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I'm trying to figure out what temperature actually is, and there seem to be a lot of answers floating around:

and any of these could be fine, except I have no idea how we could possible measure them (except maybe the first, but people generally seem to say that it's actually wrong).

That is, if I take a traditional, thermal-expansion-based thermometer and put it in contact with something, it measures something which corresponds to all of the definitions given above, which is constant at equilibrium, etc. It seems to measure "temperature".

But what is the physical process by which (for example) $\frac{\partial E}{\partial S}$ governs precisely how much the mercury will rise in a thermometer? Why do we believe that "temperature" (the thing a thermometer measures) corresponds so precisely to "temperature" (any of the four definitions given above)?

(and for that matter, if we take $\frac{\partial E}{\partial S}$ as a definition of temperature, how do we define $S$?)

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    $\begingroup$ FWIW: Strictly speaking, A thermometer measures the temperature of itself. That temperature is presumed to be the same as, or closely related to the object or fluid whose temperature you wish to know. Even a bolometer (a.k.a., infrared thermometer) is really measuring the temperature of something inside the instrument, that is heated by radiation from the surface of interest. $\endgroup$ – Solomon Slow Jul 31 at 18:29
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    $\begingroup$ Also note: When you choose a glass thermometer in a chemistry lab, it's important to know how it's meant to be used. One kind is calibrated to give the correct reading when immersed in liquid to the depth of a certain line that is etched on the glass, and it only gives its most accurate reading when the air above the liquid is some particular temperature (Maybe 20C? I forget). Another kind is meant to be immersed to where the top of the mercury column is level with the surface of the liquid-to-be-measured. A third kind gives its most accurate reading only when completely submerged. $\endgroup$ – Solomon Slow Jul 31 at 18:39
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Temperature is a measure based on the most likely value of kinetic energy possessed by the individual particles in a very large ensemble of those particles. The classic "ensemble" in this context is the so-called ideal gas, but the general principle holds for nonideal gases and solids and liquids, with the appropriate corrections which reflect the actual equations of state for those substances.

The kinetic energy of those particles is related to their speed as they vibrate. More vigorous vibrations means more speed, which we measure as an increase in temperature.

That speed increase means the particles exchange greater forces during their collisions, which tends to make them push away harder on one another as the temperature goes up.

This in turn causes the ensemble to expand upon heating.

The liquid mercury in the thermometer is there because its degree of thermal expansion is conveniently large and predictable over a significant temperature range, which makes it a good temperature indicator.

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  • $\begingroup$ In the answer to this question (physics.stackexchange.com/questions/334004/…) they say: "sometimes one hears people incorrectly saying that temperature measures the mean particle energy - this is so for ideal gasses but not in general". $\endgroup$ – Sam Jaques Jul 31 at 16:32
  • $\begingroup$ will edit my answer. -NN $\endgroup$ – niels nielsen Jul 31 at 16:34
  • $\begingroup$ Thanks, a follow-up: Is it measuring (a) the most likely value of the kinetic energy of the particles; (b) the average kinetic energy of the particles; (c) the expected value of kinetic energy over the ensemble of possible states of the system? $\endgroup$ – Sam Jaques Jul 31 at 16:38
  • $\begingroup$ In fact another follow-up: Is there a (some) simple foundational principle(s) that could derive different equations of state (en.wikipedia.org/wiki/Equation_of_state) with enough careful work, or are they all found empirically? For example I think one can assume a basic kinetic theory to get the right equation for diatomic gases; do the same principles apply to larger molecules with more complicated interactions? $\endgroup$ – Sam Jaques Jul 31 at 16:42
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    $\begingroup$ the business of writing down equations of state is an extremely messy one from the physics standpoint, especially when dealing with solids. This is because most of the simplifying assumptions that work well for ideal gases cannot be invoked here. all the second-order effects must be included in the equation if it is to produce useful predictions. Basic physics identifies those effects, but the equations themselves are curve-fitted to experimental data. $\endgroup$ – niels nielsen Jul 31 at 16:48

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