And in neither container, before or after, is there any quantity called heat sitting in storage: the heat is the transfer that happens when you open the wall.

Where the confusion comes from

Everyday English uses "heat" for at least three physics quantities, and only one of them is $Q$.

• "The heat of the oven" means its temperature.
• "The heat stored in the wall" means internal energy.
• "The heat added to the water" is the only one that means $Q$.

The CED's appendix works through the same problem from the language side. It notes that saying "the amount of heat transferred" is awkward if you understand that heat itself is a transfer of energy, that a synonym for that phrase would read "the transfer of thermal energy is transferred", and that the phrase "the amount of heat done on a system" reads strangely and is rarely if ever used, even though it is the exact parallel of "the amount of work done on a system".

One useful piece of context before you over-correct. The appendix also states that the AP Physics Exams will not directly assess student understanding of physics vocabulary, giving as examples that students will not be asked to identify the correct definition of acceleration, or the difference between a system and an object, and that students will be expected to use these definitions in contextually appropriate situations. So you will not be asked to define heat. You will be asked questions whose correct answer depends on treating it as a transfer, which is a harder version of the same requirement.

The third quantity in that list deserves its own separation, because it is the one $Q$ is genuinely close to. Internal energy is what a system has; 9.4.A.1 defines it as the sum of the kinetic energy of the objects that make up the system and the potential energy of the configuration of those objects. Heating changes it, cooling changes it, and so does work, which is what $\Delta U = Q + W$ says. Heat vs work takes that equation apart, and the PV diagram guide runs the procedure.

When it costs a mark

• Writing that an object "contains 500 J of heat". Whatever the object contains is internal energy. The CED's appendix says flatly that an object cannot have heat any more than an object may have work.
• Assuming the hotter object always holds more energy. It holds more energy per atom. A cup of boiling water and a warm bath are the standard mismatch, and the worked comparison above puts numbers on the ideal-gas version.
• Predicting the direction of transfer from size or total energy. 9.3.A.3 makes the direction depend on temperature and nothing else: spontaneously from the higher-temperature system to the lower-temperature one.
• Treating thermal equilibrium as no transfer at all. 9.3.A.4 says thermal equilibrium results when no net energy is transferred by thermal processes. 9.3.A.3.i and 9.3.A.3.ii describe the underlying picture, that in collisions between atoms from different systems energy is most likely to be transferred from higher-energy atoms to lower-energy ones, and that after many such collisions the most probable state is one in which both systems have the same temperature. Individual exchanges continue; the totals stop moving.
• Using the same units for both. $Q$ is in joules and $T$ is in kelvin, and an answer that hands back a temperature in joules has usually skipped the division by $mc$.
• Substituting a temperature where the equation wants a temperature difference, or the reverse. $Q = mc\Delta T$ and $Q/\Delta t = kA\Delta T/L$ take differences, so Celsius and kelvin give identical numbers in both. $PV = nRT$, $U = \frac{3}{2}nRT$ and $K_{\text{avg}} = \frac{3}{2}k_B T$ take an absolute temperature and break on Celsius. Celsius vs kelvin sorts out which is which.
• Reaching for a specific heat that changes with temperature. Topic 9.5's boundary statement is one sentence: AP Physics 2 will model specific heat as independent of temperature. So $c$ is a constant in every exam calculation.
• Expecting a latent-heat equation. The AP Physics 2 sheet's Thermal Physics panel prints eight equations and none of them handles a change of phase. Going through them: $P = F_{\perp}/A$, $K_{\text{avg}} = \frac{3}{2}k_B T = \frac{1}{2}mv_{\text{rms}}^2$, $Q/\Delta t = kA\Delta T/L$, $PV = nRT = Nk_B T$, $U = \frac{3}{2}nRT = \frac{3}{2}Nk_B T$, $W = -P\Delta V$, $\Delta U = Q + W$ and $Q = mc\Delta T$. Every one of those describes a single phase.

The three thermal processes, and why the list matters

Heat is defined by how the energy moves, so the mechanisms are part of the definition rather than a separate topic. 9.3.A.2 lists them: the thermal processes by which energy may be transferred between systems at different temperatures are conduction, convection, and radiation. The CED's appendix names the same three inside its definition of heat.

That is a closed list of three in AP Physics 2, and one consequence is worth naming. Energy transferred any other way is not heat, and the most important other way is work. Compress a gas in an insulated cylinder and its temperature rises with no thermal process involved at all, which is why $\Delta U = Q + W$ needs two terms rather than one.

Of the three, only conduction has a printed rate equation. 9.5.B.1 gives it as $\dfrac{Q}{\Delta t} = \dfrac{kA\Delta T}{L}$, relating the rate to the thermal conductivity, the physical dimensions of the material, and the temperature difference across it. Note which quantity appears on the right: $\Delta T$, a difference. Halve the temperature difference across a window and the transfer rate halves, whatever the absolute temperatures were.

9.5.B.2 adds that the thermal conductivity of a material is an intrinsic property of that material, depending on the arrangement and interactions of the atoms that make it up, and 9.5.A.2 says the same of specific heat. Both are properties of a substance, like temperature is a property of a system. Neither is a quantity of heat.

A warning about the letter. On the AP Physics 2 sheet, $k$ in the Thermal Physics panel is thermal conductivity, $k_B$ is Boltzmann's constant, and $k$ in the constants list is the Coulomb constant at $9.0 \times 10^9\ \text{N} \cdot \text{m}^2/\text{C}^2$. Read the panel, not the letter.

When they coincide, and why that lulls you

For a single object being warmed by a single heater, $Q$ and $\Delta T$ are proportional. $Q = mc\Delta T$ with $m$ and $c$ fixed means twice the transfer gives twice the rise, every time. So in the most common laboratory situation the two quantities move together and behave as though they were the same thing measured in different units.

That is the trap, and it holds right up to the point where any of three things changes.

1. The mass changes. Then the same $Q$ gives a different $\Delta T$, which is the first worked example below.
2. The substance changes. Then $c$ changes, and identical transfers into identical masses give different rises, which is the second.
3. Work enters. Then the temperature can change with no thermal transfer at all, and $Q$ and $\Delta T$ come apart completely. That case belongs to heat vs work and to the two processes in isothermal vs adiabatic.

There is a quieter coincidence in the wording of exam questions. "The water is heated to 80 degrees" and "2000 J is transferred to the water" both describe the same event, and a well-set question gives you one of them and asks for the other. Deciding which one you have been handed, a temperature or a transfer, before you write an equation is most of the work. A temperature belongs on the right of $Q = mc\Delta T$; a transfer belongs on the left.