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I was recently surprised to learn that defining temperature isn't easy. For a long time, it was defined operationally: how much does a thermometer expand. Also surprising, temperature isn't a measure of thermal energy: 0 C water has more thermal energy than 0 C ice, but the same temperature.

I came up with this definition, and I'd like to know if it's correct:

Temperature is the propensity of something to give off heat, that is transmit thermal energy. Equivalently, it is the resistance of something to receive heat.

Is that correct? Please justify your answer.

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No responses yet - if the question is unclear or otherwise hard to answer, let me know, and I'll revise it! –  Robert James Aug 8 '11 at 22:59
    
The question is odd. Temperature is defined already through the laws of thermodynamics (which do not reference things like molecules, kinetics, etc.) and uses only physical ideas like equilibrium and functions of state (i.e. macroscopic reproducibility). In that treatment, one discovers temperature as a side-effect of considering reversible heat engines (at the same time as discovering entropy). –  genneth Aug 9 '11 at 9:30
    
Consider two substances with identical temperature but different thermal conductivities. It might be confusing to state that they have the same propensity to transfer thermal energy. I'm also not sure I would say that water at 0 C has more thermal energy than ice at 0 C. I was taught that equivalent amount would have the same thermal energy but that the liquid would have more internal energy or phase energy. –  Noah Aug 21 '13 at 17:25
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3 Answers

I think there is another, very nice, definition of temperature given in the book Thermal Physics by Daniel Schroeder:

We know that in a closed system, the entropy is maximized. Imagine now a large system that consists of two systems A and B in contact. We define temperature as that which is the same for A and B when the system is in equilibrium. But what is the same?

Entropy is an additive thing: the total entropy $S$ is just $S_A + S_B$, the sum of the entropy of systems A and B. Since the systems are in contact, energy can move between A and B. The total energy is fixed, so $U_B = U - U_A$ and $dU_B = -dU_A$.

Moving energy around changes the entropy: $$dS = \frac{\partial S_A}{\partial U_A} dU_A - \frac{\partial S_B}{\partial U_B} dU_A$$

But in equilibrium, $S$ is already maximized and $dS = 0$, so we have $$\frac{\partial S_A}{\partial U_A} = \frac{\partial S_B}{\partial U_B}$$

This means: If two systems are in thermal equilibrium, then the quantity that is the same for them is $\partial S / \partial U$. We also see that energy wants to move from $A$ to $B$ if $\partial S_A/\partial U_A$ is smaller than $\partial S_B/\partial U_B$, so we can define temperature as $$\frac{1}{T} = \frac{\partial S}{\partial U}$$

In a way, you are right: temperature is the propensity to transmit thermal energy. More strictly speaking, temperature is a measure for the gain in entropy when the internal energy changes.

Intuitively speaking: if system B gains more entropy than A loses if energy flows from A to B, then energy will flow from A to B so that overall entropy grows. In typical systems, the curve $S(U)$ grows with $U$ and becomes flatter for larger $U$, which means that temperature also increases with $U$. That is not always the case: For some weird systems (two-state paramagnets as the simplest example), you can actually have negative or infinite temperatures.

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I think your description of temperature has intuitive merit - it is a true and important statement about temperature. However, I would not use it as a definition of temperature because, to a certain extent, that would trivialize the second law of thermodynamics.

The second law of thermodynamics says that we can't move heat from a body of low temperature to a body of higher temperature without doing some work (the amount of work required can be quantified, but that is not critical). If we define temperature by referencing which way heat spontaneously flows, then the second law loses some of its physical content.

To put it another way, if we use this proposed definition of temperature, the only way to tell which of two bodies is hotter is to put them in contact and see which way the heat flows. That's not really our goal, though. Our goal in thermodynamics is to predict which way heat will flow (and how much work we can extract from the heat flow) before we ever put the bodies in contact, and to do it based on some other property the body has. It would be better to have a concept of temperature independent of direction of heat flow.

There are two good alternative ways to think about temperature. You already mentioned the first. In classical thermodynamics, temperature is simply what a thermometer measures. If we take one mole of an ideal gas and let it come into equilibrium with a system, the system's temperature is $PV/R$, with $P$ the pressure of the gas, $V$ its volume, and $R$ a constant called the gas constant. Thus, temperature is a state variable, and a characteristic of systems in equilibrium.

This definition of temperature, based on an ideal gas, is more useful than one based on a particular mercury thermometer because formulas such as the efficiency of an ideal heat pump come out to have a simple form when we use the gas thermometer to define temperature. When temperature is measured relative to an ideal gas, it turns out that mercury and other liquids have coefficients of expansion that change slightly with the temperature, so they aren't good references for precision measurements. However, they can be calibrated to agree with an ideal gas thermometer. We must also remember that an ideal gas is only closely approximated by real gases, so even in this case we cannot get absolutely perfect temperature measurements.

The most widely-used scale of temperature in science is the Kelvin scale. It equates absolute zero with 0 K and the triple point of water with 273.16 K. Then it takes a linear interpolation between those two points, based on the ideal gas mentioned above.

The other way to think about temperature is statistically. Suppose you have a bunch of gas molecules bouncing around in a room. At any given moment, some are going fast and others are going slowly. The molecules thus have a spread of kinetic energies.

We then wonder what the distribution of these kinetic energies is. Are they mostly clustered around one value, say all between $.016$ and $.020$ electron-volts? Or are they spread out widely from $0$ to $0.1$ electron-volts?

We could imagine the energy distribution having all sorts of different shapes. It could have five different humps, or look like a bell curve, or be flat with a gap missing, etc. It turns out this isn't the case. The energy distribution plot for equilibrium systems comes out looking pretty much the same all the time; it's a decaying exponential $\propto e^{-\beta E}$. Every time you go up by a certain fixed amount of energy, the number of particles at that energy state gets cut in half. (The exact fixed amount, $\ln(2)/\beta$, changes depending on the system; what's common is the property of getting regularly cut in half.) This is called the Boltzmann distribution.

All you have to do is specify the number $\beta$ and you know everything about the distribution of energy. This is quite a remarkable fact, and not an obvious one. This distribution occurs because it maximizes entropy for a fixed energy.

In this distribution, the average energy is $1/\beta$, and we call this average energy the temperature. (If the energies do not have a Boltzmann distribution for some reason we could still identify the mean energy as the temperature.) If $\beta$ is small, then the temperature is high, and the energy distribution is widely spread and fairly flat. At high temperature, things are quite random. If $\beta$ is big, the temperature is low and energies are clustered near zero. Low temperatures are more orderly. (An interesting side note is that a perfectly-flat energy distribution has the temperature being infinite, and further if the high-energy states are statistically more-likely than the low-energy states, the temperature is negative.)

It turns out that this statistical temperature and the thermodynamic temperature defined earlier agree with one another, up to a scale factor called Boltzmann's constant. So we can think of temperature either way, and the definitions are equally good. Either definition adds content to the second law of thermodynamics by making temperature an independent concept, and the direction of the flow of heat a physical hypothesis that involves temperature.

Finally, an important property of temperature is that it is well-defined. It is an observed fact if systems A and B are in thermal equilibrium, and B and C are in thermal equilibrium, and A and C are as well. This is called the zeroth law of thermodynamics.

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""The most widely-used scale of temperature in science is the Kelvin scale."" At this point I had included some words on the life and death of Lord Kelvin .... :=) –  Georg Aug 9 '11 at 10:49
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The other answers are great, I'm just going to try to answer it more concisely.

The proposed definition is:

Temperature is the propensity of something to give off heat, that is transmit thermal energy. Equivalently, it is the resistance of something to receive heat.

My altered version (which is a half-definition of temperature) is as follows:

Heat will not flow between two bodies at the same temperature. When two bodies are at different temperatures, heat flows from the higher temperature to the lower temperature.

This describes most of the properties of the quantity called temperature. With this, we can always identify a higher and lower relative temperature, however, we can't use this and only this to attach a numeric value on temperature. For that, you can read the other answers, which gets into the science of thermodynamics itself. For many people, however, my proposed half-definition may be sufficient for what they're looking for. If someone knows of a simple alteration that will fully define temperature, I'd love to hear it.

What is wrong with "the propensity of something to give off heat"? Obviously the statement has intuitive value. The heat flow, however, is characterized by several very different kind of quantities. My favorite is the concept of thermal resistance. If you have 2 large reservoirs at different temperatures, then the heat flow from the higher temperature to the lower one can said to be inversely proportional to the thermal resistance between the two, as well as proportional to the difference in temperatures. Thus, giving off heat is proportional to temperature, but it is also proportional to other things. For that reason we can only define the direction of flow on this qualitative level (I'm giving a "no math" answer), which the reason and the function of my alteration.

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